Abstract
Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical diffusion theory is no longer suitable. The differential equations with fractional order have recently been proved to be powerful tools for describing anomalous diffusion. Nevertheless, the analysis methods and numerical methods for fractional differential equations are still in the stage of exploration. In the paper, we consider the Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition, respectively. We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme. It is found that the numerical results are in satisfactory agreement with the analytical solution. Through the robustness analysis, it is also found that the diffusion processes on fractals are more sensitive to the spectral dimension than to the anomalous diffusion exponent.
Highlights
In recent years, more and more phenomena of anomalous diffusion were found in physics, biology, and chemistry [ – ]
We have proved that the implicit numerical scheme is unconditionally stable and convergent
The robustness analysis of the parameters, which is concerned with the fractional derivative order, is not considered
Summary
More and more phenomena of anomalous diffusion were found in physics, biology, and chemistry [ – ]. Let the non-homogeneous term f (x, t) and the solution u(x, t) and the initial value ψ(x) be of the following form of a series expansion:. Proof Suppose that unj is an approximate solution of the numerical schemes ). In Figure , the numerical results for the time fractional diffusion equation with different parameters τ , h, β are provided. In Figure , the numerical results for the time fractional diffusion equation with different parameters τ , h, α are given. We assume that uα,β (x, t) is the Figure 2 Numerical results for time fractional diffusion equation at time t = 1.0, β = 0.5, α ∈ [0, 1], from top to bottom: τ = h = 0.1, 0.05, 0.025, 0.0125. Figure shows the numerical solution of the time fractional diffusion equation.
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