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.

This study used the fractional B-spline collocation technique to obtain the numerical solution of fractal-fractional differential equations. The technique was considered to solve the fractal-fractional differential equations (FFDEs) with (). In this suggested technique, the B-spline of fractional order was utilised in the collocation technique. The scheme was easily attained, efficient, and relatively precise with reduced computational work numerical findings. Via the proposed technique, FFDEs can be reduced for solving a system of linear algebraic equations using an appropriate numerical approach. The verified numerical illustrative experiments were presented will show the effectiveness of the technique proposed in this study in solving FFDEs in three cases of nonlocal integral and differential operators namely power law kernel, when the kernels are exponential and the generalization of Mittag-Leffler kernel. The approximate solution is very good and accurate to the exact solution.

In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in  and fractal parameters, (ii) different values in fractional parameter while fractal and  parameters are fixed, and (iii) different values in fractal parameter controlling fractional and  parameters.

The study of integral equations is one of the most important topics that researchers are interested in, it arises in many scientific fields for instance engineering, and mathematical and scientific analysis. The first who mentioned the term integral equations is Du Bois-Reymond (1888). As a result, a lot of interest appeared from researchers, and the most important of these researchers are Laplace, Fourier, Poission, Liouville, and Able. Upadhyay et al., (2015) provided some special types of integral equations. The quadratic integral equation is a special form of integral equations. The initial study appeared by Chandrasekhar (1947). More appearance of Quadratic integral equation was in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport, and in the traffic theory, see Argyros (1985), Banaś et al. (2007), El-Sayed et al. (2008).

 The aim of the present paper is to develop a generalized fractional kinetic equation involving generalized multi-variable Mittag-Leffler function. Using the Laplace transform, the solutions of the fractional kinetic equation are established in terms on general Mittag-Leffler function. The results obtained here are general in nature to yield a large number known and (presumably) new results as their special cases. 

In this article, an implementation of an efficient numerical method for solving the system of coupled non-linear fractional (Caputo sense) dynamical model of marriage (FDMM) is introduced. The proposed system describes the dynamics of love affair between couples. The method is based on the spectral collocation method using Legendre polynomials. The proposed method reduces FDMM to a system of algebraic equations, which solved using Newton iteration method. Special attention is given to study the convergence analysis and deduce an error upper bound of the resulting approximate solution. Numerical simulation is given to show the validity and the accuracy of the proposed method. 

In this paper we establish the existence of integrable solution of boundary value problems of (mixed type) Fredholm - volterra functional integro-differential equations  with nonlocal boundary conditions.