Elsevier

Applied Mathematics and Computation

Volume 225, 1 December 2013, Pages 62-78
Applied Mathematics and Computation

Numerical methods for solving a two-dimensional variable-order modified diffusion equation

https://doi.org/10.1016/j.amc.2013.08.064Get rights and content

Abstract

In this paper, a two-dimensional variable-order modified diffusion equation is considered. We develop the numerical methods to solve the equation. By Fourier analysis, we discuss the convergence, stability and solvability of the numerical method. The numerical method for improving temporal accuracy is also developed. Moreover, our theoretical analysis results are demonstrated by the numerical example.

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]p(x,t)t=A1-αt1-α+B1-βt1-β2p(x,t)x2for 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 equationu(x,y,t)t=A1-α(x,y,t)t1-α(x,y,t)+B1-β(x,y,t)t1-β(x,y,t)u+f(x,y,t),0<x,y<X,0<tT,u(x,y,0)=ϕ(x,y),0x,yX,u(x,0,t)=φ1(x,t),u(x,X,t)=φ2(x,t),0xX,0<tT,u(0,y,t)=ψ1(y,t),u(X,y,t)=ψ2(y,t),0yX,0<tT,where 0<α(x,y,t)<β(x,y,t)1,A and B are positive constants, and γ(x,y,t)tγ(x,y,t)q(x,y,t) is the variable-order Riemann–Liouville fractional partial derivative of order γ(x,y,t) for the q(x,y,t) defined by Lin et al. [19], Zhuang et al. [38]γ(x,y,t)tγ(x,y,t)q(x,y,t)=1Γ(m-γ(x,y,t))mξm0ξq(x,y,ν)(ξ-ν)γ(x,y,t)+1-mdνξ=t,m-1<γ(x,y,t)<m;mtmq(x,y,t),γ(x,y,t)=mN.

For a two-dimensional variable-order anomalous subdiffusion equation [4]u(x,y,t)t=0Dt1-γ(x,y,t)κ12u(x,y,t)x2+κ22u(x,y,t)y2+f(x,y,t),where the constants κ1,κ2>0 and 0<γ(x,y,t)<1, 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:p(x,y,t)=u(x,y,t)-ϕ(x,y),then the initial and boundary value problem (1), (2), (3), (4) becomes the following initial and boundary value problemp(x,y,t)t=A1-α(x,y,t)t1-α(x,y,t)+B1-β(x,y,t)t1-β(x,y,t)p+f(x,y,t),0<x,y<X,0<tT,p(x,y,0)=0,0x,yX,p(x,0,t)=φ1(x,t),p(x,X,t)=φ2(x,t),0xX,0<tT,p(0,y,t)=ψ1(y,t),p(X,y,t)=ψ2(y,t),0yX,0<tT,whereφ1(x,t)=φ1(x,t)-ϕ(x,0),φ2(x,t)=φ2(x,t)-ϕ(x,X),ψ1(y,t)=ψ1(y,t)-ϕ(0,y),ψ2(y,t)=ψ2(y,t)-ϕ(X,y),f(x,y,t)=A1-α(x,y,t)t1-α(x,y,t)+B1-β(x,y,t)t1-β(x,y,t)ϕ+f(x,y,t)=A1t1-α(x,y,t)Γ(α(x,y,t))+B1t1-β(x,y,t)Γ(β(x,y,t))ϕ+f(x,y,t).

Section snippets

A numerical method

In this paper, we letxi=ihx,i=0,1,,Mx;yj=jhy,j=0,1,,My,tk=kτ,k=0,1,,N,respectively, where hx=X/Mx,hy=X/My and τ=T/N are the spatial and temporal steps, respectively. Moreover, we also introduce notationsΩ=(x,y,t)|0x,yX,0tT,P(Ω)=p(x,y,t)|pxxxx,pyyyy,pxxt,pyyt,pttC(Ω).

In the paper, we assume p(x,y,t)P(Ω). At the grid point (xi,yj,tk), Eq. (6) becomesp(xi,yj,tk)t=A1-αi,jkt1-αi,jk+B1-βi,jkt1-βi,jkp(xi,yj,tk)+fi,jk,where αi,jkα(xi,yj,tk),βi,jkβ(xi,yj,tk),fi,jkf(xi,yj,tk).

From [37]

Convergence of the numerical method

Similar to [4], we have the following lemma:

Lemma 2

If 0<γ(x,y,t)1, for i=1,2,,Mx,j=1,2,,My,k=1,2,,N,l=0,1,, the coefficients λi,jk,l satisfy:

  • (1)

    λi,jk,0=1;λi,jk,1=γi,jk-10;λi,jk,l0,l=2,3,;

  • (2)

    l=0λi,jk,l=0;

  • (3)

    for n=1,2,,-l=1nλi,jk,l1,


where λi,jk,l=(-1)l1-γi,jkl,γi,jk=γ(xi,yj,tk).

We now discuss convergence of the numerical method (18), (19), (20), (21). Subtracting (18) from (16), we obtain the following error equationEi,jk=Ei,jk-1+μi,jkl=0kλi,jk,lδx2Ei,jk-l+μ¯i,jkl=0kλi,jk,lδy2Ei,jk-l+νi,jkl=0k

Stability of the numerical method

In this section, We discuss stability of the numerical method (18), (19), (20), (21), consider the following difference equationρi,jk=ρi,jk-1+μi,jkl=0kλi,jk,lδx2ρi,jk-l+μ¯i,jkl=0kλi,jk,lδy2ρi,jk-l+νi,jkl=0kλ¯i,jk,lδx2ρi,jk-l+ν¯i,jkl=0kλ¯i,jk,lδy2ρi,jk-l,i=1,2,,Mx-1,j=1,2,,My-1,k=1,2,,N,where ρi,jk=pi,jk-pi,jk, whereas pi,jk is the approximation for pi,jk.

For k=0,1,,N, we define the following grid function:ρk(x,y)=ρi,jk,when(x,y)Ωi,j;0,when(x,y)Ω,where Ωi,j and Ω as defined in

Solvability of the numerical method

Obviously that the corresponding homogeneous linear algebraic equations for the numerical method (18), (19), (20), (21) ispi,jk=pi,jk-1+μi,jkl=0kλi,jk,lδx2pi,jk-l+μ¯i,jkl=0kλi,jk,lδy2pi,jk-l+νi,jkl=0kλ¯i,jk,lδx2pi,jk-l+ν¯i,jkl=0kλ¯i,jk,lδy2pi,jk-l,i=1,2,,Mx-1,j=1,2,,My-1,k=1,2,,N,pi,j0=0,i=0,1,,Mx,j=0,1,,My,pi,0k=ui,Myk=0,i=1,2,,Mx-1,k=1,2,,N,p0,jk=pMx,jk=0,j=1,2,,My-1,k=1,2,,N.Clear, 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 p(t) has third order continuous derivative, then1-12tp(tk)=tp(tk)τ+O(τ2),where tp(tk)=p(tk)-p(tk-1) is the first-order backward difference.

By taking x=xi,y=yj,t=tk in (6) givesp(xi,yj,tk)t=Aαi,jkp(xi,yj,tk)+Bβi,jkp(xi,yj,tk)+f(xi,yj,tk),whereαi,jk=α(xi,yj,tk),βi,jk=β(xi,yj,tk),γp(x,y,t)=ξ1Γ(γ)0ξp(x,y,ν)dν(ξ-ν)1-γξ=t.From p(x,y,t)P(Ω) and Lemma 3 givestp(xi,yj,tk)τ+O(τ2)=Atαi,jkp(xi,yj,tk)τ+Btβi,jkp(xi,yj,tk)τ+O(τ2)+1-12tf(xi,yj,tk),

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 equationp(x,y,t)t=1-α(x,y,t)t1-α(x,y,t)+1-β(x,y,t)t1-β(x,y,t)p+f(x,y,t),0<x,y<1,0<t1,p(x,y,0)=0,0x,y1,p(x,0,t)=ext2,p(x,1,t)=e1+xt2,0x1,0<t1,p(0,y,t)=eyt2,p(1,y,t)=e1+y

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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