A study of normality and continuity for mixed integral equations

Abdou, M. A.; Nasr, Mohamed E.; Abdel–Aty, M. A.

doi:10.1007/s11784-018-0490-0

Cited by 13 publications

(4 citation statements)

References 13 publications

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“…Integral equations are used in many disciplines of applied mathematics to explore and solve problems. 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.…”

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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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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“…Galerkin methods are also one of the methods that attracted the attention of researchers and are widely used with general approximate functions, Bernstein polynomials [12], Legendre polynomials [13], Alpert's multiwavelet bases [14], and conflict-type wavelets [15]. Furthermore, among the numerical methods that have been developed are the Quadrature methods [16], Homotopy analysis methods [17], Modified homotopy perturbation methods [18], and least squares approximation methods [19]. Several typical examples of iterative approximation methods are the block-by-block method and Runge-Kutta method [20], optimal perturbation iteration method [21], and Picard iteration method [22].…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel

Abusalim,

Abdou,

Nasr

et al. 2023

Fractal Fract

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The nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of nonlinear integral equations. The existence and unique numerical solution of this system is discussed, as is the truncation error of the numerical solution. The solution of the nonlinear integral equation system is obtained using the spectral relations and techniques of the Chebyshev polynomial method. Finally, we will discuss examples of when the kernel takes various forms to demonstrate this technique’s high accuracy and simplicity. Some numerical results and estimating errors are calculated and plotted using the program Wolfram Mathematica 10.

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“…See [1,5,6,10,24,26,27] for more information on the topics of two-dimensional nonlinear integral equations, which have long been of growing interest in many fields, including medicine, biology, physics, geography, and fuzzy control. According to the references [2,3,7,[9][10][11][12], many problems in engineering, applied mathematics, and mathematical physics can be reduced into two-dimensional nonlinear integral equations. The analytical solutions to these equations are typically difficult.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

Abusalim¹,

Abdou²,

Abdel–Aty³

et al. 2023

Preprint

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This study presents the solution of the second type of a two-dimensional nonlinear integral equation in Banach space. Also, the existence and uniqueness of this equation’s solution are discussed. We utilize a numerical approach involving hybrid and block-pulse functions to obtain the approximate solution of a two-dimensional nonlinear integral equation. Nonlinear integral equation in two dimensions is reduced numerically to a system of nonlinear algebraic equations that can be solved using numerical methods. This study focuses on showing the convergence analysis for the numerical approach and obtaining an error estimate. Some numerical examples have been provided to demonstrate the approach’s viability and efficacy

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A study of normality and continuity for mixed integral equations

Cited by 13 publications

References 13 publications

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

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

An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel

Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

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