Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order

Eshkuvatov, Zainidin K.; Laadjal, Zaid; Ismail, Shahrina

doi:10.1063/5.0057120

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…The obtained linear and non-linear integro-differential equations can be solved through many numerical methods. The existence of solution and approximation of the problems have been studied by many authors via: homotopy analysis method (HAM) [1][2][3][4], modified HAM [5][6][7][8][9], optimal HAM [10][11][12], q-HAM [13][14][15][16][17], for variety of problems. A number of series numerical methods were derived for integro-differential equations, such as variational iteration method [18], Taylor-successive approximation method [19], the Adomian decomposition method (ADM) [20][21][22], differential transform method [23], Wavelet-Galerkin method [24], the Tau method [25], modified HPM [26], Laplace transform ADM [27], finite element method [28] for partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

Eshkuvatov¹

2022

Math. Model. Comput.

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Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied in many different problems of mathematical physics and engineering. In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions. Practical investigations revealed that ND-HAM leads an easy way how to find initial guess and it approaches the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM). Uniqueness solution of the problem and convergence of ND-HAM are proved in the Banach space. Finally, two examples are illustrated to show the accuracy and validity of the proposed method. Five methods are compared in each example.

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Section: Introductionmentioning

confidence: 99%

New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

Eshkuvatov¹

2022

Math. Model. Comput.

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show abstract

“…The obtained linear and non-linear integro-differential equations can be solved through many numerical methods. The existence of solution and approximation of the problems have been investigated by many authors viz: homotopy analysis method (HAM) [1,2,3,4], modified HAM [5,6,7,8,9], optimal HAM [10,11,12], q-HAM [13,14,15,16,17], for variety of problems. A number of series numerical methods were derived for integro-differential equations, such as variational iteration method [18], Taylor-successive approximation method [19], the Adomian decomposition method (ADM) [20,21,22], differential transform method [23], Wavelet-Galerkin method [24], the Tau method [25], modified HPM [26], Laplace transform ADM [27], finite element method [28] for partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

New Development of Homotopy Analysis Method for a Non-linear Integro-Differential Equations with initial conditions

Eshkuvatov

2021

Preprint

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Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied it in many different problems of mathematical-physics and engineering. In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions. Practical investigations revealed that ND-HAM leads easy way how to find initial guess and it approaches to the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM). Two examples are illustrated to show the accuracy and validity of the proposed method. Five methods are compared in each example.

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Theoretical and Numerical Studies of Fractional Volterra-Fredholm Integro-Differential Equations in Banach Space

Alsa'di,,

Nik Long,

Eshkuvatov

2024

MJMS

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This paper examines the theoretical, analytical, and approximate solutions of the Caputo fractional Volterra-Fredholm integro-differential equations (FVFIDEs). Utilizing Schaefer's fixed-point theorem, the Banach contraction theorem and the Arzel\`{a}-Ascoli theorem, we establish some conditions that guarantee the existence and uniqueness of the solution. Furthermore, the stability of the solution is proved using the Hyers-Ulam stability and Gronwall-Bellman's inequality. Additionally, the Laplace Adomian decomposition method (LADM) is employed to obtain the approximate solutions for both linear and non-linear FVFIDEs. The method's efficiency is demonstrated through some numerical examples.

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Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order

Cited by 3 publications

References 12 publications

New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

New Development of Homotopy Analysis Method for a Non-linear Integro-Differential Equations with initial conditions

Theoretical and Numerical Studies of Fractional Volterra-Fredholm Integro-Differential Equations in Banach Space

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