Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

Biazar, Jafar; Montazeri, Roya

doi:10.1155/2019/3037273

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…To deal with this issue, one can use a multistage (localized) residual power series 42 . Such an idea has been used in formulation of OHAM 43 …”

Section: Convergence Analysismentioning

confidence: 99%

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

Hussain

2022

Math Methods in App Sciences

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This article presents an effective and simple approximation scheme to analytically approximate solution behavior of a multi‐dimensional reaction–diffusion Brusselator system. This system has applications in cooperative processes of chemical kinetics and biochemical reaction processes. The proposed residual power series method is based on multiple power series expansion and residual error function formulation. The convergence properties and error bounds of the proposed scheme are theoretically established and numerically verified. Two benchmark problems in two‐ and three‐dimensional space are considered to validate the efficiency and accuracy of the proposed scheme. The obtained results verified reliability, accuracy, and simplicity of the proposed scheme against the available results in literature.

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“…To deal with this issue, one can use a multistage (localized) residual power series 42 . Such an idea has been used in formulation of OHAM 43 …”

Section: Convergence Analysismentioning

confidence: 99%

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

Hussain

2022

Math Methods in App Sciences

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“…A recent approximate analytical methods are: modified ADM for fractional optimal control problems (FOCPs) [20], optimal control of a constrained fractionally damped elastic beam [21], analyses of an optimal solutions of optimization problems from fractional gradient based system using VIM [22], conformable fractional optimal control problem of an heat conduction equations using Laplace and finite Fourier sine transforms [23], spectral Galerkin approximation [24], and direct transcription methods for FOCPs [25], but the aforementioned methods lack convergence norm that will guarantee the convergence of the series solution. In 1992, Liao proposed the homotopy analysis method (HAM) [26] for solving the nonlinear problem which was later advanced to optimal homotopy asymptotic method for noninteger order [27], new fractional homotopy method for optimal control problems (OCPs) [28], and comparisons of OHAM [29]. But OHAM has never been used to solve FOCPs, which drives this research work.…”

Section: Introductionmentioning

confidence: 99%

A New Optimal Homotopy Asymptotic Method for Fractional Optimal Control Problems

Okundalaye

Othman

2021

International Journal of Differential Equations

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Solving fractional optimal control problems (FOCPs) with an approximate analytical method has been widely studied by many authors, but to guarantee the convergence of the series solution has been a challenge. We solved this by integrating the Galerkin method of optimization technique into the whole region of the governing equations for accurate optimal values of control-convergence parameters C j s . The arbitrary-order derivative is in the conformable fractional derivative sense. We use Euler–Lagrange equation form of necessary optimality conditions for FOCPs, and the arising fractional differential equations (FDEs) are solved by optimal homotopy asymptotic method (OHAM). The OHAM technique speedily provides the convergent approximate analytical solution as the arbitrary order derivative approaches 1. The convergence of the method is discussed, and its effectiveness is verified by some illustrative test examples.

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“…This procedure give us with a convenient way to control the convergence of approximation series and demonstrates its validity and potential efficiency to solve a wide class of problems in applied science and engineering and also valid for small parameters. In the last few years, OHAM and its modifications has been applied successfully to solve many types of differential equations [12,13,14,15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

An Accurate Approximate Solutions of Multipoint Boundary Value Problems

Anakira¹,

Al-Shorman²,

Jameel³

2019

(GLM)

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The main objective of this paper is to obtain a new accurate approximate solutions for a kind of ordinary differential equations called multipoint boundary value problems by using simple modification of optimal homotopy asymptotic method (OHAM). This procedure is a well-performance for calculating a better approximate solutions using one-order of approximation comparing with other methods which need higher order of approximations to gives the same results. Some examples are presented to testify the accuracy and applicability of this procedure. Comparisons are made between the present procedure and the other methods.

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Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

Cited by 7 publications

References 24 publications

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

A New Optimal Homotopy Asymptotic Method for Fractional Optimal Control Problems

An Accurate Approximate Solutions of Multipoint Boundary Value Problems

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