An approximation scheme for the time fractional convection–diffusion equation☆
Introduction
Many natural phenomena are modeled by partial differential equations, even medical problems [1], [2] can also be modeled by partial differential equations. As a class of partial differential equations, integer order convection–diffusion equations are also have a wide range of applications. The growing number of applications of fractional derivatives in various fields [3], [4], [5], [6] such that fractional convection–diffusion equations have received an increasing attention in recent years and have been used to model a wide range of problems in sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity.
One of the key issues with fractional convection–diffusion equations is the design of efficient numerical schemes and computational efficiency is an important parameter for the numerical schemes. A variety of numerical methods for the fractional partial differential equations have been proposed, such as finite element methods [7], [8], [9], [10], [11], [12], finite difference methods [13], [14], [15], [16], [17], spectral methods [18], [19], [20], meshless methods [21], [22]. In [13], fractional advection–diffusion equation with space–time fractional derivative has been considered and the stability and convergence of the mentioned method has also been discussed. Zhang et al. [14] have proposed a new numerical method for the time mobile–immobile advection–dispersion problem with variable fractional derivative. Zhuang et al. [15] have developed some numerical methods for the nonlinear fractional advection–diffusion equation with variable-order fractional derivative. The authors of [23] have proposed an implicit meshless method based on the radial basis functions for the numerical simulation of time fractional diffusion equation.
In this paper, we aim to propose a time discrete form combined with finite difference methods for the time fractional convection–diffusion equation. We consider the time fractional convection–diffusion equation of the form: where Ω denotes a bounded domain in Rn, 0 < α < 1, f and u0(x) ≡ u0 are given smooth functions, and . The fractional derivative is Caputo fractional derivative defined by
Section snippets
Time discrete scheme
We will introduce a finite difference approximation to discrete the time fractional derivative. Let tm ≔ mτ, where τ ≔ T/M is the time step, be mesh points, where M is a positive integer.
The time fractional derivative at is estimated by Using the analysis in [18] the truncation error has the following form
Stability analysis and error estimates
In this section, we discuss the stability and convergence of Eq. (6) using the following lemma. Firstly, we define the functional spaces endowed with standard norms and inner products. The space L2(Ω) is equipped with the usual L2-scalar product which induces the L2-norm for u, v ∈ L2(Ω).
Now by simplifying Eq. (6), we have
Lemma 3.1 [24] Let Ω be a bounded domain in Rn with piecewise smooth boundary ∂Ω, if U
Numerical results
In this section, we carry out two numerical experiments and present some results to confirm our theoretical statements. In our examples, we use the method of the order of time equal to space order [13], the main purpose is to check the convergence behavior of the numerical solution with respect to the time step τ used in the calculation.
Conclusion
In this paper, we proposed a discrete scheme to solve the fractional convection–diffusion equation. The stability analysis and error estimates for the discrete scheme are discussed. Numerical examples for some case are given to illustrate the accuracy and capability of the method. Constructing more efficient algorithms to reduce the storage requirement is also our goal in future works.
Acknowledgments
Authors thank the anonymous referees and editors very much for their valuable suggestions and comments, which greatly helped us to improve this work. This work is supported by the NSFC (No.11461072), the Youth Science and Technology Education Project of Xinjiang (No.QN2016JQ0367), the Science and Technology Innovation Project for Master Graduate Student of Xinjiang Normal University (No. XSY201602011).
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This work is supported by the NSFC (No. 11461072), the Youth Science and Technology Education Project of Xinjiang (No. QN2016JQ0367) and the Science and Technology Innovation Project for Master Graduate Student of Xinjiang Normal University (No. XSY201602011).