New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

Alhazmi, Shareefa Eisa Ali

doi:10.5539/jmr.v9n5p18

Cited by 11 publications

(4 citation statements)

References 14 publications

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“…(2) Also, the series method can be used to separate position from time in the mixed integral equations, and then it is possible to obtain three Volterra integral equations of the second type that depend on time. The other part takes the form of Fredholm's integral equation, see [2][3][4]11] and the references therein for more details.…”

Section: )mentioning

confidence: 99%

“…The solution of the MIE of the first kind in one, two and three dimensions has been obtained analytically using the separation of variables method in [1]. The MIE of the first kind can be solved analytically using one of the following methods: The Cauchy method, orthogonal polynomial method, potential theory method and Krein's method, see [9][10][11][12][13][14][15][16] and the references therein for details. The relation between the MIEs and some contact problems can be found in [13][14][15][16][17] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

Abdou

Youssef

2021

ARJOM

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In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions. The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

Abdou

Youssef

2021

ARJOM

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“…Many problems in mathematical physics (Hadjadj and Dussauge [19]), theory of elasticity (Abdou et al [6,7] and Popov [35]), contact problems in two layers of elastic materials (Bugami [9]), generalized potential theory (Alhazmi [11]), spectral relationships in laser theory (Gao et al [18]), quantum mechanics (Lienert and Tumulka [24]), and mixed problems in the idea of elasticity (Aleksandrovsk and Covalence [10] and Georgiadis and Gourgiotis [40]) lead to one kind of integral equation. As a result, we have discovered that integral equations (IEs) have tight ties to various subfields within many scientific disciplines.…”

Section: Introductionmentioning

confidence: 99%

A computational technique for computing second-type mixed integral equations with singular kernels

Mahdy,

Abdou,

Mohamed

2023

J. Math. Computer Sci.

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In the present article, we establish the numerical solution for the mixed Volterra-Fredholm integral equation (MV-FIE) and it has a discontinuous kernel in position. While the Volterra integral term is considered in the class of time C[0, T ], T < 1, and has a continuous kernel in time. The necessary conditions have been established to ensure that there is a single solution in the space L 2 [−1, 1] × C[0, T ], T < 1. By utilizing the separation of variables technique, MV-FIE is transformed to Fredholm integral equation (FIE) of the second kind with variables coefficients in time. The separation technique of variables helps the authors choose the appropriate time function to establish the conditions of convergence in solving the problem and obtaining its solution. Then, using the Boubaker polynomials method, we end up with a linear algebraic system (LAS) abbreviated. The Banach fixed point (BFP) hypothesis has been presented to determine the existence and uniqueness of the solution of the LAS. The convergence of the solution and the stability of the error are discussed. The Maple 18 software is used to perform some numerical calculations once some numerical experiments have been taken into consideration.

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“…with generalized potential kernel of the fundamental problems in the theory of elasticity. Al-Hazmi [3] presented orthogonal polynomials method to discuss the solution of mixed integral equation of the first kind with generalized potential kernel.…”

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confidence: 99%

The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

Ibrahim

Abdou

2023

Preprint

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In this paper, we presented an integro differential equation (IDE), in compelx plane, with singular kernel. The IDE is investigated from the first and second boundary value problem (BVP) of a weakened infinite plate by a curvilinear hole (C,) in the presence of heat. The hole is conformally mapped inside a unit circle by a special transformation with complex constants coefficients\(z=\omega (\zeta ),\,\zeta =\xi +i\eta ,\,i=\sqrt { - 1} .\) This mapping will conform the curvilinear hole in the infinite elastic plate into a unit circle \(\gamma ,\left| \zeta \right|\;<\,\,1,\) such that inside the circle. The stress and strain compounds were calculated, in the presence of heat effect, and this was clarified by numerical results. Many applications for the problem are discussed. Maple 2019 is used for computations results. MSC (2010): 74B10, 30C20.

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New Model for Solving Mixed Integral Equation of the First Kind with Generalized Potential Kernel

Cited by 11 publications

References 14 publications

A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

A computational technique for computing second-type mixed integral equations with singular kernels

The singular complex integro differential equation via the boundary value problems of elastic plate weakened by curvilinear hole

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