Brahim Tellab

This study explores and compares two numerical methods for approximating solutions to fractional integro-differential equations with initial conditions. The first method is a modified version of the Adomian decomposition method (ADM), which improves convergence by refining the decomposition series and efficiently computing Adomian polynomials. The second approach leverages artificial neural networks (ANN), utilizing their universal approximation property to construct solutions based on optimized weight and bias parameters. By applying these methods to various test cases, we evaluate their effectiveness in handling complex nonlinear problems. The numerical results highlight the accuracy and reliability of both techniques, offering insights into their practical applications in solving fractional differential equations.

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications.

Numerous classical models of thermostats have been analyzed in the literature, and while existence and uniqueness results have been established, the investigation of other properties, such as stability, can present significant challenges. Therefore, there is a need for fractional mathematical models that are nonlocal problems and allow for the straightforward study of the asymptotic behavior of their solutions. However, solving these problems is often exceedingly complex or even impossible, necessitating the use of numerical methods to obtain approximate solutions. The focus of this research is a generalized boundary value problem that is based on a classical thermostat model. Our initial step involves determining the associated integral equation for the problem at hand. The properties of existence and uniqueness are proved by exploiting the classical contraction principle of Banach together with another kind called ‐ ‐contraction. Afterwards, thanks to the Bernstein polynomials and fractional matrix of derivative, the approximate solutions of our nonlinear fractional thermostat model problem are obtained. This approach utilizes the Legendre operational matrix of fractional derivatives in combination with a truncated Legendre series for the numerical integration of fractional differential equations. By doing so, the problem can be reduced to solving a system of algebraic equations, resulting in significant simplification. This method has been successfully used to solve both linear and nonlinear fractional differential equations. Finally, an example on this type of problem have been studied and the approximate solution are plotted.

In this work, we discussed the existence and uniqueness of the solution for a fractional boundary value problem involving the ‐Riemann–Liouville operators where we used Banach's fixed point theorem; after that, we applied an iterative method and artificial neural networks technique to solve our main problem. Finally, the approximate solution was plotted.

In this work, we investigated a nonlinear two-term boundary value problem involving the $\psi$-Caputo fractional derivative with integral boundary conditions. By a construction of its associated Green function and application of the upper and lower solutions method together with some fixed point theorems due to Banach and Schauder, we established the existence and uniqueness of positive solutions of our considered main problem. At the end some illustrative examples are provided to validate our theoretical results.