Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method

In this research, systems of linear and nonlinear differential equations of fractional order are solved analytically using the novel interesting method: ARA- Residual Power Series (ARA-RPS) technique. This approach technique is based on the combination of the residual power series scheme with the ARA transform to establish analytical approximate solutions in a fast convergent series representation using the concept of the limit. The proposed method needs less time and effort compared with the residual power series technique. To prove the simplicity, applicability, and reliability of the presented method, three numerical examples are proposed and simulated. The obtained results show that the ARA-RPS technique is applicable, simple, and effective to get solutions to linear and nonlinear engineering and physical problems.

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“…To find the coefficients of the series expansions in ( 22) and ( 23), we define the ARA-residual functions of the (18) as follows:…”

Section: Methodology Of the Ara-rpsmmentioning

confidence: 99%

“…ese techniques can be classi ed into two categories, approximate and analytical, such as transform methods, homotopy analysis methods, and residual power series methods. [17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

Qazza

Burqan

Saadeh

2022

Mathematical Problems in Engineering

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“…some of these integrals can't be solved up to now, some of them are solved numerically, and there still some integrations that cannot be determined exactly or need much effort to be solved. The importance of computing improper integrals has come from the wide usage in applied math, physics, engineering and etc., [6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

New Theorems in Solving Families of Improper Integrals

Abu-Ghuwaleh

Saadeh

Burqan

2022

Axioms

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Many improper integrals appear in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik. It is a challenge for some researchers to determine the method in which these integrations are formed or solved. In this article, we present some new theorems to solve different families of improper integrals. In addition, we establish new formulas of integrations that cannot be solved by mathematical software such as Mathematica or Maple. In this article, we present three main theorems that are essential in generating new formulas for solving improper integrals. To show the efficiency and the simplicity of the presented techniques, we present some applications and examples on integrations that cannot be solved by regular methods. Furthermore, we acquire new results for integrations and compare them to that obtained in the classical table of integrations. Some previous results, become special cases of our outcomes or generalizations to acquire new integrals.

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“…In this regard, some recent techniques are proposed for solving FDEs. Among them decomposition technique, symmetric perturbation technique, variable frequency technique, and partial differential transformation technique [20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of Nonlinear Fractional Gradient-Based System Using Fractional Power Series Method

Abu-Gdairi

2022

Int. J. Anal. Appl.

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This paper adapted a reliable treatment technique, called the fractional residual power series, to the fractional gradient-based system in solving a class of nonlinear programming model in Caputo’s sense. The gradient-based system has been constructed to transform the nonlinear programming problem with equality constraints to unconstrained optimization problem, and then the fractional residual power series method is implemented to obtain the essential behavior of underlying problem. The proposed methods have been applied effectively to produce optimal solution in rapidly convergent fractional series representations without linearization, or any limitations. To confirm the performance of the proposed methods, some optimization problems are tested. Further, numerical comparisons with other existing methods are also given. The results exhibit that the FRPS method is easy, simple, effective, and fully compatible with the complexity of such models.

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Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method

Cited by 16 publications

References 51 publications

Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations

New Theorems in Solving Families of Improper Integrals

Analytical Solution of Nonlinear Fractional Gradient-Based System Using Fractional Power Series Method

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