Dirichlet Problem in a Simply Connected Domain, Bounded by the Nontrivial Kind

Qazza, Ahmad; Hatamleh, Raéd

doi:10.17654/de017030177

Cited by 7 publications

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“…It was widely used by Ramanujan as a tool in computing definite integrals and infinite series. The theorems presented in this article are just as powerful and efficient as Ramanujan's master theorem in solving some families of improper integrals, for more about integral transforms, see [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New Theorems in Solving Families of Improper Integrals

Abu-Ghuwaleh

Saadeh

Burqan

2022

Axioms

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“…Numerous phenomena in various elds of science can be fruitfully formulated by the use of fractional derivatives. is is because the sensible modeling for a physical phenomenon depends on instantaneous time as well as on prior time history; hence, we may use fractional calculus to deal with these problems [1][2][3][4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…Many physical and engineering problems can be formulated by fractional di erential equations (FDEs) and obtaining the solutions of these equations have been the theme of many interesting investigations and have attracted the attention of researchers. Recently, there have been a large number of techniques dedicated to get solutions of FDEs [11][12][13][14][15][16]. ese techniques can be classi ed into two categories, approximate and analytical, such as transform methods, homotopy analysis methods, and residual power series methods.…”

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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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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“…Fractional calculus is a generalization of the derivatives and integrals of an arbitrary system. Recently, the subject of fractional calculus has received the attention of scientists and engineers because of its important applications in various fields, whether science or engineering [14][15][16][17][18][19]. Many real-life problems in various fields of applied science have been modeled using fractional differential equations (FDEs), which are generalizations of ODEs.…”

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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Dirichlet Problem in a Simply Connected Domain, Bounded by the Nontrivial Kind

Cited by 7 publications

References 0 publications

New Theorems in Solving Families of Improper Integrals

New Theorems in Solving Families of Improper Integrals

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

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

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