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

Volume 32, Issue 2, pp 137--151 http://dx.doi.org/10.22436/jmcs.032.02.04

Publication Date: August 18, 2023 Submission Date: January 09, 2023 Revision Date: May 30, 2023 Accteptance Date: July 01, 2023

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Authors

A. M. S. Mahdy - Department of Mathematics, Faculty of Science, Zagazig University, P. O. Box 44519, Zagazig, Egypt. - Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif, 21944, Saudi Arabia. M. A. Abdou - Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, 21526, Egypt. D. Sh. Mohamed - Department of Mathematics, Faculty of Science, Zagazig University, P. O. Box 44519, Zagazig, Egypt.

Abstract

In the present article, we establish the numerical solution for the mixed Volterra- Fredholm integral equation (MV-FIE) in (1+1) dimensional in the Banach space \(L_2[-1,1] \times C[0, T], T < 1.\) The Fredholm integral term is considered in the space \(L_2[-1,1]\) 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 \(L_2[-1,1] \times C[0, T], T < 1.\) 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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Keywords

• Volterra-Fredholm integral equations
• separation of variables
• Boubaker polynomials
• numerical solution
• linear algebraic system

MSC

•  46B45
•  65R20

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