Mohamed Adel

In this study, we present an approximate solution for a mathematical model of Lumpy Skin Disease across three time intervals, formulated within the framework of the Caputo-Katugampola fractional derivative (CKFD). The given technique is based on the [Formula: see text]-Khalouta decomposition method ([Formula: see text]-KDM), which combines the [Formula: see text]-Khalouta transformation with a decomposition technique. A comprehensive analysis is conducted, including the convergence and the error estimation of the proposed scheme. The residual error function is estimated as a key criterion for evaluating both the accuracy and efficiency of the numerical method. The approximate solutions obtained confirm that the proposed approach is straightforward, efficient, and reliable for numerical simulation of such problems. Several examples are considered to further confirm the accuracy and utility of the method.

We investigate weak solutions of a fractional Langevin inclusion subject to a nonlocal boundary condition and involving the generalized Katugampola–Caputo derivative. The problem is posed in a real Banach space and the multivalued nonlinearity is treated through Pettis-integrable selections. We first derive a precise integral representation for the associated linear boundary problem. The derivation identifies an explicit nonresonance denominator and yields a contraction condition that guarantees uniqueness of the auxiliary problem. We then construct the induced multivalued integral operator and combine the De Blasi measure of weak noncompactness with a Mönch-type fixed-point principle to obtain sufficient conditions for the existence of weak solutions. The estimates retain the contributions of both the dissipative term and the nonlocal boundary term. To illustrate the applicability of the theoretical results, three examples are provided in which all assumptions are verified, including the summability requirements in [Formula: see text].

In this paper, for some second-order singular functional differential equations (SOSFDEs), a tau-type spectral technique is built using non-origin-symmetric polynomials called shifted convolved Fibonacci polynomials as the approximation basis. Derivative formulae, integral identities required for the tau projection, power-form and inverse representations, and several operational relations are derived for this basis. The singular term and the shifted arguments can be handled directly within an algebraic framework thanks to these formulas, eliminating the need to introduce a grid on the computing interval. For the unknown coefficients, the resultant approximation yields a finite system. The uniform convergence of the truncated shifted convolved Fibonacci expansion is established, together with error estimates and a stability result for the proposed approximation. To demonstrate the procedure’s accuracy, three cases are utilized as benchmarks. As the truncation order increases, the reported errors gradually diminish, and the computed solutions correspond closely with the exact answers. Further evidence of the reliability and efficiency of the suggested shifted convolved Fibonacci tau procedure is provided by comparisons with existing numerical data.

A spectral Galerkin algorithm is developed for the numerical treatment of modified anomalous time-fractional sub-diffusion equations (MATFSDEs). The algorithm is constructed using Fibonacci coefficient polynomials, from which two space–time trial families are formed. These families are selected so that the homogeneous initial and boundary conditions are automatically incorporated after a suitable transformation of the original problem. The Galerkin formulation then reduces the model to a finite algebraic matrix system whose entries can be explicitly evaluated. To support the construction, inversion, moment, and linearization identities for the selected polynomials are obtained and then used to express the required matrices in closed form. The convergence of the expansion is investigated, and explicit error bounds are derived. Numerical tests are reported to demonstrate the accuracy, applicability, and competitiveness of the proposed scheme in comparison with some existing methods in the literature.

We give an accurate method, which is the spectral collocation method (SCM), to solve a specific category of the extremely significant linear and nonlinear fractional (Caputo sense) models of the RLC electrical circuit. The combination of the SCM and the use of Appell–Changhee polynomials (ACPs) results in a multitude of major benefits. The fractional-order derivative can be approximated with the assistance of the ACPs. Using the procedure that is offered, the model that is provided is transformed into a set of algebraic equations. Additionally, we compare the approximate solutions generated for the proposed models using the offered technique with the exact solutions obtained from the provided simulation instances. To determine the efficiency and accuracy of the provided methodology, we compute the residual error function using various values for the order of approximation.

This paper is devoted to introducing an accurate method to solve numerically the fractional Brusselator system (FBS). The proposed technique relies on combining the good features of both the Laplace transform method (LTM) and the Adams fourth-order predictor-corrector method (A4PCM). This technique involves using the Laplace transform and the linear Lagrange interpolation to derive an approximate formula of the fractional Caputo derivatives and use it to convert the FBS into a system of first-order ordinary differential equations (ODEs) and then solving these equations numerically by applying the A4PCM. The convergence and error estimations for the suggested scheme are also included by presenting the stability analysis as well as computing the order of convergence. Numerical examples illustrate the good impact of the strategies and rightness of the theoretical framework of the suggested method with high accuracy. One of the advantages of this proposed technique is its ability to provide a highly accurate numerical solution to the model under study over time intervals larger than one (as is the case with most analytical methods). Furthermore, using the proposed fractional derivative approximation eliminates the need to maintain the complete time history of the solution, resulting in a simpler and easier-to-use time-domain model.