Numerical Treating of Mixed Integral Equation Two-Dimensional in Surface Cracks in Finite Layers of Materials

Al‐Bugami, A. M.

doi:10.1155/2022/3398175

Cited by 11 publications

(10 citation statements)

References 8 publications

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“…In all previous research, the time periods in the mixed equation were divided, and this equation transformed into an algebraic system for FIEs (see [16,17]). Here, we use the separation technique method to get FIEs with time parameters that are described as an integrated operator in time.…”

Section: Separation Methodsmentioning

confidence: 99%

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

Abdou

Basseem

2023

Journal of Applied Mathematics

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This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

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Section: Separation Methodsmentioning

confidence: 99%

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

Abdou

Basseem

2023

Journal of Applied Mathematics

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“…After applying the conformal mapping (1) in the boundary conditions (6), we can write the expression 1 () ()…”

Section: Gaursat Functionsmentioning

confidence: 99%

“…Abdou et al [5] applied orthogonal polynomials method, Chebechev polynomials, to obtain numerically the singular quadratic integral equation. Al-Bigamy [6] applied Toeplitz matrix method to obtain the numerical solution of mixed Integral equation of two-dimensional in surface cracks in finite layers of materials.…”

mentioning

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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“…See [1][2][3][4][5][6] for more information on the topic of two-dimensional nonlinear integral equations, which have long been of growing interest in many fields, including medicine [7], biology [8], physics [9], geography and fuzzy control [10]. According to the references [11][12][13][14][15][16], many problems in engineering [17], applied mathematics and mathematical physics [18] can be reduced to two-dimensional nonlinear integral equations with a symmetric and nonsymmetrical kernel. The analytical solutions for these equations are typically difficult.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

et al. 2023

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The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L2[0,1]×L2[0,1]. Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach.

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Numerical Treating of Mixed Integral Equation Two-Dimensional in Surface Cracks in Finite Layers of Materials

Cited by 11 publications

References 8 publications

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

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

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

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