Numerical simulations of systems of PDEs by variational iteration method

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

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“…Example Consider the following Lane‐Emden type equation

\{\begin{matrix} y^{''} false(x false) = - false(\frac{2}{x} y^{'} false(x false) + e^{y false(x false)} false), x \in false[0, 1 false], \\ y false(0 false) = 0, y^{'} false(0 false) = 0, \end{matrix}

This problem models the isothermal gas spheres . For a thorough discussion of this formulation, see Davis.…”

Section: Numerical Examplesmentioning

confidence: 99%

A collocation method for differential equations with one‐point Taylor initial conditions

Ghabarpoor

Kajani

Allame

2019

Math Methods in App Sciences

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“…Wazwaz 1 used the Adomian Decomposition method to handle some system of partial di®erential equations and the reactionÀdi®usion Brusselator model. On the other hand Batiha et al 2 improved the results obtained by Wazwaz 3 by the application of Variational Iteration method to solve some linear and nonlinear systems of partial di®erential equations. Another method which is equally e®ective in solving various linear and nonlinear ordinary di®erential equations and partial di®erential equations is the Di®erential Transform method (DTM).This method (DTM), which is based on Taylor series expansion, was introduced by Zhou 4 in a study of electrical circuits.…”

Section: Introductionmentioning

confidence: 96%

Solution of Higher-Order Abel Equations by Differential Transform Method

Mukherjee

Goswami

Roy

2012

Int. J. Mod. Phys. C

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“…Very recently, Ahmad Bataineh et al ( [2]) presented two modifications of HAM to solve linear and non linear ODEs. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.…”

Section: Introductionmentioning

confidence: 99%

“…Systems of partial differential equations (PDEs) arise in many scientific models such as the propagation of shallow water waves and the Brusselator model of the chemical reaction-diffusion model. Very recently, Batiha et al [2] improved Wazwazs [9] results on the application of the variational iteration method (VIM) to solve some linear and non linear systems of PDEs. In [8], Saha Ray implemented the modified Adomian decomposition method (ADM) for solving the coupled sine-Gordon equation.…”

Section: Introductionmentioning

confidence: 99%