Hybrid Numerical-Analytical Approach for Linear and Nonlinear Elliptic Partial Differential Equations Based on the Two-Dimensional Differential Transformation Method


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Abstract


In this paper, we study the ability of obtaining approximate analytical solutions corresponding to initial value problems associated with linear and nonlinear elliptic partial differential equations, by proposing the two-dimensional differential transformation method (DTM). The approximate solutions for these equations have the form of convergent power series. The tiny relative error occurs after few numbers of iterations shows that, the proposed technique is simple and convenient, revealing the efficiency and the high performance in reducing the size of computational work through obtaining the results
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Keywords


Differential Transform Method; Elliptic Partial Differential Equation; Series Solution; Numerical Solution; Closed Form Solution

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References


V.D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations (Hindawi Publishing Corporation, Inc., 2008).
http://dx.doi.org/10.1155/9789774540394

A. Sommerfeld, Partial Differential Equations in Physics, (1st Edition, Elsevier Academic Press, Inc., 1949 )

A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC Press, Inc. 2002).
http://dx.doi.org/10.1201/9781420035322

R. Abazari, M. Abazari, Numerical simulation of generalized Hirota--Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun Nonlinear Sci Numer Simulat, vol. 17, 2012, pp. 619--629.
http://dx.doi.org/10.1016/j.cnsns.2011.05.022

J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits (Huazhong University Press, Inc., 1986).

P.K. Gupta, Approximate analytical solutions of fractional Benney--Lin equation by reduced differential transform method and the homotopy perturbation method, Computers and Mathematics with Applications, vol. 61, 2011, pp. 2829--2842.
http://dx.doi.org/10.1016/j.camwa.2011.03.057

H. Jafari, M. Alipour, H. Tajadodi, Two-Dimensional Differential Transform Method For Solving NonLinear Partial Differential Equations, International Journal of Research and Reviews in Applied Sciences, vol. 2 n. 1, 2010, pp. 47-52.

N.H. AbuGhannam, I. Suwan, Semi-Numerical Analytical Solution to Linear and Nonlinear Heat Equations via the Reduced Differential Transformation Method, International Journal on Heat and Mass Transfer Theory and Applications, vol. 1 n. 4, August 2013, pp. 235-241.

C.K. Chen, S.H. Ho, Application of Differential Transformation to Eigenvalue Problems, Journal of Applied Mathematics and Computation, vol. 79, 1996, pp. 173-188.
http://dx.doi.org/10.1016/0096-3003(95)00253-7

E.A. Spence, Boundary Value Problems for Linear Elliptic PDEs, Ph.D. dissertation, Trinity Hall, University of Cambridge, 2010.

D.H. Knipl, Fundamental properties of differential equations with dynamically defined delayed feedback, Electronic Journal of Qualitative Theory of Differential Equations, vol. 17, 2013, pp. 1-18.

Ü. Cansu, O. Özkan, Differential Transform Solution of Some Linear Wave Equations with Mixed Nonlinear Boundary Conditions and its Blow up, Applied Mathematical Sciences, vol. 4 n. 10, 2010, pp. 467 -- 475.

F. Ayaz, Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation, vol. 147 n. 2, 2004, pp. 547--567.
http://dx.doi.org/10.1016/s0096-3003(02)00794-4

J.K. Hunter, Lecture notes on PDEs, (Department of Mathematics University of California, 2013)

M.A. Shubin, Elliptic operator (Encyclopedia of Mathematics, Springer, Inc. 2001)

J. Rauch, L. C. Evans, Partial differential equations, Journal of the American Mathematical Society, vol. 37 n. 3, 2000, pp. 363--367.
http://dx.doi.org/10.1090/s0273-0979-00-00868-5

D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order (2nd Edition, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1983)
http://dx.doi.org/10.1007/978-3-642-96379-7_1

J.D. Fairs & R.L. Burden, Numerical Methods, (3rd Edition, Brooks Cole, Inc. 2002).

A.D. Polyanin, V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Chapman & Hall/CRC Press, Inc. 2004).
http://dx.doi.org/10.1201/9780203489659


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