Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials

AIMS Mathematics, Journal Year: 2024, Volume and Issue: 9(9), P. 25457 - 25481

Published: Jan. 1, 2024

<p>This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select basis functions technique be appropriate orthogonal combinations of second kind Chebyshev polynomials (CPs). By implementing approach, FRSE, with its governing conditions, is converted into matrix system whose entries can obtained explicitly. This by expressing derivatives in terms second-kind CPs and after computing some definite integrals based on properties kind. A thorough investigation carried out convergence analysis. demonstrate that approach applicable accurate providing numerical examples.</p>

Language: Английский

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Approximate Petrov–Galerkin Solution for the Time Fractional Diffusion Wave Equation

Mathematical Methods in the Applied Sciences, Journal Year: 2025, Volume and Issue: unknown

Published: April 20, 2025

ABSTRACT This paper discusses the Petrov–Galerkin method's application in solving time fractional diffusion wave equation (TFDWE). The method is based on using two modified sets of shifted fourth‐kind Chebyshev polynomials (FKCPs) as basis functions. explicit forms all spectral matrices were reported. These are essential to transforming TFDWE and its underlying homogeneous conditions into a matrix system. An appropriate algorithm can be used solve this system obtain desired approximate solutions. error analysis was studied depth. Four numerical examples provided that included comparisons with other existing methods literature.

Language: Английский

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A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

Mathematical Methods in the Applied Sciences, Journal Year: 2024, Volume and Issue: unknown

Published: Aug. 6, 2024

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity with two‐term fractional derivatives. The involved derivative operators are given in the form Liouville–Caputo. developed algorithm solving SBVPs consists two main stages. first stage is devoted to iterative quasilinearization method (QLM) conquer (strong) governing SBVPs. Secondly, we employ Genocchi polynomials (GGPs) treat resulting sequence linearized numerically. An upper error estimate series solution norm obtained via rigorous analysis. benefit presented QLM‐GGPs procedure that required number iteration within few steps, polynomial through computer implementations second stage. Three widely applicable test cases investigated observe efficacy as well high‐order accuracy algorithm. comparable robustness validated by doing comparisons results some well‐established available computational methods. It apparently shown provides promising tool solve strongly nonlinear derivatives accurately efficiently.

Language: Английский

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New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems

Fractal and Fractional, Journal Year: 2024, Volume and Issue: 8(4), P. 199 - 199

Published: March 29, 2024

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages class GSJPs that possess crucial property satisfying given By establishing OMs both ODs VOFDs GSJPs, integrate them into SCM, enabling efficient accurate computations. An error analysis convergence study are conducted to validate efficacy proposed algorithm. We demonstrate applicability accuracy our through eight examples. Comparative analyses prior research highlight improved efficiency achieved by approach. The recommended exhibits excellent agreement between approximate precise results in tables graphs, demonstrating its high accuracy. This contributes advancement methods BCs, providing reliable tool solving complex BVPs exceptional

Language: Английский

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Bivariate Jacobi polynomials depending on four parameters and their effect on solutions of time-fractional Burgers’ equations

Journal of Computational Science, Journal Year: 2024, Volume and Issue: unknown, P. 102450 - 102450

Published: Sept. 1, 2024

Language: Английский

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New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations

Boundary Value Problems, Journal Year: 2024, Volume and Issue: 2024(1)

Published: Oct. 30, 2024

Abstract The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). utilizes set of generalized shifted Jacobi polynomials (GSJPs) that adhere to specified and Our approach involves constructing operational matrices (OMs) both ordinary derivatives (ODs) fractional (FDs) GSJPs we employ. We subsequently employ collocation spectral using these OMs. This successfully converts TFCKdVEs into algebraic equations, greatly simplifying task. In order assess efficiency precision proposed numerical technique, utilized it solve two distinct instances.

Language: Английский

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A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations

Algorithms, Journal Year: 2024, Volume and Issue: 18(1), P. 2 - 2

Published: Dec. 26, 2024

This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. uses the operational of derivatives applies collocation method to convert with their underlying conditions into algebraic systems can be numerically treated. The convergence error bounds are examined deeply. Some examples shown demonstrate efficiency applicability proposed algorithms.

Language: Английский

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Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm

Journal of Applied Mathematics and Computing, Journal Year: 2024, Volume and Issue: unknown

Published: Oct. 15, 2024

Language: Английский

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