A Local Radial Basis Function Method for Numerical Approximation of Multidimensional Multi-Term Time-Fractional Mixed Wave-Diffusion and Subdiffusion Equation Arising in Fluid Mechanics

This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation featu...

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Bibliographic Details
Published in:Fractal and fractional Vol. 8; no. 11; p. 639
Main Authors: Kamran, Gul, Ujala, Khan, Zareen A., Haque, Salma, Mlaiki, Nabil
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2024
Subjects:
Approximation
Boundary conditions
Caputo fractional derivative
Computing time
Fluid mechanics
Ill-conditioned problems (mathematics)
Laplace transformation
Laplace transforms
Mathematical analysis
Meshless methods
Methods
mixed subdiffusion and wave-diffusion equation
MQ-RBF
Numerical models
Parameter sensitivity
Partial differential equations
Radial basis function
Stehfest’s algorithm
Talbot’s algorithm
Time domain analysis
ISSN:2504-3110, 2504-3110
Online Access:Get full text
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Summary:This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously incorporating both wave-diffusion and subdiffusion terms. Our proposed method follows three basic steps: (i) The given equation is transformed into a time-independent form using the Laplace transform (LT); (ii) the LMM is then used to solve the transformed equation in the LT domain; (iii) finally, the time domain solution is obtained by inverting the LT. We use the improved Talbot method and the Stehfest method to invert the LT. The LMM is used to circumvent the shape parameter sensitivity and ill-conditioning of interpolation matrices that commonly arise in global mesh-free methods. Traditional time-stepping methods achieve accuracy only with very small time steps, significantly increasing the computational time. To overcome these shortcomings, the LT is used to provide a more powerful alternative by removing the need for fine temporal discretization. Additionally, the Ulam–Hyers stability of the considered model is analyzed. Four numerical examples are presented to illustrate the effectiveness and practical applicability of the method.
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ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract8110639