Numerical methods for solving a two-dimensional variable-order modified diffusion equation
Introduction
Fractional diffusion equations are widely used to describe anomalous diffusion processes and have attracted attention of numerical researchers [1], [2], [6], [7], [8], [9], [11], [12], [13], [16], [24], [25], [26], [27], [32]. By the inclusion of a secondary fractional time derivative acting on a diffusion operator, a model for describing processes that become less anomalous as time progresses recently has been presented [1], [31], [32]for this modified fractional diffusion equation, Langlands et al. proposed the solution with the form of an infinite series of Fox functions on an infinite domain [17]; Liu et al. discussed the numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term [20]; Liu et al. also researched the finite element approximation for a modified anomalous subdiffusion equation [21].
Variable-order calculus operator in recent years has been developed to more accurately characterize the evolution of a system [5], [10], [14], [15], [18], [22], [23], [28], [29], [30], [33]. Sun et al. discussed variable-order fractional differential operators in anomalous diffusion modeling, also analyzed mean square displacement behaviors of anomalous diffusions with variable and random orders, and researched random-order fractional differential equation models [34], [35], [36]. However, the work for involving numerical method and numerical analysis of variable-order fractional differential equations so far is very few. Lin et al. discussed stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation [19]; Zhuang et al. presented the numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term [38]; Chen et al. researched numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation [3]. To our knowledge, research for involving numerical methods and numerical analysis of variable-order differential equations is still in an embryonic state, we need to constantly strive to tap.
In this paper, we will study numerical methods for solving the initial and boundary value problem of a two-dimensional variable-order modified diffusion equationwhere and B are positive constants, and is the variable-order Riemann–Liouville fractional partial derivative of order for the defined by Lin et al. [19], Zhuang et al. [38]
For a two-dimensional variable-order anomalous subdiffusion equation [4]where the constants and , Chen et al. developed the implicit and explicit numerical methods, by Fourier method and Lemma 2.2 in [4], their stability, convergence and solvability are investigated. It is not difficult to see that Eq. (1) is more complicated than Eq. (5), and so far, we are unaware of any other published work on Eq. (1). This work is our effort to remedy the situation.
Obviously, by the following transformation:then the initial and boundary value problem (1), (2), (3), (4) becomes the following initial and boundary value problemwhere
Section snippets
A numerical method
In this paper, we letrespectively, where and are the spatial and temporal steps, respectively. Moreover, we also introduce notations
In the paper, we assume . At the grid point , Eq. (6) becomeswhere .
From [37]
Convergence of the numerical method
Similar to [4], we have the following lemma: Lemma 2 If , for , the coefficients satisfy: ; ; for ,
where .
We now discuss convergence of the numerical method (18), (19), (20), (21). Subtracting (18) from (16), we obtain the following error equation
Stability of the numerical method
In this section, We discuss stability of the numerical method (18), (19), (20), (21), consider the following difference equationwhere , whereas is the approximation for .
For , we define the following grid function:where and as defined in
Solvability of the numerical method
Obviously that the corresponding homogeneous linear algebraic equations for the numerical method (18), (19), (20), (21) isClear, similar to the proof of Theorem 2, we can also certificate that the solution of the
The numerical method for improving temporal accuracy
Firstly, we have the following lemma [3]: Lemma 3 If has third order continuous derivative, thenwhere is the first-order backward difference.
By taking in (6) giveswhereFrom and Lemma 3 gives
Numerical example
In this section, in order to demonstrate the theoretical analysis results, we apply the numerical method (18), (19), (20), (21) and the numerical method for improving temporal accuracy (53), (54), (55), (56) to solve the following initial-boundary value problem of the two-dimensional variable-order modified diffusion equation
Conclusion
In this paper, a numerical method for solving a two-dimensional variable-order modified diffusion equation have been developed. The convergence, stability and solvability of the numerical method have been discussed by Fourier analysis. The numerical method for improving temporal accuracy has also been developed. Moreover, the theoretical analysis results have been demonstrated by the numerical example. Finally, we pointed out that: the numerical method and numerical analysis skills in this
Acknowledgements
This research was supported by the Natural Science Foundation of Fujian province Grant (2009J01014).
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