Document Zbl 0713.65051

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Comput. Modelling</a> <a href="/?q=in%3A175112" title="Articles in this Issue">13, No. 7, 17-43 (1990)</a>. </div> <div class="abstract">An analytical method for approximate solution of nonlinear ordinary and partial differential equations (initial and boundary value problems) is briefly described and broadly illustrated on a considerable number of examples. In short, the method presupposes knowledge of the Green’s function of the highest derivative and starts from expanding the solution into a series $u_ 0+\epsilon u_ 1+\epsilon^ 2u_ 2+...$, then expanding the nonlinearity into a Taylor series about $u_ 0$ and putting then $\epsilon =1$. A convergence proof for ordinary differential equations is announced.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4361">G.Stoyan</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A4174253">Cited in <strong>7</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A4174253">Cited in <strong>162</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L10" title="MSC2020">65L10</a> </td> <td class="space"> Numerical solution of boundary value problems involving ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65J99" title="MSC2020">65J99</a> </td> <td class="space"> Numerical analysis in abstract spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34B15" title="MSC2020">34B15</a> </td> <td class="space"> Nonlinear boundary value problems for ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adecomposition+method">decomposition method</a>; <a href="/?q=ut%3Anonlinear+equations">nonlinear equations</a>; <a href="/?q=ut%3AGreen%27s+function">Green’s function</a>; <a href="/?q=ut%3ATaylor+series">Taylor series</a>; <a href="/?q=ut%3Aconvergence">convergence</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0713.65051" data-ciurl="/ci/04174253" data-biburl="/bibtex/04174253.bib" data-amsurl="/amsrefs/04174253.bib" data-xmlurl="/xml/04174253.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04174253.pdf" title="Zbl 0713.65051 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/0895-7177(90)90125-7" aria-label="DOI for “A review of the decomposition method and some recent results for nonlinear equations”" title="10.1016/0895-7177(90)90125-7">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic Press: Academic Press New York · <a href="/0614.35013" class="nowrap">Zbl 0614.35013</a></td> </tr><tr> <td>[2]</td> <td class="space">Adomian, G., Theory of random systems, (Trans. of 4th Prague Conf. on Information Theory, Statistical Decision and Random Processes (1967), Prague Publ. House) · <a href="/0556.93005" class="nowrap">Zbl 0556.93005</a></td> </tr><tr> <td>[3]</td> <td class="space">Adomian, G., Stochastic Systems (1983), Academic Press: Academic Press New York · <a href="/0504.60066" class="nowrap">Zbl 0504.60066</a></td> </tr><tr> <td>[4]</td> <td class="space">Adomian, G., Applications of Nonlinear Stochastic Systems Theory to Physics (1988), Kluwer: Kluwer New York · <a href="/0666.60061" class="nowrap">Zbl 0666.60061</a></td> </tr><tr> <td>[5]</td> <td class="space">Rach, R.; Adomian, G., Smooth polynomial expansions of piecwise-differentiable functions, Appl. Math. Lett., 2, 377-380 (1989) · <a href="/0721.41043" class="nowrap">Zbl 0721.41043</a></td> </tr><tr> <td>[6]</td> <td class="space">Adomian, G.; Rach, R., Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Computers Math. Applic., 19, 12, 9-12 (1990) · <a href="/0702.35058" class="nowrap">Zbl 0702.35058</a></td> </tr><tr> <td>[7]</td> <td class="space">Adomian, G., Vibration of offshore structures, Part I, Maths Comput. Simuln, 29, 122-199 (1987)</td> </tr><tr> <td>[8]</td> <td class="space">Adomian, G., Vibration in offshore structures, Part II, Maths Comput. Simuln, 29, 1-6 (1987)</td> </tr><tr> <td>[9]</td> <td class="space">Adomian, G., A new approach to the Efinger model for a nonlinear quantum theory for gravitating particles, Fdns Phys., 17, 419-424 (1987)</td> </tr><tr> <td>[10]</td> <td class="space">Adomian, G., Decomposition solution for Duffing and Van der Pol oscillators, Int. J. math. Sci, 9, 732-732 (1986) · <a href="/0605.34036" class="nowrap">Zbl 0605.34036</a></td> </tr><tr> <td>[11]</td> <td class="space">Adomian, G., Convergent series solution of nonlinear equations, J. Comput. appl. Math., 11, 2 (1984) · <a href="/0549.65034" class="nowrap">Zbl 0549.65034</a></td> </tr><tr> <td>[12]</td> <td class="space">Adomian, G., On the convergence region for decomposition solutions, J. Comput. appl. Math, 11 (1984) · <a href="/0547.65053" class="nowrap">Zbl 0547.65053</a></td> </tr><tr> <td>[13]</td> <td class="space">Adomian, G., Nonlinear stochastic dynamical systems in physical problems, J. math. Analysis Applic., 111, 1 (1985) · <a href="/0582.60067" class="nowrap">Zbl 0582.60067</a></td> </tr><tr> <td>[14]</td> <td class="space">Adomian, G., On composite nonlinearities and the decomposition method, J. math. Analysis Applic., 114, 1 (1986) · <a href="/0617.65046" class="nowrap">Zbl 0617.65046</a></td> </tr><tr> <td>[15]</td> <td class="space">Adomian, G., Linear stochastic operators, (Ph.D. (Physics) Dissertation (1963), University of California: University of California Los Angeles) · <a href="/0114.08503" class="nowrap">Zbl 0114.08503</a></td> </tr><tr> <td>[16]</td> <td class="space">G. Adomian, Stochastic Green’s functions. In <span class="zbmathjax-textit">Stochastic Processes in Mathematical Physics and Engineering</span>; G. Adomian, Stochastic Green’s functions. In <span class="zbmathjax-textit">Stochastic Processes in Mathematical Physics and Engineering</span> · <a href="/0139.34205" class="nowrap">Zbl 0139.34205</a></td> </tr><tr> <td>[17]</td> <td class="space">Adomian, G., Stochastic operators and dynamical systems, (Wang, P. C.C., Information Linkage Between Applied Mathematics and Industry (1979), Academic Press: Academic Press New York) · <a href="/0582.60067" class="nowrap">Zbl 0582.60067</a></td> </tr><tr> <td>[18]</td> <td class="space">Adomian, G., New results in stochastic equations—the nonlinear case, (Lakshmikanthum, V., Nonlinear Equations in Abstract Spaces (1978), Academic Press: Academic Press New York) · <a href="/0453.60062" class="nowrap">Zbl 0453.60062</a></td> </tr><tr> <td>[19]</td> <td class="space">Adomian, G., The solution of general linear and nonlinear stochastic systems, (Rose, J., Modern Trends in Cybernetics and Systems (1976), Editura Technica: Editura Technica Romania) · <a href="/0426.93048" class="nowrap">Zbl 0426.93048</a></td> </tr><tr> <td>[20]</td> <td class="space">Adomian, G., Solution of nonlinear stochastic physical problems, Rc. Semin. Mat. Stochastic Problems in Mechanics (1982), Torino, Italy · <a href="/0491.60066" class="nowrap">Zbl 0491.60066</a></td> </tr><tr> <td>[21]</td> <td class="space">Adomian, G., On the Green’s function in higher-order stochastic differential equations, J. math. Analysis Applic., 88, 2 (1982) · <a href="/0493.60064" class="nowrap">Zbl 0493.60064</a></td> </tr><tr> <td>[22]</td> <td class="space">Adomian, G., Stochastic model for colored noise, J. math. Analysis Applic., 88, 2 (1982) · <a href="/0493.60065" class="nowrap">Zbl 0493.60065</a></td> </tr><tr> <td>[23]</td> <td class="space">(Adomian, G., Stochastic systems analysis. Stochastic systems analysis, Applied Stochastic Processes (1980), Academic Press: Academic Press New York), 1-18 · <a href="/0474.60050" class="nowrap">Zbl 0474.60050</a></td> </tr><tr> <td>[24]</td> <td class="space">Adomian, G.; Adomian, G. E., Solution of the Marchuk model of infectious disease and immune response, (Witten, M., Mathematical Models in Medicine Diseases and Epidemics, 7 (1986), Pergamon Press: Pergamon Press New York), 803-807, Mathl Modelling · <a href="/0604.92006" class="nowrap">Zbl 0604.92006</a></td> </tr><tr> <td>[25]</td> <td class="space">Adomian, G.; Bellman, R. E., The stochastic Riccati equation, J. nonlinear Analysis Theory Meth. Applic., 4, 6 (1980) · <a href="/0447.60044" class="nowrap">Zbl 0447.60044</a></td> </tr><tr> <td>[26]</td> <td class="space">Adomian, G.; Bellomo, N., On the Tricomi problems, (Witten, M., Hyperbolic Partial Differential Equations, Vol. 3 (1986), Pergamon Press: Pergamon Press New York) · <a href="/0597.35086" class="nowrap">Zbl 0597.35086</a></td> </tr><tr> <td>[27]</td> <td class="space">Adomian, G.; Bellomo, N.; Riganti, R., Semilinear stochastic systems: analysis with the method of stochastic Green’s function and application to mechanics, J. math. Analysis Applic., 96, 2 (1983) · <a href="/0523.60057" class="nowrap">Zbl 0523.60057</a></td> </tr><tr> <td>[28]</td> <td class="space">Adomian, G.; Bigi, D.; Riganti, R., On the solutions of stochastic initial value problems in continuum mechanics, J. math. Analysis Applic., 110, 2 (1985) · <a href="/0582.60066" class="nowrap">Zbl 0582.60066</a></td> </tr><tr> <td>[29]</td> <td class="space">Adomian, G.; Elrod, M., Generation of a stochastic process with desired first- and second-order statistics, Kybernetes, 10, 1 (1981) · <a href="/0444.60048" class="nowrap">Zbl 0444.60048</a></td> </tr><tr> <td>[30]</td> <td class="space">Adomian, G.; Rach, R., Coupled differential equations and coupled boundary conditions, J. math. Analysis Applic., 112, 1, 129-135 (1985) · <a href="/0579.60057" class="nowrap">Zbl 0579.60057</a></td> </tr><tr> <td>[31]</td> <td class="space">G. Adomian and R. Rach, A new computational approach for inversion of very large matrices. Int. J. math. Modelling; G. Adomian and R. Rach, A new computational approach for inversion of very large matrices. Int. J. math. Modelling · <a href="/0613.65023" class="nowrap">Zbl 0613.65023</a></td> </tr><tr> <td>[32]</td> <td class="space">G. Adomian and R. Rach, Solving nonlinear differential equations with decimal power nonlinearities. J. Math. Analysis Applic.; G. Adomian and R. Rach, Solving nonlinear differential equations with decimal power nonlinearities. J. Math. Analysis Applic. · <a href="/0591.60052" class="nowrap">Zbl 0591.60052</a></td> </tr><tr> <td>[33]</td> <td class="space">Adomian, G.; Rach, R., Algebraic computation and the decomposition method, Kybernetes, 15, 1 (1986) · <a href="/0604.60064" class="nowrap">Zbl 0604.60064</a></td> </tr><tr> <td>[34]</td> <td class="space">Adomian, G.; Rach, R., Algebraic equations with exponential terms, J. math. Analysis Applic., 112, 1 (1985) · <a href="/0579.60058" class="nowrap">Zbl 0579.60058</a></td> </tr><tr> <td>[35]</td> <td class="space">Adomian, G.; Rach, R., Nonlinear differential equations with negative power nonlinearities, J. math. Analysis Applic., 112, 2 (1985) · <a href="/0579.60059" class="nowrap">Zbl 0579.60059</a></td> </tr><tr> <td>[36]</td> <td class="space">Adomian, G.; Rach, R., Applications of decomposition method to inversion of matrices, J. math. Analysis Applic., 108, 2 (1985) · <a href="/0598.65011" class="nowrap">Zbl 0598.65011</a></td> </tr><tr> <td>[37]</td> <td class="space">Adomian, G.; Rach, R., Polynomial nonlinearities in differential equations, J. math. Analysis Applic., 109, 1 (1985) · <a href="/0606.34009" class="nowrap">Zbl 0606.34009</a></td> </tr><tr> <td>[38]</td> <td class="space">Adomian, G.; Rach, R., Nonlinear stochastic differential-delay equations, J. math. Analysis Applic., 91, 1 (1983) · <a href="/0504.60067" class="nowrap">Zbl 0504.60067</a></td> </tr><tr> <td>[39]</td> <td class="space">Adomian, G.; Rach, R.; Sarafyan, D., On the solution of equations containing radicals by the decomposition method, J. math. Analysis Applic., 111, 2 (1985) · <a href="/0579.60060" class="nowrap">Zbl 0579.60060</a></td> </tr><tr> <td>[40]</td> <td class="space">Adomian, G.; Sibul, L. H., On the control of stochastic systems, J. math. Analysis Applic., 83, 2 (1981) · <a href="/0476.93077" class="nowrap">Zbl 0476.93077</a></td> </tr><tr> <td>[41]</td> <td class="space">Adomian, G.; Sibul, L. H.; Rach, R., Coupled nonlinear stochastic differential equations, J. math. Analysis Applic., 92, 2 (1983) · <a href="/0517.60064" class="nowrap">Zbl 0517.60064</a></td> </tr><tr> <td>[42]</td> <td class="space">Bellman, R. E.; Adomian, G., Partial Differential Equations—New Methods for Their Treatment and Applications (1985), Reidel: Reidel Dordrecht, The Netherlands · <a href="/0557.35003" class="nowrap">Zbl 0557.35003</a></td> </tr><tr> <td>[43]</td> <td class="space">Bellomo, N.; Monaco, R., A comparison between Adomian’s decomposition methods and perturbation techniques for nonlinear random differential equations, J. Math. Analysis Applic., 110, 495-502 (1985) · <a href="/0575.60064" class="nowrap">Zbl 0575.60064</a></td> </tr><tr> <td>[44]</td> <td class="space">Bellomo, N.; Riganti, R., Nonlinear Stochastic Systems in Physics and Mechanics (1987), World Scientific: World Scientific Singapore · <a href="/0623.60084" class="nowrap">Zbl 0623.60084</a></td> </tr><tr> <td>[45]</td> <td class="space">Bellomo, N.; Sarafyan, D., On a comparison between Adomian’s decomposition method and Picard iteration, J. math. Analysis Applic, 123 (1987) · <a href="/0624.60079" class="nowrap">Zbl 0624.60079</a></td> </tr><tr> <td>[46]</td> <td class="space">Bigi, D.; Riganti, R., Solution of nonlinear boundary value problems by the decomposition method, Appl. Math. Modelling, 10, 48-52 (1986) · <a href="/0592.60048" class="nowrap">Zbl 0592.60048</a></td> </tr><tr> <td>[47]</td> <td class="space">Rach, R., A convenient computational form for the Adomian polynomials, J. math. Analysis Applic., 102, 2, 415-419 (1984) · <a href="/0552.60061" class="nowrap">Zbl 0552.60061</a></td> </tr><tr> <td>[48]</td> <td class="space">Adomian, G.; Rach, R., Purely nonlinear equations, Computers Math Applic., 20, 1, 1-3 (1990) · <a href="/0698.34013" class="nowrap">Zbl 0698.34013</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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