Space-Time Fractional Diffusion with Memory Effects: Stability Analysis and Numerical Approximation
Author(s): Rajesh Kumar
Publication #: 2609038
Date of Publication: 10.12.2016
Country: India
Pages: 1-14
Published In: Volume 2 Issue 6 December-2016
DOI: https://doi.org/10.62970/IJIRCT.v2.i6.2609038
Abstract
Fractional diffusion equations provide an interesting mathematical framework for the problem of anomalous transport processes for which diffusion assumptions are insufficient to describe nonlocal spatial interactions and memory-dependent diffusion. In this paper, we have developed a space-time fractional diffusion model to explore the combined effects of memory and spatial nonlocality on diffusion dynamics. The temporal component is represented by a Caputo fractional derivative of order (0<α≤1), the spatial component by a fractional diffusion operator of order (1<β≤2). The model can now be viewed in the same light as before where the temporal fractional order describes the memory effect of the system and the spatial fractional order describes the nonlocal diffusion effect. We study the core analytical properties of the model, namely stability and the dependence of the solution on the fractional parameters. We then develop a numerical approximation to solve the space-time fractional diffusion equation. The numerical approximation is designed to provide an approximate representation of the temporal memory term and the nonlocal spatial operator. We perform numerical experiments in order to understand the effect of the fractional orders, diffusion parameters and discretization parameters on the solution evolution. We evaluate the agreement and accuracy of the numerical approximation with the classical diffusion model by error analysis and comparisons. We find that temporal and spatial fractionalities play an important role on the diffusion rate, profile and long-term behavior. In particular, the fractional model exhibits slower relaxation and persistent memory effects in time and nonlocal spatial spreading, and the classical diffusion behavior is recovered in the appropriate integer order limit. The present framework offers a mathematically consistent and computationally appealing approach to understand anomalous diffusion processes with both temporal memory and spatial nonlocality.
Keywords: Space-time fractional diffusion; fractional calculus; temporal memory; spatial nonlocality; Caputo fractional derivative; fractional Laplacian; stability analysis; numerical approximation; anomalous diffusion; convergence analysis.
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