An Algorithm for Solving Phase‐Lag Nonlinear Mixed Integral Equation With Discontinuous Generalized Kernel

In this work, a nonlinear fractional integrodifferential equation (NFIo‐DE) with discontinuous generalized kernel in position and time is explored in space L 2 ( Ω ) × C [0, T ], T < 1, with respect to the phase‐lag time. Here, Ω is the domain of integration with respect to position, Ω ∈ (−1, 1),...

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Bibliographic Details
Published in:Advances in Mathematical Physics Vol. 2025; no. 1
Main Authors: Al-Bugami, Abeer M., Abdou, M. A.
Format: Journal Article
Language:English
Published: John Wiley & Sons, Inc 01.01.2025
Wiley
Subjects:
Algorithms
ISSN:1687-9120, 1687-9139
Online Access:Get full text
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Summary:In this work, a nonlinear fractional integrodifferential equation (NFIo‐DE) with discontinuous generalized kernel in position and time is explored in space L 2 ( Ω ) × C [0, T ], T < 1, with respect to the phase‐lag time. Here, Ω is the domain of integration with respect to position, Ω ∈ (−1, 1), while T is the time. The integral equation (IE) under discussion has a general kernel. The equation can be transformed into a Volterra‐Fredholm integral equation (V‐FIE) of mixed type by applying the integration technique and using initial conditions. The study examines the presence of a singular solution, the convergence of the solution, and the stability of the error under specific circumstances. A Fredholm integral equation (FIE) with a general singular kernel could be obtained by applying a particular technique to separate the variables. This method was shown to relate the time variables to the IE’s kernel in terms of the existence of a unique solution. Using the Toeplitz matrix method (TMM) it was possible to obtain an algebraic system that was studied in terms of the existence of a unique solution and its convergence. Finally, some numerical results are calculated, when the kernel takes a general form of logarithmic kernel, Carleman function, and Cauchy kernel. Furthermore, the error estimate is calculated for each case.
ISSN:1687-9120
1687-9139
DOI:10.1155/admp/5558147