A hybrid technique of deep learning neural networks with finite difference method for higher order fractional Volterra-Fredholm integro-differential equations with \(\varphi\)-Caputo operator

Volume 40, Issue 3, pp 310--329 https://dx.doi.org/10.22436/jmcs.040.03.02

Publication Date: July 08, 2025 Submission Date: December 29, 2024 Revision Date: March 05, 2025 Accteptance Date: May 22, 2025

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Authors

K. Alsa'di - Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia. N. M. A. Nik Long - Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia. - Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia.

Abstract

This paper deals with the theoretical and numerical aspects for higher order Volterra-Fredholm fractional integro-differential equations (VF-IDEs) under \(\varphi\)-Caputo operator. Using Liptchiz conditions, Krasnoselskii's fixed point theorem, and Gronwall inequality with respect to the function \(\varphi\), existence and uniqueness of the solution are investigated. The stability of the solution is analyzed through the continuity of the parameters. Moreover, a new hybrid technique which is the combination of deep learning artificial neural network and finite difference method (FDL-ANN) is developed to approximate the solution of higher order VF-IDEs. This technique uses the Adaptive Moment Estimation Method (Adam) as an optimization algorithm with feed-forward deep learning to minimize the error function and training the model using five layers with different activation functions. The numerical analysis for the error bound and the computation complexity are provided for FDL-ANN. The numerical examples demonstrated the efficiency of the proposed method in solving the complicated higher order fractional problems of linear and non-linear terms.

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Keywords

• Fractional integro-differential equation
• \(\varphi\)-Caputo operator
• artificial neural network
• Adam optimization
• deep learning
• feed-forward networks

MSC

•  26A33
•  45J05
•  65L60
•  65M06
•  68T07

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