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B-splines</title> <meta name="description" content="Search results for: Quartic B-splines"> <meta name="keywords" content="Quartic B-splines"> <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> <button class="d-block d-lg-none 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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="Quartic B-splines"> <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> 10</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: Quartic B-splines</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">10</span> Collocation Method Using Quartic B-Splines for Solving the Modified RLW Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20A.%20Soliman">A. A. Soliman</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The Modified Regularized Long Wave (MRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=collocation%20method" title="collocation method">collocation method</a>, <a href="https://publications.waset.org/abstracts/search?q=MRLW%20equation" title=" MRLW equation"> MRLW equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Quartic%20B-splines" title=" Quartic B-splines"> Quartic B-splines</a>, <a href="https://publications.waset.org/abstracts/search?q=solitons" title=" solitons"> solitons</a> </p> <a href="https://publications.waset.org/abstracts/7664/collocation-method-using-quartic-b-splines-for-solving-the-modified-rlw-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/7664.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">304</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">9</span> Quartic Spline Method for Numerical Solution of Self-Adjoint Singularly Perturbed Boundary Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Reza%20Mohammadi">Reza Mohammadi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=second-order%20ordinary%20differential%20equation" title="second-order ordinary differential equation">second-order ordinary differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=singularly-perturbed" title=" singularly-perturbed"> singularly-perturbed</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20spline" title=" quartic spline"> quartic spline</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence%20analysis" title=" convergence analysis"> convergence analysis</a> </p> <a href="https://publications.waset.org/abstracts/46459/quartic-spline-method-for-numerical-solution-of-self-adjoint-singularly-perturbed-boundary-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/46459.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">361</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">8</span> Basins of Attraction for Quartic-Order Methods </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Young%20Hee%20Geum">Young Hee Geum</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We compare optimal quartic order method for the multiple zeros of nonlinear equations illustrating the basins of attraction. To construct basins of attraction effectively, we take a 600脳600 uniform grid points at the origin of the complex plane and paint the initial values on the basins of attraction with different colors according to the iteration number required for convergence. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=basins%20of%20attraction" title="basins of attraction">basins of attraction</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence" title=" convergence"> convergence</a>, <a href="https://publications.waset.org/abstracts/search?q=multiple-root" title=" multiple-root"> multiple-root</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20equation" title=" nonlinear equation"> nonlinear equation</a> </p> <a href="https://publications.waset.org/abstracts/52045/basins-of-attraction-for-quartic-order-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/52045.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">252</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">7</span> Finite Element Method for Solving the Generalized RLW Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Abdel-Maksoud%20Abdel-Kader%20Soliman">Abdel-Maksoud Abdel-Kader Soliman</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The General Regularized Long Wave (GRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=generalized%20RLW%20equation" title="generalized RLW equation">generalized RLW equation</a>, <a href="https://publications.waset.org/abstracts/search?q=solitons" title=" solitons"> solitons</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20b-spline" title=" quartic b-spline"> quartic b-spline</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20partial%20differential%20equations" title=" nonlinear partial differential equations"> nonlinear partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=difference%20equations" title=" difference equations"> difference equations</a> </p> <a href="https://publications.waset.org/abstracts/9023/finite-element-method-for-solving-the-generalized-rlw-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9023.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">489</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">6</span> Spline Solution of Singularly Perturbed Boundary Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Reza%20Mohammadi">Reza Mohammadi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=second-order%20ordinary%20differential%20equation" title="second-order ordinary differential equation">second-order ordinary differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=singularly-perturbed" title=" singularly-perturbed"> singularly-perturbed</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20spline" title=" quartic spline"> quartic spline</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence%20analysis" title=" convergence analysis"> convergence analysis</a> </p> <a href="https://publications.waset.org/abstracts/56960/spline-solution-of-singularly-perturbed-boundary-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/56960.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">295</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">5</span> Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Khosrow%20Maleknejad">Khosrow Maleknejad</a>, <a href="https://publications.waset.org/abstracts/search?q=Yaser%20Rostami"> Yaser Rostami</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=%C4%B1ntegro-differential%20equations" title="谋ntegro-differential equations">谋ntegro-differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20B-spline%20wavelet" title=" quartic B-spline wavelet"> quartic B-spline wavelet</a>, <a href="https://publications.waset.org/abstracts/search?q=operational%20matrices" title=" operational matrices"> operational matrices</a>, <a href="https://publications.waset.org/abstracts/search?q=dual%20functions" title=" dual functions"> dual functions</a> </p> <a href="https://publications.waset.org/abstracts/5002/numerical-solution-for-integro-differential-equations-by-using-quartic-b-spline-wavelet-and-operational-matrices" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/5002.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">456</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">4</span> Unsteady Similarity Solution for a Slender Dry Patch in a Thin Newtonian Fluid Film</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=S.%20S.%20Abas">S. S. Abas</a>, <a href="https://publications.waset.org/abstracts/search?q=Y.%20M.%20Yatim"> Y. M. Yatim</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper the unsteady, slender, symmetric dry patch in an infinitely wide and thin liquid film of Newtonian fluid draining under gravity down an inclined plane in the presence of strong surface-tension effect is considered. A similarity transformation, named a travelling-wave similarity solution is used to reduce the governing partial differential equation into the ordinary differential equation which is then solved numerically using a shooting method. The introduction of surface-tension effect on the flow leads to a fourth-order ordinary differential equation. The solution obtained predicts that the dry patch has a quartic shape and the free surface has a capillary ridge near the contact line which decays in an oscillatory manner far from it. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=dry%20patch" title="dry patch">dry patch</a>, <a href="https://publications.waset.org/abstracts/search?q=Newtonian%20fluid" title=" Newtonian fluid"> Newtonian fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=similarity%20solution" title=" similarity solution"> similarity solution</a>, <a href="https://publications.waset.org/abstracts/search?q=surface-tension%20effect" title=" surface-tension effect"> surface-tension effect</a>, <a href="https://publications.waset.org/abstracts/search?q=travelling-wave" title=" travelling-wave"> travelling-wave</a>, <a href="https://publications.waset.org/abstracts/search?q=unsteady%20thin-film%20flow" title=" unsteady thin-film flow"> unsteady thin-film flow</a> </p> <a href="https://publications.waset.org/abstracts/10714/unsteady-similarity-solution-for-a-slender-dry-patch-in-a-thin-newtonian-fluid-film" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/10714.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">304</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">3</span> Analytical Solution of Specific Energy Equation in Exponential Channels</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Abdulrahman%20Abdulrahman">Abdulrahman Abdulrahman</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The specific energy equation has many applications in practical channels, such as exponential channels. In this paper, the governing equation of alternate depth ratio for exponential channels, in general, was investigated towards obtaining analytical solution for the alternate depth ratio in three exponential channel shapes, viz., rectangular, triangular, and parabolic channels. The alternate depth ratio for rectangular channels is quadratic; hence it is very simple to solve. While for parabolic and triangular channels, the alternate depth ratio is cubic and quartic equations, respectively, analytical solution for these equations may be achieved easily for a given Froud number. Different examples are solved to prove the efficiency of the proposed solution. Such analytical solution can be easily used in natural rivers and most of practical channels. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=alternate%20depth" title="alternate depth">alternate depth</a>, <a href="https://publications.waset.org/abstracts/search?q=analytical%20solution" title=" analytical solution"> analytical solution</a>, <a href="https://publications.waset.org/abstracts/search?q=specific%20energy" title=" specific energy"> specific energy</a>, <a href="https://publications.waset.org/abstracts/search?q=parabolic%20channel" title=" parabolic channel"> parabolic channel</a>, <a href="https://publications.waset.org/abstracts/search?q=rectangular%20channel" title=" rectangular channel"> rectangular channel</a>, <a href="https://publications.waset.org/abstracts/search?q=triangular%20channel" title=" triangular channel"> triangular channel</a>, <a href="https://publications.waset.org/abstracts/search?q=open%20channel%20flow" title=" open channel flow"> open channel flow</a> </p> <a href="https://publications.waset.org/abstracts/121104/analytical-solution-of-specific-energy-equation-in-exponential-channels" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/121104.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">199</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">2</span> Numerical Solution of Manning's Equation in Rectangular Channels</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Abdulrahman%20Abdulrahman">Abdulrahman Abdulrahman</a> </p> <p class="card-text"><strong>Abstract:</strong></p> When the Manning equation is used, a unique value of normal depth in the uniform flow exists for a given channel geometry, discharge, roughness, and slope. Depending on the value of normal depth relative to the critical depth, the flow type (supercritical or subcritical) for a given characteristic of channel conditions is determined whether or not flow is uniform. There is no general solution of Manning's equation for determining the flow depth for a given flow rate, because the area of cross section and the hydraulic radius produce a complicated function of depth. The familiar solution of normal depth for a rectangular channel involves 1) a trial-and-error solution; 2) constructing a non-dimensional graph; 3) preparing tables involving non-dimensional parameters. Author in this paper has derived semi-analytical solution to Manning's equation for determining the flow depth given the flow rate in rectangular open channel. The solution was derived by expressing Manning's equation in non-dimensional form, then expanding this form using Maclaurin's series. In order to simplify the solution, terms containing power up to 4 have been considered. The resulted equation is a quartic equation with a standard form, where its solution was obtained by resolving this into two quadratic factors. The proposed solution for Manning's equation is valid over a large range of parameters, and its maximum error is within -1.586%. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=channel%20design" title="channel design">channel design</a>, <a href="https://publications.waset.org/abstracts/search?q=civil%20engineering" title=" civil engineering"> civil engineering</a>, <a href="https://publications.waset.org/abstracts/search?q=hydraulic%20engineering" title=" hydraulic engineering"> hydraulic engineering</a>, <a href="https://publications.waset.org/abstracts/search?q=open%20channel%20flow" title=" open channel flow"> open channel flow</a>, <a href="https://publications.waset.org/abstracts/search?q=Manning%27s%20equation" title=" Manning's equation"> Manning's equation</a>, <a href="https://publications.waset.org/abstracts/search?q=normal%20depth" title=" normal depth"> normal depth</a>, <a href="https://publications.waset.org/abstracts/search?q=uniform%20flow" title=" uniform flow"> uniform flow</a> </p> <a href="https://publications.waset.org/abstracts/72618/numerical-solution-of-mannings-equation-in-rectangular-channels" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/72618.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">221</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">1</span> Damping and Stability Evaluation for the Dynamical Hunting Motion of the Bullet Train Wheel Axle Equipped with Cylindrical Wheel Treads</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Barenten%20Suciu">Barenten Suciu</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Classical matrix calculus and Routh-Hurwitz stability conditions, applied to the snake-like motion of the conical wheel axle, lead to the conclusion that the hunting mode is inherently unstable, and its natural frequency is a complex number. In order to analytically solve such a complicated vibration model, either the inertia terms were neglected, in the model designated as geometrical, or restrictions on the creep coefficients and yawing diameter were imposed, in the so-called dynamical model. Here, an alternative solution is proposed to solve the hunting mode, based on the observation that the bullet train wheel axle is equipped with cylindrical wheels. One argues that for such wheel treads, the geometrical hunting is irrelevant, since its natural frequency becomes nil, but the dynamical hunting is significant since its natural frequency reduces to a real number. Moreover, one illustrates that the geometrical simplification of the wheel causes the stabilization of the hunting mode, since the characteristic quartic equation, derived for conical wheels, reduces to a quadratic equation of positive coefficients, for cylindrical wheels. Quite simple analytical expressions for the damping ratio and natural frequency are obtained, without applying restrictions into the model of contact. Graphs of the time-depending hunting lateral perturbation, including the maximal and inflexion points, are presented both for the critically-damped and the over-damped wheel axles. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bullet%20train" title="bullet train">bullet train</a>, <a href="https://publications.waset.org/abstracts/search?q=creep" title=" creep"> creep</a>, <a href="https://publications.waset.org/abstracts/search?q=cylindrical%20wheels" title=" cylindrical wheels"> cylindrical wheels</a>, <a href="https://publications.waset.org/abstracts/search?q=damping" title=" damping"> damping</a>, <a href="https://publications.waset.org/abstracts/search?q=dynamical%20hunting" title=" dynamical hunting"> dynamical hunting</a>, <a href="https://publications.waset.org/abstracts/search?q=stability" title=" stability"> stability</a>, <a href="https://publications.waset.org/abstracts/search?q=vibration%20analysis" title=" vibration analysis"> vibration analysis</a> </p> <a href="https://publications.waset.org/abstracts/96999/damping-and-stability-evaluation-for-the-dynamical-hunting-motion-of-the-bullet-train-wheel-axle-equipped-with-cylindrical-wheel-treads" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/96999.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">153</span> </span> </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" 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