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fractional derivatives</title> <meta name="description" content="Search results for: caputo fractional derivatives"> <meta name="keywords" content="caputo fractional derivatives"> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> </a> 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action="https://publications.waset.org/abstracts/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value="caputo fractional derivatives"> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 800</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: caputo fractional derivatives</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">800</span> Caputo-Type Fuzzy Fractional Riccati Differential Equations with Fuzzy Initial Conditions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Trilok%20Mathur">Trilok Mathur</a>, <a href="https://publications.waset.org/abstracts/search?q=Shivi%20Agarwal"> Shivi Agarwal</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper deals with the solutions of fuzzy-fractional-order Riccati equations under Caputo-type fuzzy fractional derivatives. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhura difference and strongly generalized fuzzy differentiability. The Laplace-Adomian-Pade method is used for solving fractional Riccati-type initial value differential equations of fractional order. Moreover, we also displayed some examples to illustrate our methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Caputo-type%20fuzzy%20fractional%20derivative" title="Caputo-type fuzzy fractional derivative">Caputo-type fuzzy fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=Fractional%20Riccati%20differential%20equations" title=" Fractional Riccati differential equations"> Fractional Riccati differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Laplace-Adomian-Pade%20method" title=" Laplace-Adomian-Pade method"> Laplace-Adomian-Pade method</a>, <a href="https://publications.waset.org/abstracts/search?q=Mittag%20Leffler%20function" title=" Mittag Leffler function"> Mittag Leffler function</a> </p> <a href="https://publications.waset.org/abstracts/51080/caputo-type-fuzzy-fractional-riccati-differential-equations-with-fuzzy-initial-conditions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/51080.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">395</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">799</span> Linear fractional differential equations for second kind modified Bessel functions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Olivares">Jorge Olivares</a>, <a href="https://publications.waset.org/abstracts/search?q=Fernando%20Maass"> Fernando Maass</a>, <a href="https://publications.waset.org/abstracts/search?q=Pablo%20Martin"> Pablo Martin</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Caputo" title="Caputo">Caputo</a>, <a href="https://publications.waset.org/abstracts/search?q=modified%20Bessel%20functions" title=" modified Bessel functions"> modified Bessel functions</a>, <a href="https://publications.waset.org/abstracts/search?q=hypergeometric" title=" hypergeometric"> hypergeometric</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20fractional%20differential%20equations" title=" linear fractional differential equations"> linear fractional differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=transform%20Laplace" title=" transform Laplace"> transform Laplace</a> </p> <a href="https://publications.waset.org/abstracts/91374/linear-fractional-differential-equations-for-second-kind-modified-bessel-functions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/91374.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">343</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">798</span> Hypergeometric Solutions to Linear Nonhomogeneous Fractional Equations with Spherical Bessel Functions of the First Kind</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Pablo%20Martin">Pablo Martin</a>, <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Olivares"> Jorge Olivares</a>, <a href="https://publications.waset.org/abstracts/search?q=Fernando%20Maass"> Fernando Maass</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=caputo%20fractional%20derivatives" title="caputo fractional derivatives">caputo fractional derivatives</a>, <a href="https://publications.waset.org/abstracts/search?q=hypergeometric%20functions" title=" hypergeometric functions"> hypergeometric functions</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20differential%20equations" title=" linear differential equations"> linear differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=spherical%20Bessel%20functions" title=" spherical Bessel functions"> spherical Bessel functions</a> </p> <a href="https://publications.waset.org/abstracts/91343/hypergeometric-solutions-to-linear-nonhomogeneous-fractional-equations-with-spherical-bessel-functions-of-the-first-kind" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/91343.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">325</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">797</span> Nonhomogeneous Linear Fractional Differential Equations Will Bessel Functions of the First Kind Giving Hypergeometric Functions Solutions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fernando%20Maass">Fernando Maass</a>, <a href="https://publications.waset.org/abstracts/search?q=Pablo%20Martin"> Pablo Martin</a>, <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Olivares"> Jorge Olivares</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Caputo" title="Caputo">Caputo</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculation" title=" fractional calculation"> fractional calculation</a>, <a href="https://publications.waset.org/abstracts/search?q=hypergeometric" title=" hypergeometric"> hypergeometric</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20differential%20equations" title=" linear differential equations"> linear differential equations</a> </p> <a href="https://publications.waset.org/abstracts/91373/nonhomogeneous-linear-fractional-differential-equations-will-bessel-functions-of-the-first-kind-giving-hypergeometric-functions-solutions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/91373.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">197</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">796</span> Modified Fractional Curl Operator</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Rawhy%20Ismail">Rawhy Ismail </a> </p> <p class="card-text"><strong>Abstract:</strong></p> Applying fractional calculus in the field of electromagnetics shows significant results. The fractionalization of the conventional curl operator leads to having additional solutions to an electromagnetic problem. This work restudies the concept of the fractional curl operator considering fractional time derivatives in Maxwell’s curl equations. In that sense, a general scheme for the wave loss term is introduced and the degree of freedom of the system is affected through imposing the new fractional parameters. The conventional case is recovered by setting all fractional derivatives to unity. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=curl%20operator" title="curl operator">curl operator</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title=" fractional calculus"> fractional calculus</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20curl%20operators" title=" fractional curl operators"> fractional curl operators</a>, <a href="https://publications.waset.org/abstracts/search?q=Maxwell%20equations" title=" Maxwell equations"> Maxwell equations</a> </p> <a href="https://publications.waset.org/abstracts/35772/modified-fractional-curl-operator" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/35772.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">487</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">795</span> A Dynamical Study of Fractional Order Obesity Model by a Combined Legendre Wavelet Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hakiki%20Kheira">Hakiki Kheira</a>, <a href="https://publications.waset.org/abstracts/search?q=Belhamiti%20Omar"> Belhamiti Omar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we propose a new compartmental fractional order model for the simulation of epidemic obesity dynamics. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. We also present some fractional differential illustrative examples to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Caputo%20derivative" title="Caputo derivative">Caputo derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=epidemiology" title=" epidemiology"> epidemiology</a>, <a href="https://publications.waset.org/abstracts/search?q=Legendre%20wavelet%20method" title=" Legendre wavelet method"> Legendre wavelet method</a>, <a href="https://publications.waset.org/abstracts/search?q=obesity" title=" obesity "> obesity </a> </p> <a href="https://publications.waset.org/abstracts/42731/a-dynamical-study-of-fractional-order-obesity-model-by-a-combined-legendre-wavelet-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/42731.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">421</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">794</span> Effects of the Fractional Order on Nanoparticles in Blood Flow through the Stenosed Artery</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mohammed%20Abdulhameed">Mohammed Abdulhameed</a>, <a href="https://publications.waset.org/abstracts/search?q=Sagir%20M.%20Abdullahi"> Sagir M. Abdullahi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, based on the applications of nanoparticle, the blood flow along with nanoparticles through stenosed artery is studied. The blood is acted by periodic body acceleration, an oscillating pressure gradient and an external magnetic field. The mathematical formulation is based on Caputo-Fabrizio fractional derivative without singular kernel. The model of ordinary blood, corresponding to time-derivatives of integer order, is obtained as a limiting case. Analytical solutions of the blood velocity and temperature distribution are obtained by means of the Hankel and Laplace transforms. Effects of the order of Caputo-Fabrizio time-fractional derivatives and three different nanoparticles i.e. Fe3O4, TiO4 and Cu are studied. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. It is observed that the addition of Fe3O4 nanoparticle reduced the resistance impedance of the blood flow and temperature distribution through bell shape stenosed arteries as compared to TiO4 and Cu nanoparticles. On entering in the stenosed area, blood temperature increases slightly, but, increases considerably and reaches its maximum value in the stenosis throat. The shears stress has variation from a constant in the area without stenosis and higher in the layers located far to the longitudinal axis of the artery. This fact can be an important for some clinical applications in therapeutic procedures. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=nanoparticles" title="nanoparticles">nanoparticles</a>, <a href="https://publications.waset.org/abstracts/search?q=blood%20flow" title=" blood flow"> blood flow</a>, <a href="https://publications.waset.org/abstracts/search?q=stenosed%20%20artery" title=" stenosed artery"> stenosed artery</a>, <a href="https://publications.waset.org/abstracts/search?q=mathematical%20models" title=" mathematical models"> mathematical models</a> </p> <a href="https://publications.waset.org/abstracts/58237/effects-of-the-fractional-order-on-nanoparticles-in-blood-flow-through-the-stenosed-artery" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/58237.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">267</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">793</span> A New Study on Mathematical Modelling of COVID-19 with Caputo Fractional Derivative</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sadia%20Arshad">Sadia Arshad</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The new coronavirus disease or COVID-19 still poses an alarming situation around the world. Modeling based on the derivative of fractional order is relatively important to capture real-world problems and to analyze the realistic situation of the proposed model. Weproposed a mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework. The new model is formulated in the Caputo sense and employs a nonlinear time-varying transmission rate. The existence and uniqueness solutions of the fractional order derivative have been studied using the fixed-point theory. The associated dynamical behaviors are discussed in terms of equilibrium, stability, and basic reproduction number. For the purpose of numerical implementation, an effcient approximation scheme is also employed to solve the fractional COVID-19 model. Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic. According to the comparative results with real data, we find the best value of fractional orderand justify the use of the fractional concept in the mathematical modelling, for the new fractional modelsimulates the reality more accurately than the other classical frameworks. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title="fractional calculus">fractional calculus</a>, <a href="https://publications.waset.org/abstracts/search?q=modeling" title=" modeling"> modeling</a>, <a href="https://publications.waset.org/abstracts/search?q=stability" title=" stability"> stability</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20solution" title=" numerical solution"> numerical solution</a> </p> <a href="https://publications.waset.org/abstracts/151862/a-new-study-on-mathematical-modelling-of-covid-19-with-caputo-fractional-derivative" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/151862.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">111</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">792</span> A Coupled System of Caputo-Type Katugampola Fractional Differential Equations with Integral Boundary Conditions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yacine%20Arioua">Yacine Arioua</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we investigate the existence and uniqueness of solutions for a coupled system of nonlinear Caputo-type Katugampola fractional differential equations with integral boundary conditions. Based upon a contraction mapping principle, Schauders fixed point theorems, some new existence and uniqueness results of solutions for the given problems are obtained. For application, some examples are given to illustrate the usefulness of our main results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20differential%20equations" title="fractional differential equations">fractional differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=coupled%20system" title=" coupled system"> coupled system</a>, <a href="https://publications.waset.org/abstracts/search?q=Caputo-Katugampola%20derivative" title=" Caputo-Katugampola derivative"> Caputo-Katugampola derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=fixed%20point%20theorems" title=" fixed point theorems"> fixed point theorems</a>, <a href="https://publications.waset.org/abstracts/search?q=existence" title=" existence"> existence</a>, <a href="https://publications.waset.org/abstracts/search?q=uniqueness" title=" uniqueness"> uniqueness</a> </p> <a href="https://publications.waset.org/abstracts/124953/a-coupled-system-of-caputo-type-katugampola-fractional-differential-equations-with-integral-boundary-conditions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/124953.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">264</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">791</span> Spectral Quasi Linearization Techniques for the Solution of Time Fractional Diffusion Wave Equations in Boundary Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kizito%20Ugochukwu%20Nwajeria">Kizito Ugochukwu Nwajeria</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents a spectral quasi-linearization technique (SQLT) for solving time fractional diffusion wave equations in boundary value problems. The proposed method integrates spectral approximations for spatial derivatives with a quasi-linearization approach to address the nonlinearity introduced by fractional time derivatives. Time fractional differential equations typically formulated using Caputo or Riemann-Liouville derivatives, model complex phenomena such as anomalous diffusion and wave propagation, which are not captured by classical integer-order models. The SQLT method iteratively linearizes the nonlinear terms at each time step, transforming the original problem into a series of linear subproblems, which can be efficiently solved. Using high-order spectral methods such as Chebyshev or Legendre polynomials for spatial discretization, the technique achieves high accuracy in approximating the solution. A convergence analysis is provided, demonstrating the method's efficiency and establishing error bounds. Numerical experiments on a range of test problems confirm the effectiveness of SQLT in solving fractional diffusion wave equations with various boundary conditions. The method offers a robust framework for addressing time fractional differential equations in diverse fields, including materials science, bioengineering, and anomalous transport phenomena. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=spectral%20methods" title="spectral methods">spectral methods</a>, <a href="https://publications.waset.org/abstracts/search?q=quasilinearization" title=" quasilinearization"> quasilinearization</a>, <a href="https://publications.waset.org/abstracts/search?q=time-fractional%20diffusion-wave%20equations" title=" time-fractional diffusion-wave equations"> time-fractional diffusion-wave equations</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20value%20problems" title=" boundary value problems"> boundary value problems</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title=" fractional calculus"> fractional calculus</a> </p> <a href="https://publications.waset.org/abstracts/195665/spectral-quasi-linearization-techniques-for-the-solution-of-time-fractional-diffusion-wave-equations-in-boundary-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/195665.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">1</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">790</span> Fractional Order Differentiator Using Chebyshev Polynomials</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Koushlendra%20Kumar%20Singh">Koushlendra Kumar Singh</a>, <a href="https://publications.waset.org/abstracts/search?q=Manish%20Kumar%20Bajpai"> Manish Kumar Bajpai</a>, <a href="https://publications.waset.org/abstracts/search?q=Rajesh%20Kumar%20Pandey"> Rajesh Kumar Pandey</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A discrete time fractional orderdifferentiator has been modeled for estimating the fractional order derivatives of contaminated signal. The proposed approach is based on Chebyshev’s polynomials. We use the Riemann-Liouville fractional order derivative definition for designing the fractional order SG differentiator. In first step we calculate the window weight corresponding to the required fractional order. Then signal is convoluted with this calculated window’s weight for finding the fractional order derivatives of signals. Several signals are considered for evaluating the accuracy of the proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20order%20derivative" title="fractional order derivative">fractional order derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=chebyshev%0D%0Apolynomials" title=" chebyshev polynomials"> chebyshev polynomials</a>, <a href="https://publications.waset.org/abstracts/search?q=signals" title=" signals"> signals</a>, <a href="https://publications.waset.org/abstracts/search?q=S-G%20differentiator" title=" S-G differentiator"> S-G differentiator</a> </p> <a href="https://publications.waset.org/abstracts/21346/fractional-order-differentiator-using-chebyshev-polynomials" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/21346.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">648</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">789</span> Reduced Differential Transform Methods for Solving the Fractional Diffusion Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yildiray%20Keskin">Yildiray Keskin</a>, <a href="https://publications.waset.org/abstracts/search?q=Omer%20Acan"> Omer Acan</a>, <a href="https://publications.waset.org/abstracts/search?q=Murat%20Akkus"> Murat Akkus</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, the solution of fractional diffusion equations is presented by means of the reduced differential transform method. Fractional partial differential equations have special importance in engineering and sciences. Application of reduced differential transform method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results show that the approach is easy to implement and accurate when applied to fractional diffusion equations. The method introduces a promising tool for solving many fractional partial differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20diffusion%20equations" title="fractional diffusion equations">fractional diffusion equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Caputo%20fractional%20derivative" title=" Caputo fractional derivative"> Caputo fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=reduced%20differential%20transform%20method" title=" reduced differential transform method"> reduced differential transform method</a>, <a href="https://publications.waset.org/abstracts/search?q=partial" title=" partial"> partial</a> </p> <a href="https://publications.waset.org/abstracts/17526/reduced-differential-transform-methods-for-solving-the-fractional-diffusion-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/17526.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">525</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">788</span> Multiple Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20Guezane-Lakoud">A. Guezane-Lakoud</a>, <a href="https://publications.waset.org/abstracts/search?q=S.%20Bensebaa"> S. Bensebaa</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=positive%20solution" title="positive solution">positive solution</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20caputo%20derivative" title=" fractional caputo derivative"> fractional caputo derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=Banach%20contraction%20principle" title=" Banach contraction principle"> Banach contraction principle</a>, <a href="https://publications.waset.org/abstracts/search?q=Avery%20and%20Peterson%20fixed%20point%20theorem" title=" Avery and Peterson fixed point theorem"> Avery and Peterson fixed point theorem</a> </p> <a href="https://publications.waset.org/abstracts/17545/multiple-positive-solutions-for-boundary-value-problem-of-nonlinear-fractional-differential-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/17545.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">414</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">787</span> Solutions of Fractional Reaction-Diffusion Equations Used to Model the Growth and Spreading of Biological Species</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kamel%20Al-Khaled">Kamel Al-Khaled</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some ﬁelds such as cell biology, chemistry, physics, and ﬁnance, makes it necessary to apply the results reported here to some numerical examples. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20partial%20differential%20equations" title="fractional partial differential equations">fractional partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=reaction-di%EF%AC%80usion%20equations" title=" reaction-diﬀusion equations"> reaction-diﬀusion equations</a>, <a href="https://publications.waset.org/abstracts/search?q=adomian%20decomposition" title=" adomian decomposition"> adomian decomposition</a>, <a href="https://publications.waset.org/abstracts/search?q=biological%20species" title=" biological species"> biological species</a> </p> <a href="https://publications.waset.org/abstracts/55994/solutions-of-fractional-reaction-diffusion-equations-used-to-model-the-growth-and-spreading-of-biological-species" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/55994.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">375</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">786</span> Fractional Calculus into Structural Dynamics</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Lopez">Jorge Lopez</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work, we introduce fractional calculus in order to study the dynamics of a damped multistory building with some symmetry. Initially we make a review of the dynamics of a free and damped multistory building. Then we introduce those concepts of fractional calculus that will be involved in our study. It has been noticed that fractional calculus provides models with less parameters than those based on classical calculus. In particular, a damped classical oscilator is more naturally described by using fractional derivatives. Accordingly, we model our multistory building as a set of coupled fractional oscillators and compare its dynamics with the results coming from traditional methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=coupled%20oscillators" title="coupled oscillators">coupled oscillators</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title=" fractional calculus"> fractional calculus</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20oscillator" title=" fractional oscillator"> fractional oscillator</a>, <a href="https://publications.waset.org/abstracts/search?q=structural%20dynamics" title=" structural dynamics"> structural dynamics</a> </p> <a href="https://publications.waset.org/abstracts/124822/fractional-calculus-into-structural-dynamics" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/124822.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">243</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">785</span> Operational Matrix Method for Fuzzy Fractional Reaction Diffusion Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sachin%20Kumar">Sachin Kumar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20PDE" title="fractional PDE">fractional PDE</a>, <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20valued%20function" title=" fuzzy valued function"> fuzzy valued function</a>, <a href="https://publications.waset.org/abstracts/search?q=diffusion%20equation" title=" diffusion equation"> diffusion equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Legendre%20polynomial" title=" Legendre polynomial"> Legendre polynomial</a>, <a href="https://publications.waset.org/abstracts/search?q=spectral%20method" title=" spectral method"> spectral method</a> </p> <a href="https://publications.waset.org/abstracts/125273/operational-matrix-method-for-fuzzy-fractional-reaction-diffusion-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/125273.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">201</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">784</span> Analytical Solutions of Time Space Fractional, Advection-Dispersion and Whitham-Broer-Kaup Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Muhammad%20Danish%20Khan">Muhammad Danish Khan</a>, <a href="https://publications.waset.org/abstracts/search?q=Imran%20Naeem"> Imran Naeem</a>, <a href="https://publications.waset.org/abstracts/search?q=Mudassar%20Imran"> Mudassar Imran</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this article, we study time-space Fractional Advection-Dispersion (FADE) equation and time-space Fractional Whitham-Broer-Kaup (FWBK) equation that have a significant role in hydrology. We introduce suitable transformations to convert fractional order derivatives to integer order derivatives and as a result these equations transform into Partial Differential Equations (PDEs). Then the Lie symmetries and corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=modified%20Riemann-Liouville%20fractional%20derivative" title="modified Riemann-Liouville fractional derivative">modified Riemann-Liouville fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=lie-symmetries" title=" lie-symmetries"> lie-symmetries</a>, <a href="https://publications.waset.org/abstracts/search?q=optimal%20system" title=" optimal system"> optimal system</a>, <a href="https://publications.waset.org/abstracts/search?q=invariant%20solutions" title=" invariant solutions"> invariant solutions</a> </p> <a href="https://publications.waset.org/abstracts/2191/analytical-solutions-of-time-space-fractional-advection-dispersion-and-whitham-broer-kaup-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/2191.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">431</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">783</span> Magnetohydrodynamic Couette Flow of Fractional Burger’s Fluid in an Annulus</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sani%20Isa">Sani Isa</a>, <a href="https://publications.waset.org/abstracts/search?q=Ali%20Musa"> Ali Musa</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Burgers’ fluid with a fractional derivatives model in an annulus was analyzed. Combining appropriately the basic equations, with the fractionalized fractional Burger’s fluid model allow us to determine the velocity field, temperature and shear stress. The governing partial differential equation was solved using the combine Laplace transformation method and Riemann sum approximation to give velocity field, temperature and shear stress on the fluid flow. The influence of various parameters like fractional parameters, relaxation time and retardation time, are drawn. The results obtained are simulated using Mathcad software and presented graphically. From the graphical results, we observed that the relaxation time and time helps the flow pattern, on the other hand, other material constants resist the fluid flow while fractional parameters effect on fluid flow is opposite to each other. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=sani%20isa" title="sani isa">sani isa</a>, <a href="https://publications.waset.org/abstracts/search?q=Ali%20musaburger%E2%80%99s%20fluid" title=" Ali musaburger’s fluid"> Ali musaburger’s fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=Laplace%20transform" title=" Laplace transform"> Laplace transform</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20derivatives" title=" fractional derivatives"> fractional derivatives</a>, <a href="https://publications.waset.org/abstracts/search?q=annulus" title=" annulus"> annulus</a> </p> <a href="https://publications.waset.org/abstracts/190150/magnetohydrodynamic-couette-flow-of-fractional-burgers-fluid-in-an-annulus" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/190150.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">24</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">782</span> A New Fuzzy Fractional Order Model of Transmission of Covid-19 With Quarantine Class</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Asma%20Hanif">Asma Hanif</a>, <a href="https://publications.waset.org/abstracts/search?q=A.%20I.%20K.%20Butt"> A. I. K. Butt</a>, <a href="https://publications.waset.org/abstracts/search?q=Shabir%20Ahmad"> Shabir Ahmad</a>, <a href="https://publications.waset.org/abstracts/search?q=Rahim%20Ud%20Din"> Rahim Ud Din</a>, <a href="https://publications.waset.org/abstracts/search?q=Mustafa%20Inc"> Mustafa Inc</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper is devoted to a study of the fuzzy fractional mathematical model reviewing the transmission dynamics of the infectious disease Covid-19. The proposed dynamical model consists of susceptible, exposed, symptomatic, asymptomatic, quarantine, hospitalized and recovered compartments. In this study, we deal with the fuzzy fractional model defined in Caputo’s sense. We show the positivity of state variables that all the state variables that represent different compartments of the model are positive. Using Gronwall inequality, we show that the solution of the model is bounded. Using the notion of the next-generation matrix, we find the basic reproduction number of the model. We demonstrate the local and global stability of the equilibrium point by using the concept of Castillo-Chavez and Lyapunov theory with the Lasalle invariant principle, respectively. We present the results that reveal the existence and uniqueness of the solution of the considered model through the fixed point theorem of Schauder and Banach. Using the fuzzy hybrid Laplace method, we acquire the approximate solution of the proposed model. The results are graphically presented via MATLAB-17. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Caputo%20fractional%20derivative" title="Caputo fractional derivative">Caputo fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=existence%20and%20uniqueness" title=" existence and uniqueness"> existence and uniqueness</a>, <a href="https://publications.waset.org/abstracts/search?q=gronwall%20inequality" title=" gronwall inequality"> gronwall inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=Lyapunov%20theory" title=" Lyapunov theory"> Lyapunov theory</a> </p> <a href="https://publications.waset.org/abstracts/147667/a-new-fuzzy-fractional-order-model-of-transmission-of-covid-19-with-quarantine-class" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/147667.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">106</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">781</span> Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kazem%20Ghanbari">Kazem Ghanbari</a>, <a href="https://publications.waset.org/abstracts/search?q=Yousef%20Gholami"> Yousef Gholami</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper deals with study about fractional order impulsive Hamiltonian systems and fractional impulsive Sturm-Liouville type problems derived from these systems. The main purpose of this paper devotes to obtain so called Lyapunov type inequalities for mentioned problems. Also, in view point on applicability of obtained inequalities, some qualitative properties such as stability, disconjugacy, nonexistence and oscillatory behaviour of fractional Hamiltonian systems and fractional Sturm-Liouville type problems under impulsive conditions will be derived. At the end, we want to point out that for studying fractional order Hamiltonian systems, we will apply recently introduced fractional Conformable operators. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20derivatives%20and%20integrals" title="fractional derivatives and integrals">fractional derivatives and integrals</a>, <a href="https://publications.waset.org/abstracts/search?q=Hamiltonian%20system" title=" Hamiltonian system"> Hamiltonian system</a>, <a href="https://publications.waset.org/abstracts/search?q=Lyapunov-type%20inequalities" title=" Lyapunov-type inequalities"> Lyapunov-type inequalities</a>, <a href="https://publications.waset.org/abstracts/search?q=stability" title=" stability"> stability</a>, <a href="https://publications.waset.org/abstracts/search?q=disconjugacy" title=" disconjugacy"> disconjugacy</a> </p> <a href="https://publications.waset.org/abstracts/48806/lyapunov-type-inequalities-for-fractional-impulsive-hamiltonian-systems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/48806.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">356</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">780</span> Globally Attractive Mild Solutions for Non-Local in Time Subdiffusion Equations of Neutral Type</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez%20Camus">Jorge Gonzalez Camus</a>, <a href="https://publications.waset.org/abstracts/search?q=Carlos%20Lizama"> Carlos Lizama</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work is proved the existence of at least one globally attractive mild solution to the Cauchy problem, for fractional evolution equation of neutral type, involving the fractional derivate in Caputo sense. An almost sectorial operator on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Hausdorff measure of noncompactness and fixed point theorems, specifically Darbo-type. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the analytic integral resolvent operator, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, each mild solution is globally attractive, a property that is desired in asymptotic behavior for that solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=attractive%20mild%20solutions" title="attractive mild solutions">attractive mild solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20Volterra%20equations" title=" integral Volterra equations"> integral Volterra equations</a>, <a href="https://publications.waset.org/abstracts/search?q=neutral%20type%20equations" title=" neutral type equations"> neutral type equations</a>, <a href="https://publications.waset.org/abstracts/search?q=non-local%20in%20time%20equations" title=" non-local in time equations"> non-local in time equations</a> </p> <a href="https://publications.waset.org/abstracts/99925/globally-attractive-mild-solutions-for-non-local-in-time-subdiffusion-equations-of-neutral-type" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/99925.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">160</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">779</span> Analytical Soliton Solutions of the Fractional Jaulent-Miodek System</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sajeda%20Elbashabsheh">Sajeda Elbashabsheh</a>, <a href="https://publications.waset.org/abstracts/search?q=Kamel%20Al-Khaled"> Kamel Al-Khaled</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper applies a modified Laplace Adomian decomposition method to solve the time-fractional JaulentMiodek system. The method produce convergent series solutions with easily compatible components. This paper considers the Caputo fractional derivative. The effectiveness and applicability of the method are demonstrated by comparing its results with those of prior studies. Results are presented in tables and figures. These solutions might be imperative and significant for the explanation of some practical physical phenomena. All computations and figures in the work are done using MATHEMATICA. The numerical results demonstrate that the current methods are effective, reliable, and simple to i implement for nonlinear fractional partial differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=approximate%20solutions" title="approximate solutions">approximate solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=Jaulent-Miodek%20system" title=" Jaulent-Miodek system"> Jaulent-Miodek system</a>, <a href="https://publications.waset.org/abstracts/search?q=Adomian%20decomposition%20method" title=" Adomian decomposition method"> Adomian decomposition method</a>, <a href="https://publications.waset.org/abstracts/search?q=solitons" title=" solitons"> solitons</a> </p> <a href="https://publications.waset.org/abstracts/186620/analytical-soliton-solutions-of-the-fractional-jaulent-miodek-system" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/186620.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">44</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">778</span> Fundamental Solutions for Discrete Dynamical Systems Involving the Fractional Laplacian</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez%20Camus">Jorge Gonzalez Camus</a>, <a href="https://publications.waset.org/abstracts/search?q=Valentin%20Keyantuo"> Valentin Keyantuo</a>, <a href="https://publications.waset.org/abstracts/search?q=Mahamadi%20Warma"> Mahamadi Warma</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20fractional%20Laplacian" title="discrete fractional Laplacian">discrete fractional Laplacian</a>, <a href="https://publications.waset.org/abstracts/search?q=explicit%20representation%20of%20solutions" title=" explicit representation of solutions"> explicit representation of solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20heat%20and%20wave%20equations" title=" fractional heat and wave equations"> fractional heat and wave equations</a>, <a href="https://publications.waset.org/abstracts/search?q=fundamental" title=" fundamental"> fundamental</a> </p> <a href="https://publications.waset.org/abstracts/99922/fundamental-solutions-for-discrete-dynamical-systems-involving-the-fractional-laplacian" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/99922.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">209</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">777</span> Numerical Solution of Magneto-Hydrodynamic Flow of a Viscous Fluid in the Presence of Nanoparticles with Fractional Derivatives through a Cylindrical Tube</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Muhammad%20Abdullah">Muhammad Abdullah</a>, <a href="https://publications.waset.org/abstracts/search?q=Asma%20Rashid%20Butt"> Asma Rashid Butt</a>, <a href="https://publications.waset.org/abstracts/search?q=Nauman%20Raza"> Nauman Raza</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Biomagnetic fluids like blood play key role in different applications of medical science and bioengineering. In this paper, the magnetohydrodynamic flow of a viscous fluid with magnetic particles through a cylindrical tube is investigated. The fluid is electrically charged in the presence of a uniform external magnetic field. The movement in the fluid is produced due to the cylindrical tube. Initially, the fluid and tube are at rest and at time t=0⁺, the tube starts to move along its axis. To obtain the mathematical model of flow with fractional derivatives fractional calculus approach is used. The solution of the flow model is obtained by using Laplace transformation. The Simon's numerical algorithm is employed to obtain inverse Laplace transform. The hybrid technique, we are employing has less computational effort as compared to other methods. The numerical calculations have been performed with Mathcad software. As the special cases of our problem, the solution of flow model with ordinary derivatives and flow without magnetic particles has been procured. Finally, the impact of non-integer fractional parameter alpha, Hartmann number Ha, and Reynolds number Re on flow and magnetic particles velocity is analyzed and depicted by graphs. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=viscous%20fluid" title="viscous fluid">viscous fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=magnetic%20particles" title=" magnetic particles"> magnetic particles</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title=" fractional calculus"> fractional calculus</a>, <a href="https://publications.waset.org/abstracts/search?q=laplace%20transformation" title=" laplace transformation"> laplace transformation</a> </p> <a href="https://publications.waset.org/abstracts/90032/numerical-solution-of-magneto-hydrodynamic-flow-of-a-viscous-fluid-in-the-presence-of-nanoparticles-with-fractional-derivatives-through-a-cylindrical-tube" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/90032.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">207</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">776</span> Robust Fractional Order Controllers for Minimum and Non-Minimum Phase Systems – Studies on Design and Development</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Anand%20Kishore%20Kola">Anand Kishore Kola</a>, <a href="https://publications.waset.org/abstracts/search?q=G.%20Uday%20Bhaskar%20Babu"> G. Uday Bhaskar Babu</a>, <a href="https://publications.waset.org/abstracts/search?q=Kotturi%20Ajay%20Kumar"> Kotturi Ajay Kumar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The modern dynamic systems used in industries are complex in nature and hence the fractional order controllers have been contemplated as a fresh approach to control system design that takes the complexity into account. Traditional integer order controllers use integer derivatives and integrals to control systems, whereas fractional order controllers use fractional derivatives and integrals to regulate memory and non-local behavior. This study provides a method based on the maximumsensitivity (Ms) methodology to discover all resilient fractional filter Internal Model Control - proportional integral derivative (IMC-PID) controllers that stabilize the closed-loop system and deliver the highest performance for a time delay system with a Smith predictor configuration. Additionally, it helps to enhance the range of PID controllers that are used to stabilize the system. This study also evaluates the effectiveness of the suggested controller approach for minimum phase system in comparison to those currently in use which are based on Integral of Absolute Error (IAE) and Total Variation (TV). <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=modern%20dynamic%20systems" title="modern dynamic systems">modern dynamic systems</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20order%20controllers" title=" fractional order controllers"> fractional order controllers</a>, <a href="https://publications.waset.org/abstracts/search?q=maximum-sensitivity" title=" maximum-sensitivity"> maximum-sensitivity</a>, <a href="https://publications.waset.org/abstracts/search?q=IMC-PID%20controllers" title=" IMC-PID controllers"> IMC-PID controllers</a>, <a href="https://publications.waset.org/abstracts/search?q=Smith%20predictor" title=" Smith predictor"> Smith predictor</a>, <a href="https://publications.waset.org/abstracts/search?q=IAE%20and%20TV" title=" IAE and TV"> IAE and TV</a> </p> <a href="https://publications.waset.org/abstracts/181141/robust-fractional-order-controllers-for-minimum-and-non-minimum-phase-systems-studies-on-design-and-development" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/181141.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">66</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">775</span> A Fractional Derivative Model to Quantify Non-Darcy Flow in Porous and Fractured Media</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Golden%20J.%20Zhang">Golden J. Zhang</a>, <a href="https://publications.waset.org/abstracts/search?q=Dongbao%20Zhou"> Dongbao Zhou</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Darcy’s law is the fundamental theory in fluid dynamics and engineering applications. Although Darcy linearity was found to be valid for slow, viscous flow, non-linear and non-Darcian flow has been well documented under both small and large velocity fluid flow. Various classical models were proposed and used widely to quantify non-Darcian flow, including the well-known Forchheimer, Izbash, and Swartzendruber models. Applications, however, revealed limitations of these models. Here we propose a general model built upon the Caputo fractional derivative to quantify non-Darcian flow for various flows (laminar to turbulence).Real-world applications and model comparisons showed that the new fractional-derivative model, which extends the fractional model proposed recently by Zhou and Yang (2018), can capture the non-Darcian flow in the relatively small velocity in low-permeability deposits and the relatively high velocity in high-permeability sand. A scale effect was also identified for non-Darcian flow in fractured rocks. Therefore, fractional calculus may provide an efficient tool to improve classical models to quantify fluid dynamics in aquatic environments. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20derivative" title="fractional derivative">fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=darcy%E2%80%99s%20law" title=" darcy’s law"> darcy’s law</a>, <a href="https://publications.waset.org/abstracts/search?q=non-darcian%20flow" title=" non-darcian flow"> non-darcian flow</a>, <a href="https://publications.waset.org/abstracts/search?q=fluid%20dynamics" title=" fluid dynamics"> fluid dynamics</a> </p> <a href="https://publications.waset.org/abstracts/154329/a-fractional-derivative-model-to-quantify-non-darcy-flow-in-porous-and-fractured-media" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/154329.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">126</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">774</span> Existence of Minimal and Maximal Mild Solutions for Non-Local in Time Subdiffusion Equations of Neutral Type</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez-Camus">Jorge Gonzalez-Camus</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work is proved the existence of at least one minimal and maximal mild solutions to the Cauchy problem, for fractional evolution equation of neutral type, involving a general kernel. An operator A generating a resolvent family and integral resolvent family on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Kuratowski measure of noncompactness and fixed point theorems, specifically Darbo-type, and an iterative method of lower and upper solutions, based in an order in X induced by a normal cone P. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the theory of resolvent families, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, the existence of minimal and maximal mild solutions was proved through in an iterative method of lower and upper solutions, using the Azcoli-Arzela Theorem, and the Gronwall’s inequality. Finally, we recovered the case derivate in Caputo sense. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20evolution%20equations" title="fractional evolution equations">fractional evolution equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Volterra%20integral%20equations" title=" Volterra integral equations"> Volterra integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=minimal%20and%20maximal%20mild%20solutions" title=" minimal and maximal mild solutions"> minimal and maximal mild solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=neutral%20type%20equations" title=" neutral type equations"> neutral type equations</a>, <a href="https://publications.waset.org/abstracts/search?q=non-local%20in%20time%20equations" title=" non-local in time equations"> non-local in time equations</a> </p> <a href="https://publications.waset.org/abstracts/105179/existence-of-minimal-and-maximal-mild-solutions-for-non-local-in-time-subdiffusion-equations-of-neutral-type" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/105179.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">176</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">773</span> Global Mittag-Leffler Stability of Fractional-Order Bidirectional Associative Memory Neural Network with Discrete and Distributed Transmission Delays</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Swati%20Tyagi">Swati Tyagi</a>, <a href="https://publications.waset.org/abstracts/search?q=Syed%20Abbas"> Syed Abbas</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Fractional-order Hopfield neural networks are generally used to model the information processing among the interacting neurons. To show the constancy of the processed information, it is required to analyze the stability of these systems. In this work, we perform Mittag-Leffler stability for the corresponding Caputo fractional-order bidirectional associative memory (BAM) neural networks with various time-delays. We derive sufficient conditions to ensure the existence and uniqueness of the equilibrium point by using the theory of topological degree theory. By applying the fractional Lyapunov method and Mittag-Leffler functions, we derive sufficient conditions for the global Mittag-Leffler stability, which further imply the global asymptotic stability of the network equilibrium. Finally, we present two suitable examples to show the effectiveness of the obtained results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bidirectional%20associative%20memory%20neural%20network" title="bidirectional associative memory neural network">bidirectional associative memory neural network</a>, <a href="https://publications.waset.org/abstracts/search?q=existence%20and%20uniqueness" title=" existence and uniqueness"> existence and uniqueness</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional-order" title=" fractional-order"> fractional-order</a>, <a href="https://publications.waset.org/abstracts/search?q=Lyapunov%20function" title=" Lyapunov function"> Lyapunov function</a>, <a href="https://publications.waset.org/abstracts/search?q=Mittag-Leffler%20stability" title=" Mittag-Leffler stability"> Mittag-Leffler stability</a> </p> <a href="https://publications.waset.org/abstracts/52374/global-mittag-leffler-stability-of-fractional-order-bidirectional-associative-memory-neural-network-with-discrete-and-distributed-transmission-delays" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/52374.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">365</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">772</span> On Boundary Value Problems of Fractional Differential Equations Involving Stieltjes Derivatives</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Baghdad%20Said">Baghdad Said</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Differential equations of fractional order have proved to be important tools to describe many physical phenomena and have been used in diverse fields such as engineering, mathematics as well as other applied sciences. On the other hand, the theory of differential equations involving the Stieltjes derivative (SD) with respect to a non-decreasing function is a new class of differential equations and has many applications as a unified framework for dynamic equations on time scales and differential equations with impulses at fixed times. The aim of this paper is to investigate the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability (UHRS) of solutions for a boundary value problem of sequential fractional differential equations (SFDE) containing (SD). This study is based on the technique of noncompactness measures (MNCs) combined with Monch-Krasnoselski fixed point theorems (FPT), and the results are proven in an appropriate Banach space under sufficient hypotheses. We also give an illustrative example. In this work, we introduced a class of (SFDE) and the results are obtained under a few hypotheses. Future directions connected to this work could focus on another problem with different types of fractional integrals and derivatives, and the (SD) will be assumed under a more general hypothesis in more general functional spaces. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=SFDE" title="SFDE">SFDE</a>, <a href="https://publications.waset.org/abstracts/search?q=SD" title=" SD"> SD</a>, <a href="https://publications.waset.org/abstracts/search?q=UHRS" title=" UHRS"> UHRS</a>, <a href="https://publications.waset.org/abstracts/search?q=MNCs" title=" MNCs"> MNCs</a>, <a href="https://publications.waset.org/abstracts/search?q=FPT" title=" FPT"> FPT</a> </p> <a href="https://publications.waset.org/abstracts/187408/on-boundary-value-problems-of-fractional-differential-equations-involving-stieltjes-derivatives" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/187408.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">41</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">771</span> Hermite–Hadamard Type Integral Inequalities Involving k–Riemann–Liouville Fractional Integrals and Their Applications</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Artion%20Kashuri">Artion Kashuri</a>, <a href="https://publications.waset.org/abstracts/search?q=Rozana%20Liko"> Rozana Liko</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, some generalization integral inequalities of Hermite–Hadamard type for functions whose derivatives are s–convex in modulus are given by using k–fractional integrals. Some applications to special means are obtained as well. Some known versions are recovered as special cases from our results. We note that our inequalities can be viewed as new refinements of the previous results. Finally, our results have a deep connection with various fractional integral operators and interested readers can find new interesting results using our idea and technique as well. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hermite-Hadamard%27s%20inequalities" title="Hermite-Hadamard's inequalities">Hermite-Hadamard's inequalities</a>, <a href="https://publications.waset.org/abstracts/search?q=H%C3%B6lder%27s%20inequality" title=" Hölder's inequality"> Hölder's inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=k-Riemann-Liouville%20fractional%20integral" title=" k-Riemann-Liouville fractional integral"> k-Riemann-Liouville fractional integral</a>, <a href="https://publications.waset.org/abstracts/search?q=special%20means" title=" special means"> special means</a> </p> <a href="https://publications.waset.org/abstracts/127761/hermite-hadamard-type-integral-inequalities-involving-k-riemann-liouville-fractional-integrals-and-their-applications" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/127761.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">128</span> </span> </div> </div> <ul class="pagination"> <li class="page-item disabled"><span class="page-link">&lsaquo;</span></li> <li class="page-item active"><span class="page-link">1</span></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=caputo%20fractional%20derivatives&page=2">2</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=caputo%20fractional%20derivatives&page=3">3</a></li> <li class="page-item"><a class="page-link" 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