M. A. Abdel-Aty - Profile on Academia.edu (original) (raw)

Papers by M. A. Abdel-Aty

Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Springer Nature, 2024

Filomat, 2023

In this work, we study the existence of at least one solution of the Quadratic integral equation ... more In this work, we study the existence of at least one solution of the Quadratic integral equation with Phase-lag term. Our proof depends on a suitable combination of the Darbo's fixed point principle and the technique of measures of noncompactness. Homotopy perturbation method is presented to obtain an approximate solution of Quadratic integral equation with Phase-lag term. Convergence and error estimate of Homotopy perturbation method are obtained. Homotopy perturbation method is a powerful device for solving a wide variety of problems. It gives excellent flexibility to the expression of the solution and how the solution is explicitly obtained, and provides great freedom in choosing the base functions of the desired solution and the corresponding auxiliary linear operator of homotopy. These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with two examples that show simplicity and effectiveness.

Fractal and fractional, Oct 1, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Analytical and Numerical Discussion for the Phase-Lag Volterra-Fredholm Integral Equation with Singular Kernel

Journal of Applied Analysis & Computation

The journal of mathematics and computer science, Aug 28, 2023

In this study, the nonlinear integro-differential equation (NIDE) of the second kind is resolved ... more In this study, the nonlinear integro-differential equation (NIDE) of the second kind is resolved using the Adomian decomposition method (ADM). The term non-linearity can be dealt with easily if used techniques of Adomian polynomials. The existence of at least one positive continuous solution to the nonlinear integro-differential equation is ensured by sufficient conditions. Both the Arzelà-Ascoli theorem and the Tychonoff fixed point principle are used in this method. These types of equations are solved using the Adomian decomposition method and the repeated trapezoidal method. The method presented at the end of the article has been tested on many examples and has proven its efficiency after discussing the results.

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

Fractal Fract, 2023

Analytical and Numerical Discussion for the Phase-Lag Volterra-Fredholm Integral Equation with Singular Kernel

Journal of Applied Analysis & Computation, 2023

Application of Adomian polynomials for solving nonlinear integro-differential equations

Journal of Mathematics and Computer Science, 2024

New algorithms for solving nonlinear mixed integral equations

AIMS Mathematics

In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral eq... more In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.

Symmetry

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmet... more 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.

Springer Nature, 2021

The purpose of this paper is to establish the general solution of a Volterra–Fredholm integral eq... more The purpose of this paper is to establish the general solution of a Volterra–Fredholm integral equation with discontinuous kernel in a Banach space. Banach’s fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to a Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed. Mathematics Subject Classification (2010): 45L05; 46B45; 65R20. Key–Words: Banach space, Volterra–Fredholm integral equation, Separation of variables method.

Springer Nature, 2021

In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that i... more In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that is the coupling of Laplace transform and Adomian decomposition method. The modified Laplace Adomian decomposition method is applied to solve the Fredholm-Volterra integro-differential equations of the second kind in the space L2[a, b]. The nonlinear term will simply be handled with the help of Adomian polynomials. The Laplace decomposition technique is found to be fast and correct. Several examples are tested and also the results of the study are discussed. The obtained results expressly reveal the complete reliability, efficiency, and accuracy of the proposed algorithmic rule for solving the Fredholm-Volterra integro-differential equations and therefore will be extended to other problems of numerous nature.

MDPI, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Faculty of Sciences and Mathematics, University of Niˇ s, Serbia, 2023

In this work, we study the existence of at least one solution of the Quadratic integral equation ... more In this work, we study the existence of at least one solution of the Quadratic integral equation with Phase-lag term. Our proof depends on a suitable combination of the Darbo's fixed point principle and the technique of measures of noncompactness. Homotopy perturbation method is presented to obtain an approximate solution of Quadratic integral equation with Phase-lag term. Convergence and error estimate of Homotopy perturbation method are obtained. Homotopy perturbation method is a powerful device for solving a wide variety of problems. It gives excellent flexibility to the expression of the solution and how the solution is explicitly obtained, and provides great freedom in choosing the base functions of the desired solution and the corresponding auxiliary linear operator of homotopy. These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with two examples that show simplicity and effectiveness.

Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2022

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic in... more In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

Solvability of quadratic integral equations with singular kernel

Proceedings of NAS RA. Mathematics

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic in... more In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

J. of Applied Analysis & Computation, 2020

In the present paper, we are concerning with a quadratic integral equation with phase–lag term. I... more In the present paper, we are concerning with a quadratic integral equation with phase–lag term. In the following pages, sufficient conditions are given for the existence of positive continuous solution to quadratic integral equations. The method used here depends on both Tychonoff fixed point principle and Arzelà–Ascoli theorem. A concrete example illustrating the mentioned applicability is also included.

International Journal of Mathematical Analysis

In this paper, the existence and uniqueness solution of the Fredholm-Volterra integral equations ... more In this paper, the existence and uniqueness solution of the Fredholm-Volterra integral equations (F-VIEs) are considered in the space L 2 [0, 1]× C n [0, T ], 0 ≤ T < 1. Using a numerical technique, F-VIEs lead to a system of linear Fredholm integral equations (SLFIEs). Also, the normality and the continuity of integral operator are discussed. The Trapezoidal Rule is used to get the solution of SLFIEs. Finally, numerical results are discussed and the error estimate is computed.

Journal of the Egyptian Mathematical Society, Feb 28, 2020

The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral eq... more The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral equation with discontinuous kernel in a Banach space. Banach's fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed.

Springer International Publishing AG, part of Springer Nature 2018, 2018

This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation ... more This paper presents a numerical method for the solution of a
Volterra–Fredholm integral equation in a Banach space. Banachs fixed
point theorem is used to prove the existence and uniqueness of the solution.
To find the numerical solution, the integral equation is reduced to
a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.

Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Springer Nature, 2024

Filomat, 2023

In this work, we study the existence of at least one solution of the Quadratic integral equation ... more In this work, we study the existence of at least one solution of the Quadratic integral equation with Phase-lag term. Our proof depends on a suitable combination of the Darbo's fixed point principle and the technique of measures of noncompactness. Homotopy perturbation method is presented to obtain an approximate solution of Quadratic integral equation with Phase-lag term. Convergence and error estimate of Homotopy perturbation method are obtained. Homotopy perturbation method is a powerful device for solving a wide variety of problems. It gives excellent flexibility to the expression of the solution and how the solution is explicitly obtained, and provides great freedom in choosing the base functions of the desired solution and the corresponding auxiliary linear operator of homotopy. These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with two examples that show simplicity and effectiveness.

Fractal and fractional, Oct 1, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Analytical and Numerical Discussion for the Phase-Lag Volterra-Fredholm Integral Equation with Singular Kernel

Journal of Applied Analysis & Computation

The journal of mathematics and computer science, Aug 28, 2023

In this study, the nonlinear integro-differential equation (NIDE) of the second kind is resolved ... more In this study, the nonlinear integro-differential equation (NIDE) of the second kind is resolved using the Adomian decomposition method (ADM). The term non-linearity can be dealt with easily if used techniques of Adomian polynomials. The existence of at least one positive continuous solution to the nonlinear integro-differential equation is ensured by sufficient conditions. Both the Arzelà-Ascoli theorem and the Tychonoff fixed point principle are used in this method. These types of equations are solved using the Adomian decomposition method and the repeated trapezoidal method. The method presented at the end of the article has been tested on many examples and has proven its efficiency after discussing the results.

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

Fractal Fract, 2023

Analytical and Numerical Discussion for the Phase-Lag Volterra-Fredholm Integral Equation with Singular Kernel

Journal of Applied Analysis & Computation, 2023

Application of Adomian polynomials for solving nonlinear integro-differential equations

Journal of Mathematics and Computer Science, 2024

New algorithms for solving nonlinear mixed integral equations

AIMS Mathematics

In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral eq... more In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.

Symmetry

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmet... more 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.

Springer Nature, 2021

The purpose of this paper is to establish the general solution of a Volterra–Fredholm integral eq... more The purpose of this paper is to establish the general solution of a Volterra–Fredholm integral equation with discontinuous kernel in a Banach space. Banach’s fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to a Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed. Mathematics Subject Classification (2010): 45L05; 46B45; 65R20. Key–Words: Banach space, Volterra–Fredholm integral equation, Separation of variables method.

Springer Nature, 2021

In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that i... more In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that is the coupling of Laplace transform and Adomian decomposition method. The modified Laplace Adomian decomposition method is applied to solve the Fredholm-Volterra integro-differential equations of the second kind in the space L2[a, b]. The nonlinear term will simply be handled with the help of Adomian polynomials. The Laplace decomposition technique is found to be fast and correct. Several examples are tested and also the results of the study are discussed. The obtained results expressly reveal the complete reliability, efficiency, and accuracy of the proposed algorithmic rule for solving the Fredholm-Volterra integro-differential equations and therefore will be extended to other problems of numerous nature.

MDPI, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Faculty of Sciences and Mathematics, University of Niˇ s, Serbia, 2023

In this work, we study the existence of at least one solution of the Quadratic integral equation ... more In this work, we study the existence of at least one solution of the Quadratic integral equation with Phase-lag term. Our proof depends on a suitable combination of the Darbo's fixed point principle and the technique of measures of noncompactness. Homotopy perturbation method is presented to obtain an approximate solution of Quadratic integral equation with Phase-lag term. Convergence and error estimate of Homotopy perturbation method are obtained. Homotopy perturbation method is a powerful device for solving a wide variety of problems. It gives excellent flexibility to the expression of the solution and how the solution is explicitly obtained, and provides great freedom in choosing the base functions of the desired solution and the corresponding auxiliary linear operator of homotopy. These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with two examples that show simplicity and effectiveness.

Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2022

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic in... more In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

Solvability of quadratic integral equations with singular kernel

Proceedings of NAS RA. Mathematics

In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic in... more In this paper, we discussed the existence and uniqueness of solution of the singular Quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular Quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed.

J. of Applied Analysis & Computation, 2020

In the present paper, we are concerning with a quadratic integral equation with phase–lag term. I... more In the present paper, we are concerning with a quadratic integral equation with phase–lag term. In the following pages, sufficient conditions are given for the existence of positive continuous solution to quadratic integral equations. The method used here depends on both Tychonoff fixed point principle and Arzelà–Ascoli theorem. A concrete example illustrating the mentioned applicability is also included.

International Journal of Mathematical Analysis

In this paper, the existence and uniqueness solution of the Fredholm-Volterra integral equations ... more In this paper, the existence and uniqueness solution of the Fredholm-Volterra integral equations (F-VIEs) are considered in the space L 2 [0, 1]× C n [0, T ], 0 ≤ T < 1. Using a numerical technique, F-VIEs lead to a system of linear Fredholm integral equations (SLFIEs). Also, the normality and the continuity of integral operator are discussed. The Trapezoidal Rule is used to get the solution of SLFIEs. Finally, numerical results are discussed and the error estimate is computed.

Journal of the Egyptian Mathematical Society, Feb 28, 2020

The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral eq... more The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral equation with discontinuous kernel in a Banach space. Banach's fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed.

Springer International Publishing AG, part of Springer Nature 2018, 2018

This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation ... more This paper presents a numerical method for the solution of a
Volterra–Fredholm integral equation in a Banach space. Banachs fixed
point theorem is used to prove the existence and uniqueness of the solution.
To find the numerical solution, the integral equation is reduced to
a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.

IntechOpen, 2019

The purpose of this chapter is to state definitions of some important spaces and classes of vecto... more The purpose of this chapter is to state definitions of some important spaces and classes of vectors and functions. The basic definitions and relations for some special functions like Gamma, Beta, Bessel and Mathieu functions are stated. Moreover, the Fourier and Laplace transformations with its basic relations are written. Also, some types of integral equations are stated. Finally, the integral operator is considered.