Numerical study for the fractional model of banks' competition using two efficient computational methods

Volume 39, Issue 4, pp 437--452 https://dx.doi.org/10.22436/jmcs.039.04.03

Publication Date: April 29, 2025 Submission Date: June 22, 2024 Revision Date: November 02, 2024 Accteptance Date: January 17, 2025

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Authors

M. Adel - Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia. M. M. Khader - Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA. M. B. Riaz - IT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic. I. Alraddadi - Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia. H. Ahmad - Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia. - Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138 Mersin 10, Turkey. - Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea. - Department of Technical Sciences, Western Caspian University, Baku 1001, Azerbaijan.

Abstract

In this paper, we examine and analyze how Egyptian banks compete for profits. Four fractional differential equations make up this model. As a first step toward mitigating its sudden negative impact, we present the appropriate optimal control technique for bank profits during crises (for instance, the Covid-19 crisis). The proposed system is then solved using the spectral collocation method (SCM). This approach efficiently solves the model (with Caputo-sense of fractional derivative) by approximating the solution using Gegenbauer wavelet polynomials (GWPs). In addition, we consider the same model but with another type of derivative known as Caputo-Fabrizio (CF), and use its properties to convert the proposed model into a system of fractional integral equations which are numerically estimated with the help of the Simpson's-\(1/3\) rule as an efficient numerical technique for integration. By contrasting the findings of the given techniques with those of the RK4 method, we can confirm their efficiency and accuracy. The results show that the techniques are efficient tools for simulating the solution to such a problem. The sudden drop in bank profits that occurred in 2020 as a result of the Covid-19 crisis may have been offset by the control mechanism in place. Special attention is given to studying the effect of some parameters in the system to provide a complete numerical simulation of the system which can be used in making decisions related to banking operations.

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Keywords

• Banks' competition model
• optimal control
• fractional derivative
• Gegenbauer wavelet polynomials
• SCM
• numerical integration
• RK4 method

MSC

•  34A12
•  41A30
•  47H10
•  65N20

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