Document Type : Research Article
Authors
1 Department of Mathematics, University of Mazandaran, Babolsar, Iran.
2 Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran.
10.30473/coam.2025.72692.1267
Abstract
The advection-dispersion, variable-order differential equations have a vast application in fluid physics and energy systems. In this study, we propose a Ritz-approximation method using shifted Legendre polynomials to construct approximate numerical solutions for these equations. The proposed method discretizes the original problem, converting it into a system of nonlinear algebraic equations that can be solved numerically at selected points. We discuss the error characteristics of the proposed method. For validation, the presented examples are compared with exact solutions and with prior results. The results indicate that the proposed method is highly effective.
Highlights
- Introduces a Ritz-approximation numerical method using Shifted Legendre Polynomials (SLPs) to solve advection-dispersion equations with variable-order fractional operators in a mobile-immobile framework (VOF).
- Demonstrates discretization of the governing problem into a small, tractable system of nonlinear algebraic equations, enabling efficient numerical solution at selected collocation/basis points.
- Provides a formal error analysis or discussion of the method’s error characteristics, establishing accuracy considerations for the proposed approach.
- Validates the method through several numerical examples, comparing results against exact solutions and prior methods to demonstrate accuracy and effectiveness.
- Establishes a flexible framework extendable to other variable-order fractional operators and/or alternative basis function sets beyond SLPs.
Keywords
- Caputo fractional derivative
- Mobile-immobile advection-dispersion
- Polynomial basis functions
- Time variable fractional order
- Satisfiers function
Main Subjects
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