Approximate Analytical Solution by Residual Power Series Method for System of Fredholm Integral Equations

Komashynska, Iryna; Al‐Smadi, Mohammed; Ateiwi, Ali Mahmud; Al-Obaidy, Sadoon Abdullah Ibrahim

doi:10.18576/amis/100315

Cited by 41 publications

(34 citation statements)

References 14 publications

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“…As in [25][26][27], we have Res u i (t) = 0, and lim k→∞ Res u i,k (t) = Res u i (t), for each t ≥ 0. As a matter of fact, this yields D nβ i Res u i,k (t) = 0 for n = 0, 1, 2, .…”

Section: Fractional Residual Power Series Methodsmentioning

confidence: 91%

“…Also, the method was applied successfully in solving composite and non-composite fractional DEs, and in predicting and representing multiplicity solutions to fractional boundary value problems [24,25]. Furthermore, [26][27][28][29] assert that the RPS method is easy and powerful to construct power series solution for strongly linear and nonlinear equations without terms of perturbation, discretization, and linearization. Unlike the classical power series method, the FRPS method distinguishes itself in several important aspects such that it does not require making a comparison between the coefficients of corresponding terms and a recursion relation is not needed and provides a direct way to ensure the rate of convergence for series solution by minimizing the residual error related.…”

Section: Introductionmentioning

confidence: 99%

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Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

et al. 2019

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A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.

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“…As in [25][26][27], we have Res u i (t) = 0, and lim k→∞ Res u i,k (t) = Res u i (t), for each t ≥ 0. As a matter of fact, this yields D nβ i Res u i,k (t) = 0 for n = 0, 1, 2, .…”

Section: Fractional Residual Power Series Methodsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

et al. 2019

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“…The residual power series (RPS) method developed in [18] is considered as an effective optimization technique to determine and define the power series solution's values of coefficients of first-and second-order fuzzy differential equations [19][20][21][22]. Furthermore, the RPS is characterized as an applicable and easy technique to create power series solutions for strongly linear and nonlinear equations without being linearized, discretized, or exposed to perturbation [23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Alaroud

Al‐Smadi

Ahmad

et al. 2018

International Journal of Differential Equations

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This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1 < ≤ 2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem's nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

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“…The objective of the present paper is to use the HAM and Laplace transform to provide optimal solutions for a fractional order differential system model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4 + T-cells. However, other category of methods to handle large amount of fractional problem can be found in [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Simulation as an alternative to linear programming of human T-cell lymphotropic virus I (HTLV-I) model infection of CD4+ T-cells

Alabedalhadi¹

2018

imf

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Simulation is a tradition of operating a process or system in the real world. The act of simulating something first requires the development of a model; this model represents the main characteristics of the selected behaviors, processes, and functions. The model represents the system itself, while the simulation represents the operation of the system over time. In this paper, the optimal solutions of fractional human T-cell lymphotropic virus model infection of CD4+ T-cells will selected by using an efficient technique, called the LHAM, which is a series solution method based on the HAM and Laplace transform in obtaining the solutions for a wide class of problem. The Pade approximation technique to enlarge region of convergence for the solutions. Results obtained using the shame presented here are in good agreement with the numerical results obtained before. Our work confirms the efficiency of LHAM as a tool for solving linear and nonlinear fractional differential equations. The numerical method proposed in this thesis can be utilized to solve other problems in field of nonlinear fractional differential equations.

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Approximate Analytical Solution by Residual Power Series Method for System of Fredholm Integral Equations

Cited by 41 publications

References 14 publications

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Simulation as an alternative to linear programming of human T-cell lymphotropic virus I (HTLV-I) model infection of CD4+ T-cells

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