ANALYTICAL STUDY OF A FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATION

Mohammed Abdel-Aty; Jihan Alahmadi; Mohammed Abdou

doi:10.11948/20250192

2026 Volume 16 Issue 3

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(3): 1109-1129

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Mohammed Abdel-Aty, Jihan Alahmadi, Mohammed Abdou. ANALYTICAL STUDY OF A FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1109-1129. doi: 10.11948/20250192

Citation:

Mohammed Abdel-Aty, Jihan Alahmadi, Mohammed Abdou. ANALYTICAL STUDY OF A FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1109-1129. doi: 10.11948/20250192

ANALYTICAL STUDY OF A FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATION

1.
Department of Mathematics and Computer Science, Faculty of Science, Benha University, Benha 13518, Egypt
2.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
3.
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21511, Egypt

Author Bio: Email: j.alahmadi@psau.edu.sa(J. Alahmadi); Email: abdella_777@yahoo.com(M. Abdou)
Corresponding author: Email: mohammed.abdallah@fsc.bu.edu.eg(M. Abdel-Aty)

Abstract

Abstract

In this paper, we consider a fractional partial integro-differential equation (frPI-DE) associated with a quadratic integral equation of order $\gamma>0$. By employing the properties of fractional integrals, the frPI-DE is shown to be equivalent to a nonlinear quadratic Volterra–Fredholm integral equation. We investigate the existence of solutions to such nonlinear functional integral equations in the Banach space $L_{2}[0, 1]\times C[0, T]$. The existence results are established via a generalized form of Darbo's fixed-point theorem, which is based on the measure of noncompactness. To illustrate the applicability and effectiveness of the theoretical findings, two examples are provided. In addition, a numerical scheme is introduced to demonstrate the convergence of approximate solutions to the nonlinear quadratic Volterra–Fredholm integral problem. The uniqueness and stability of the error are also analyzed.
- Fractional partial integro-differential equation /
- Caputo fractional derivative /
- quadratic Volterra–Fredholm integral equation /
- measure of noncompactness /
- Darbo's fixed point theorem /
- Riemann integral
MSC: 26A33, 35R09, 45K05, 47H08, 47H10

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References

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