In the past few years, the area of mathematical study has made considerable advances, owing largely to the introduction of artificial intelligence (AI) tools. Among these, artificial neural networks (ANNs) have played an important role in modernizing several mathematical techniques and problem-solving approaches. ANNs have recently become popular as a powerful mathematical research tool, providing an effective alternative to established approaches for solving fractal-fractional differential equations (FFDEs). This paper describes the use of a feed-forward ANN with a hidden layer to address systems resulting from the fractal-fractional Bagley-Torvik differential equation (FFBTDE). In addition, a power series (PS) technique is introduced to increase efficiency. The paper looks at for solving FFBTDE with variable and constant coefficients. The numerical findings show that the suggested strategy not only produces results that closely match exact and reference solutions, but also outperforms existing methods in terms of accuracy.

A highly efficient and accurate numerical method for systems of fractional differential equations (FDEs) with rational order is presented in this paper. Rational power functions and rational Taylor series projection are utilized to obtain approximate solutions. Rational semi-smooth spaces are introduced, and the regularity of solutions in these spaces is established. A series of theoretical results, such as the existence and uniqueness of solutions, properties of the rational Taylor series and its remainder term, and an operational matrix approach, are derived. It is proven that the numerical solution is exact when the exact solution is a rational power series, and the approximate solution is shown to be the rational Taylor series projection of the exact solution. The convergence of the method is analyzed. The efficiency of the proposed method is demonstrated through numerical experiments, which show significant improvements in computational time compared to existing methods.

In this study, we numerically investigate two significant medical models, Ebola Viral Disease (EVD) and Blood Ethanol Concentration (BEC) models-both formulated using Caputo-fractional derivatives. We develop and apply the Modified Fractional Euler Method (MFEM) for their solution, with a specific focus on error analysis. Comparative studies with the classical Runge-Kutta fourth-order method (RK4M) demonstrate that MFEM provides a computationally efficient and accurate alternative for solving such systems. The major features of the given procedure are its ease of application to this type of problem and other systems in various fields, in addition to the absence of numerical errors accumulating. Finally, we can control the increase in the convergence rate and the stability of the simulation process. The convergence examination and error estimation for the suggested scheme are also included. The importance of this study also lies in its contribution to our understanding of the dynamics of these two models in their fractional form. In addition, those numerical investigations demonstrate how control parameters affect specific components within these models.

 We give the approximate solution of the Riccati/Logistic differential equations (RDE/LDE). The suggested approach depends on the homotopy perturbation method developed with the Chebyshev series (CHPM). A study of the convergence analysis of CHPM is presented. The residual error function is calculated and used as a basic criterion in evaluating the accuracy and efficiency of the given numerical technique. We use the exact solution and the Runge-Kutta method of fourth order for comparison with the results of the method used. Through these results, we can confirm that the applied method is an easy and effective tool for the numerical simulation of such models. Illustrative models are given to confirm the validity and usefulness of the proposed procedure. 

In this research, we investigated the Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral constraints. We presented novel sufficient conditions for the uniqueness of the solution. Moreover, we analyzed the continuous dependence of the solution on some functions and parameters. Additionally, we proved the Hyers-Ulam stability of the problem. To demonstrate the applicability of our results, we included several examples. The present study was located in the space L1[0, T]. The techniques of Schauder’s fixed point theorem and Kolmogorov’s compactness criterion were the primary tools utilized in this work. These contributions offer a comprehensive framework for understanding the qualitative behavior of the fractional-order pantograph equation.

This paper presents theoretical proof of the existence of a unique solution to a constrained problem of the Riemann-Liouville fractional differential equation with time delay functions by utilizing the Schauder fixed point theorem. Moreover, we analyzed the continuous dependence of the solution on the initial conditions and other parameters. Further, we investigate the Hyers-Ulam stability of the problem. We introduce some examples and special cases to illustrate our results.

Artificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal-fractional differential equations (FFDEs) of high-order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs-HOL, the suggested technique is simple, highly efficient, and resilient.

The second stage, in which the star uses nuclear fuel in its interior, represents the helium burning phase. At that stage, three elements are synthesised: carbon, oxygen, and neon. This paper aims to establish a numerical solution for the helium burning system (HBN) fractal-fractional differential equations (FFDEs). The extended operative matrix method (OM) is employed in the solution of a system of differential equations. The product abundances of the four elements (helium, carbon, oxygen and neon) were obtained in a form of divergent series. These divergent series are then accelerated using Euler-Abell transformation (EUAT) and Pade approximation (EUAT-PA) to obtain more reliable results. Nine fractal-fractional (FF) gas models are calculated, and fractal-fractional parameters’ influence on product abundances is discussed. The findings show that modeling nuclear burning networks with the OM fractal-fractional derivative produces excellent results, establishing it as an accurate, resilient, and trustworthy approach, and the fractional HB models can have a considerable impact on stellar model calculations.

This study will introduce a new differentiation operator, the Hilfer fractional-fractal derivative (H-FFD). The new proposed derivative aims to attract more non-local problems that show with the same time fractal behaviors. For numerical settlement of initial value problems, we use the shifted Legendre operational matrix. The main advantage of this method is that it reduces both linear and non-linear problems alike in solving the problem into a system of linear and non-linear algebraic equations. In addition, the numerical approximation of this new operator also offers some applications to systems of linear and non-linear problems.

This paper attempts to create an artificial neural networks (ANNs) technique for solving well-known fractal-fractional differential equations (FFDEs). FFDEs have the advantage of being able to help explain a variety of real-world physical problems. The technique implemented in this paper converts the original differential equation into a minimization problem using a suggested truncated power series of the solution function. Next, answer to the problem is obtained via computing the parameters with highly precise neural network model. We can get a good approximate solution of FFDEs by combining the initial conditions with the ANNs performance. Examples are provided to portray the efficiency and applicability of this method. Comparison with similar existing approaches are also conducted to demonstrate the accuracy of the proposed approach.