Multi-term Fractional Diffusion Equations Research Articles

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Equivalence of definitions of solutions for some class of fractional diffusion equations

AbstractWe study the unique existence of weak solutions for initial boundary value problems associated with different class of fractional diffusion equations including variable order, distributed order, and multiterm fractional diffusion equations. So far, different definitions of weak solutions have been considered for these class of problems. This includes definition of solutions in a variational sense and definition of solutions from properties of their Laplace transform in time. The goal of this article is to unify these two approaches by showing the equivalence of these two definitions. Such a property allows also to show that the weak solutions under consideration combine the advantages of these two classes of solutions, which include representation of solutions by a Duhamel type of formula, suitable properties of Laplace transform of solutions, resolution of the equation in the sense of distributions, and explicit link with the initial condition.

A novel finite volume method for the nonlinear two-sided space distributed-order diffusion equation with variable coefficients

Fractional differential equations have been proved to be powerful tools for modelling anomalous diffusion in many fields of science and engineering. However, when comes to the anomalous diffusion characterized by two or more scaling exponents in the mean squared displacement (MSD), or even by logarithmic time dependency of the MSD, distributed-order diffusion equations are shown to be more useful than the general single or multi-term fractional diffusion equations. In this paper, we construct a novel finite volume method for solving a nonlinear two-sided space distributed-order diffusion equation with variable coefficients. Firstly, we apply the modified Gaussian integral formula to approximate the distributed-order integral. Secondly, we propose the finite volume method based on a piecewise-linear polynomial to discretize the problem and establish the Crank–Nicolson scheme. Furthermore, we prove that the proposed method is stable and convergent with second order accuracy in both space and time. Finally, some numerical examples are given to show the efficiency of the proposed method.

Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

In this paper, we study Galerkin finite element methods for a class of higher dimensional multi-term fractional diffusion equations. The finite difference approximation of Caputo derivative on non-uniform meshes is used in temporal direction and Galerkin finite element method is used in spatial direction. We prove that semi-discrete and fully discrete finite element schemes are unconditionally stable. Meanwhile, -norm convergence properties of the two schemes are proved rigorously. To confirm our theoretical analysis, we give some numerical examples in both two-dimensional (2D) and three-dimensional (3D) spaces. Finally, a moving local refinement technique in temporal direction is used to improve the accuracy of numerical solution.

Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives

In this paper, the initial-boundary-value problems for linear and non-linear multi-term fractional diffusion equations with the Riemann–Liouville time-fractional derivatives are considered. To guarantee the uniqueness of solutions, we employ the weak and the strong maximum principles for the equations under consideration that are formulated and proved in this paper for the first time. An essential element of our proof of the maximum principles is an estimation for the value of the Riemann–Liouville fractional derivative of a function at its maximum point that is established in this paper for a wider space of functions compared to those used in our previous publications. In the linear case, the solutions to the problems under consideration are constructed in form of the Fourier series with respect to the eigenfunctions of the corresponding eigenvalue problems.

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Abstract This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1. Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.

A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues

In this article, we propose a new mathematical model, namely a multi-term fractional diffusion equation, for oxygen delivery through a capillary to tissues. Fractional calculus is applied to describe the phenomenon of subdiffusion of oxygen in both transverse and longitudinal directions. A new iterative method (NIM) and a modified Adomian decomposition method (MDM) are used to solve the multi-term fractional diffusion equation for different conditions. The results thus obtained are compared and presented graphically. It is observed that the order of the diffusion equation affects the delivery of oxygen significantly.