International Journal of Differential Equations

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Fractional Differential Equations (2012)

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Volume 2012 |Article ID 495202 | https://doi.org/10.1155/2012/495202

Fenghui Huang, "A Time-Space Collocation Spectral Approximation for a Class of Time Fractional Differential Equations", International Journal of Differential Equations, vol. 2012, Article ID 495202, 19 pages, 2012. https://doi.org/10.1155/2012/495202

A Time-Space Collocation Spectral Approximation for a Class of Time Fractional Differential Equations

Academic Editor: Fawang Liu
Received22 May 2012
Revised26 Jul 2012
Accepted29 Jul 2012
Published09 Sep 2012

Abstract

A numerical scheme is presented for a class of time fractional differential equations with Dirichlet's and Neumann's boundary conditions. The model solution is discretized in time and space with a spectral expansion of Lagrange interpolation polynomial. Numerical results demonstrate the spectral accuracy and efficiency of the collocation spectral method. The technique not only is easy to implement but also can be easily applied to multidimensional problems.

1. Introduction

Fractional differential equations have attracted in recent years considerable interest because of their ability to model complex phenomena. For example, fractional derivatives have been used successfully to model frequency-dependent damping behavior of many viscoelastic materials. They are also used in modeling of many chemical processes, mathematical biology, and many other problems in engineering. Related equations of importance are fractional diffusion equations, the fractional advection-diffusion equation for anomalous diffusion with sources and sinks, and the fractional Fokker-Planck equation for anomalous diffusion in an external field, and so forth.

In this paper, we consider the following time fractional differential equation (TFDE) 𝐷𝜇𝑡𝑢(𝑥,𝑡)=−𝜆2𝑢(𝑥,𝑡)−𝜈𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑥+𝐷2𝑢(𝑥,𝑡)𝜕𝑥2+𝑓(𝑥,𝑡)=ℒ𝑢(𝑥,𝑡)+𝑓(𝑥,𝑡),ğ‘Ž<𝑥<𝑏,𝑡>0,(1.1) where ℒ𝑢(𝑥,𝑡)=−𝜆2𝑢(𝑥,𝑡)−(𝜈(𝜕𝑢(𝑥,𝑡)/𝜕𝑥))+(𝐷(𝜕2𝑢(𝑥,𝑡)/𝜕𝑥2)) is a linear differential operator. 𝜆,𝜈≥0, 𝐷>0 are given constants, 0<𝜇≤1, 𝑓(𝑥,𝑡) is a given continuous function, 𝐷𝜇𝑡𝑢(𝑥,𝑡) is a time fractional derivative which is defined in the Caputo sense ğ·ğ›¼ğ‘¡âŽ§âŽªâŽ¨âŽªâŽ©ğœ•ğ‘“(𝑡)=𝑚𝑓(𝑡)𝜕𝑡𝑚1,𝛼=𝑚∈ℕ,Γ∫(𝑚−𝛼)𝑡0(𝑡−𝜏)𝑚−𝛼−1𝜕𝑚𝑓(𝜏)𝑑𝜏𝑚𝑑𝜏,𝑚−1<𝛼<𝑚.(1.2) The use of Caputo derivative in the above equation is partly because of the convenience to specify the initial conditions [1].

The TFDE (1.1) includes a few special cases: time fractional diffusion equation, time fractional reaction-diffusion equation, time fractional advection-diffusion equations, and their respective corresponding integer-order partial differential equations.

There are many analytical techniques for dealing with the TFDE, such as integral transformation method (including Laplace’s transform, Fourier’s transform, and Mellin’s transform) [1–5], operational calculus method [6], Adomian decomposition method [7], iteration method and series method [8], and the method of separating variables [9].

One of the key issues with numerical solution of the TFDE (1.1) is design of efficient numerical schemes for time fractional derivative. Until now, most numerical algorithms have relied on the finite difference (FD) methods to discretize the fractional derivatives, and the numerical accuracy always dependent on the order of the fractional derivatives. On the other hand, those FD methods have been generally limited to simple cases (low dimension or small integration) and are very difficult to improve the numerical accuracy [10–14]. Some numerical schemes using low-order finite elements (FE) have also been proposed [15–17]. The fractional derivatives are defined using integrals, so they are nonlocal operators. This nonlocal property means that the next state of a system not only depends on its current state but also on its historical states starting from the initial time. This nonlocal property is good for modeling reality, but they require a large number of operations and a large memory storage capacity when discretized with low-order FD and FE schemes. From this point, the “global method”—the nonlocal methods, like the spectral method—is well suited to discretize the nonlocal operators like fractional-order derivatives. These methods naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from an integer order to a fractional-order model. For example, Hanert has proposed a pseudospectral method based on Chebyshev basis functions in space and Mittag-Leffler basis functions in time to discretize the time-space fractional diffusion equation [18, 19]. Li and Xu have proposed a Galerkin spectral method based on Lagrangian basis functions in space and Jacobi basis functions in time for time fractional diffusion equation [20].

In this paper, we propose a time-space collocation spectral method to discretize the TFDEs (1.1), which is easier to implement and apply to multidimensional problems than the existing Galerkin spectral. Another advantage of the present scheme is that the method can easily handle all kinds of boundary conditions.

2. Analytical Solution of the TFDE in a Bounded Domain

In this section, we present some analytical solutions of the TFDE which will be found helpful in the comprehension of the nature of such a problem.

We consider the TFED (1.1) with initial condition 𝑢(𝑥,0)=𝜙(𝑥),𝑥∈(ğ‘Ž,𝑏),(2.1) and Dirichlet boundary conditions 𝑢(ğ‘Ž,𝑡)=𝜑1(𝑡),𝑢(𝑏,𝑡)=𝜑2(𝑡),𝑡∈(0,𝑇),(2.2) or Neumann boundary conditions 𝑢𝑥(ğ‘Ž,𝑡)=𝜑1(𝑡),𝑢𝑥(𝑏,𝑡)=𝜑2(𝑡),𝑡∈(0,𝑇).(2.3)

For the case that 𝑓≡0 and ğ‘Ž=0,𝜑1(𝑡)=𝜑2(𝑡)=0, by applying the finite sine (cosine) and Laplace transforms to (1.1) with initial condition (2.1), the analytical solutions for the problem can be obtained [5] as 𝑢(𝑥,𝑡)=2𝑒𝜈𝑥/2ğ·ğ‘âˆšğ·âˆžî“ğ‘›=1𝐸𝜇−𝜈24𝐷+𝜆2+𝑛𝜋𝑏2𝑡𝜇sin𝑛𝜋𝑥𝑏√𝐷×𝑏√𝐷0𝜙(𝑦)sin𝑛𝜋𝑦𝑏√𝐷𝑒𝜈𝑦/2𝐷𝑑𝑦,(2.4) for homogeneous Dirichlet boundary conditions, and 𝑢(𝑥,𝑡)=2𝑒𝜈𝑥/2ğ·ğ‘âˆšğ·âˆžî“ğ‘›=1𝐸𝜇−𝜈24𝐷+𝜆2+𝑛𝜋𝑏2𝑡𝜇cos𝑛𝜋𝑥𝑏√𝐷×𝑏√𝐷0𝜙(𝑦)cos𝑛𝜋𝑦𝑏√𝐷𝑒𝜈𝑦/2𝐷𝑑𝑦,(2.5) for homogeneous Neumann boundary conditions. Where 𝐸𝛼(𝑧) denotes a one-parameter Mittag-Leffler function which is defined by the series expansion 𝐸𝛼(𝑧)∶=âˆžî“ğ‘˜=0𝑧𝑘Γ(𝛼𝑘+1),𝑧∈ℂ,(𝛼>0).(2.6)

Obviously, if we fix the variable 𝑥=𝑥∗, that is, 𝑢(⋅,𝑡) is a function of the variable 𝑡, we can see the solution 𝑢(⋅,𝑡) is not smooth on [0,𝑇]. According to (2.4) and (2.5), its first derivative behaves like 𝑢′(⋅,𝑡)∼𝑡𝜇−1 and the high-order derivative behaves like 𝑢(𝑚)(⋅,𝑡)∼𝑡𝜇−𝑚 near 𝑡=0+.

3. Collocation Spectral Method

First, we give the properties of the Caputo fractional derivative [1] as 𝐽𝛽𝐷𝛽𝑔(𝑡)=𝑔(𝑡)−𝑛−1𝑘=0𝑡𝑔(0)𝑘𝑘!,0≤𝑛−1<𝛽<𝑛,(3.1) where 𝐽𝛽 is the Riemann-Liouville fractional integral of order 𝛽 which is defined by 𝐽𝛽1𝑔(𝑡)∶=Γ(𝛽)𝑡0(𝑡−𝜏)𝛽−1𝑔(𝜏)𝑑𝜏.(3.2)

By the above properties, we can transform the initial value problem (1.1) into the following Volterra integral equation equivalently: 1𝑢(𝑥,𝑡)−𝑢(𝑥,0)=Γ(𝜇)𝑡0ℒ𝑢(𝑥,𝜏)(𝑡−𝜏)1−𝜇1𝑑𝜏+Γ(𝜇)𝑡0𝑓(𝑥,𝜏)(𝑡−𝜏)1−𝜇𝑑𝜏.(3.3)

For the singular behavior of the exact solution near 𝑡=0+ which we have mentioned in the special case (the exact solution (2.4) or (2.5) behaves 𝑢′(⋅,𝑡)∼𝑡𝜇−1 near 𝑡=0+), the direct application of the spectral methods is difficult. To overcome this difficulty, we use the technique in [21], that is, applying the transformation 𝑢(𝑥,𝑡)=𝑡1−𝜇[],𝑢(𝑥,𝑡)−𝑢(𝑥,0)(3.4) to make the solution smooth. Then (3.3) is transformed to the equation 𝑢(𝑥,𝑡)=𝑡𝑓(𝑥,𝑡)+1−𝜇Γ(𝜇)𝑡0ℒ𝑢(𝑥,𝜏)(𝑡−𝜏)1−𝜇𝜏1−𝜇𝑑𝜏,(3.5) where 𝑡𝑓(𝑥,𝑡)=1−𝜇Γ(𝜇)𝑡0𝑓(𝑥,𝜏)𝑑𝜏(𝑡−𝜏)1−𝜇+ℒ𝑢(𝑥,0)Γ(1+𝜇)𝑡.(3.6)

To apply the theory of orthogonal polynomials, we set 𝑇𝑡=2[]𝑇(1+𝑦),𝑦∈−1,1;𝜏=2[],(1+𝑠),𝑠∈−1,𝑦(3.7) then the singular problems (3.5) can be rewritten as 𝑣(𝑥,𝑦)=𝑔(𝑥,𝑦)+(𝑇(1+𝑦)/2)1−𝜇𝑇Γ(𝜇)22𝜇−1𝑦−1ℒ𝑣(𝑥,𝑠)(𝑦−𝑠)1−𝜇(1+𝑠)1−𝜇𝑑𝑠,(3.8) where 𝑦∈[−1,1], and 𝑣(𝑥,𝑦)=𝑢𝑇𝑥,2(1+𝑦),𝑔(𝑥,𝑦)=𝑓𝑇𝑥,2.(1+𝑦)(3.9)

For the collocation methods, (3.8) holds at the Gauss-Lobatto collocation points {𝑥𝑖}𝑁𝑖=0 and Jacobi collocation points {𝑦𝑗}𝑁𝑗=0 with Jacobi weight functions 𝜔(𝑦)=(1−𝑦2)𝜇−1 on [−1,1], namely, 𝑣𝑥𝑖,𝑦𝑗𝑥=𝑔𝑖,𝑦𝑗+𝑇1+𝑦𝑗/21−𝜇𝑇Γ(𝜇)22𝜇−1𝑦𝑗−1𝑥ℒ𝑣𝑖,𝑠𝑑𝑠𝑦𝑗−𝑠1−𝜇(1+𝑠)1−𝜇.(3.10)

By using the following variable change: 𝑠=𝑠𝑗(𝜃)=1+𝑦𝑗2𝑦𝜃+𝑗−12[],,𝜃∈−1,1(3.11) we can rewrite (3.10) as follows: 𝑣𝑥𝑖,𝑦𝑗𝑥=𝑔𝑖,𝑦𝑗+𝑇1+𝑦𝑗/21−𝜇Γ𝑇(𝜇)1+𝑦𝑗/222𝜇−11−1𝑥ℒ𝑣𝑖,𝑠𝑗(𝜃)1−𝜃21−𝜇𝑥𝑑𝜃=𝑔𝑖,𝑦𝑗+𝑇1+𝑦𝑗/2𝜇22𝜇−1𝑥Γ(𝜇)ℒ𝑣𝑖,𝑠𝑗(⋅),1𝜔.(3.12)

We first use 𝑣𝑗𝑖, 0≤𝑖≤𝑁; 0≤𝑗≤𝑀 to indicate the approximate value for 𝑣(𝑥𝑖,𝑦𝑗), then we can use 𝑣𝑀𝑁(𝑥,𝑦)=𝑁𝑀𝑛=0𝑚=0ğ‘£ğ‘šğ‘›â„Žğ‘›(𝑥)𝐹𝑚(𝑦),(3.13) to approximate the function 𝑣(𝑥,𝑦), where â„Žğ‘›(𝑥) is the 𝑛th Lagrange interpolation polynomial associated with the collocation points {𝑥𝑖}𝑁𝑖=0 and 𝐹𝑚(𝑦) is the 𝑚th Lagrange interpolation polynomial associated with the collocation points {𝑦𝑖}𝑀𝑖=0.

Using a (𝑀+1)-point Gauss quadrature formula relative to the Jacobi weights {𝜔𝑗}𝑀𝑗=0, (3.12) can be approximated by 𝑣𝑗𝑖𝑥=𝑔𝑖,𝑦𝑗+𝑇1+𝑦𝑗/2𝜇22𝜇−1Γ(𝜇)ℒ𝑣𝑀𝑁𝑥𝑖,𝑠𝑗(⋅),1𝑀𝑥=𝑔𝑖,𝑦𝑗+𝑇1+𝑦𝑗/2𝜇22𝜇−1×Γ(𝜇)𝑀𝑘=0𝑁𝑀𝑛=0𝑚=0𝑣𝑚𝑛−𝜆2â„Žğ‘›î€·ğ‘¥ğ‘–î€¸âˆ’ğœˆâ„Žî…žğ‘›î€·ğ‘¥ğ‘–î€¸+ğ·â„Žğ‘›î…žî…žî€·ğ‘¥ğ‘–ğ¹î€¸î€¸ğ‘šî€·ğ‘ ğ‘—î€·ğœƒğ‘˜îƒªğœ”î€¸î€¸ğ‘˜,∀𝑖=1,…,𝑁−1;𝑗=0,1,…,𝑀,(3.14) where the set {𝜃𝑘}𝑀𝑘=0 coincides with the collocation points {𝑦𝑗}𝑀𝑗=0 on [−1,1].

Then the collocation spectral method is to seek 𝑣𝑀𝑁(𝑥,𝑦) of the form (3.13) such that 𝑣𝑗𝑖 satisfies the above collocation equations (3.14) for 1≤𝑖≤𝑁−1,0≤𝑗≤𝑀.

4. Numerical Results with a Collocation Spectral Approximation

In order to demonstrate the effectiveness of the proposed time-space collocation spectral method, some examples are now presented with Dirichlet boundary conditions, Neumann boundary conditions, and mixed boundary conditions.

For completeness sake, the implementation is briefly described here. To simplify the computation, we rewrite the above collocation equations (3.14) into the following: 𝑣𝑗𝑖=𝑔𝑗𝑖+𝑁𝑀𝑛=0𝑚=0𝑣𝑚𝑛−𝜆2â„Žğ‘›î€·ğ‘¥ğ‘–î€¸âˆ’ğœˆâ„Žî…žğ‘›î€·ğ‘¥ğ‘–î€¸+ğ·â„Žğ‘›î…žî…žî€·ğ‘¥ğ‘–îƒ©ğ‘î€¸î€¸ğ‘—ğ‘€î“ğ‘˜=0𝐹𝑚𝑠𝑗𝜃𝑘𝜔𝑘=𝑔𝑗𝑖+𝑁𝑀𝑛=0𝑚=0ğ‘‘ğ‘–ğ‘›ğ‘Žğ‘—ğ‘šğ‘£ğ‘šğ‘›=𝑔𝑗𝑖+𝑁−1𝑀𝑛=1𝑚=0ğ‘‘ğ‘–ğ‘›ğ‘Žğ‘—ğ‘šğ‘£ğ‘šğ‘›+𝑀𝑚=0𝑑𝑖0𝑣𝑚0+ğ‘‘ğ‘–ğ‘ğ‘£ğ‘šğ‘î€¸ğ‘Žğ‘—ğ‘š,∀𝑖=1,…,𝑁−1;𝑗=0,1,…,𝑀,(4.1) where 𝑏𝑗=𝑇1+𝑦𝑗/2𝜇22𝜇−1Γ(𝜇),ğ‘Žğ‘—ğ‘š=𝑏𝑗𝑀𝑘=0𝐹𝑚𝑠𝑗𝜃𝑘𝜔𝑘,𝑔𝑗𝑖𝑥=𝑔𝑖,𝑦𝑗,𝑑𝑖𝑛=−𝜆2â„Žğ‘›î€·ğ‘¥ğ‘–î€¸âˆ’ğœˆâ„Žî…žğ‘›î€·ğ‘¥ğ‘–î€¸+ğ·â„Žğ‘›î…žî…žî€·ğ‘¥ğ‘–î€¸.(4.2)

In our numerical tests, we use the Chebyshev Gauss-Lobatto collocation points 𝑥𝑖1=ğ‘Ž+21−cos𝑖𝜋𝑁(ğ‘âˆ’ğ‘Ž),𝑖=0,1,…,𝑁,(4.3) with the associated weights 𝜔0=𝜔𝑁=(ğ‘âˆ’ğ‘Ž)𝜋,4𝑁𝜔𝑖=(ğ‘âˆ’ğ‘Ž)𝑖𝜋2𝑁,𝑖=1,…,𝑁−1,(4.4) in the space. The other kinds Gauss-Lobatto collocation points (such as Legendre Gauss-Lobatto collocation points) also can be used. The advantage of Gauss-Lobatto points is that they include the boundary points, which means we can apply boundary conditions there. For the time, the Jacobi Gauss collocation points are used for {𝑦𝑗}𝑀𝑗=0 with the associated weights {𝜔𝑗}𝑀𝑗=0 and other kinds of the Jacobi collocation points also suit to be used.

Let us set 𝐕𝐌𝐍=𝑣01,𝑣02,…,𝑣0𝑁−1,𝑣11,…,𝑣1𝑁−1,…,𝑣𝑀𝑁−1𝑇;𝐆𝐌𝐍=𝑔01,𝑔02,…,𝑔0𝑁−1,𝑔11,…,𝑔1𝑁−1,…,𝑔𝑀𝑁−1𝑇;(4.5) let 𝐀=(ğ‘Žğ‘–ğ‘—) be a matrix of (𝑀+1) by (𝑀+1), and 𝐃=(𝑑𝑖𝑗) is a matrix of (𝑁−1) by (𝑁−1).

4.1. Implementation of Dirichlet Boundary Conditions

The Dirichlet boundary conditions are directly applied in (4.1) and give numerical solutions on boundary in the following way: 𝑣𝑚0=𝑇1+𝑦𝑚21−𝜇𝜑1𝑇1+𝑦𝑚2,𝑣−𝜙(ğ‘Ž)𝑚𝑁=𝑇1+𝑦𝑚21−𝜇𝜑2𝑇1+𝑦𝑚2.−𝜙(𝑏)(4.6)

We set 𝐆𝐌𝐍=𝑔01,𝑔02,…,𝑔0𝑁−1,𝑔11,…,𝑔1𝑁−1,…,𝑔𝑀𝑁−1𝑇,(4.7) where 𝑔𝑗𝑖=𝑀𝑚=0𝑑𝑖0𝑣𝑚0+ğ‘‘ğ‘–ğ‘ğ‘£ğ‘šğ‘î€¸ğ‘Žğ‘—ğ‘š.(4.8)

Thus, the numerical scheme (4.1) leads to a system of equation of the form 𝐕𝐌𝐍=𝐅𝐌𝐍+𝐂𝐕𝐌𝐍,(4.9) where 𝐅𝐌𝐍=𝐆𝐌𝐍+𝐆𝐌𝐍, 𝐂=[ğ‘Žğ‘–ğ‘—ğƒ] is a matrix of (𝑁−1)×(𝑀+1) by (𝑁−1)×(𝑀+1).

4.2. Implementation of Neumann Boundary Conditions

The Neumann boundary conditions (2.3) at 𝑥=ğ‘Ž and 𝑥=𝑏 can be approximated as 𝜕𝑥𝑣𝑀𝑁𝑥0,𝑦𝑚=𝑁𝑛=0ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥0=𝑇1+𝑦𝑚21−𝜇𝜑1𝑇1+𝑦𝑚2îƒªâˆ’ğœ™î…žî€·ğ‘¥0≐𝑟𝑚(1),𝜕𝑥𝑣𝑀𝑁𝑥𝑁,𝑦𝑚=𝑁𝑛=0ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸=𝑇1+𝑦𝑚21−𝜇𝜑2𝑇1+𝑦𝑚2îƒªâˆ’ğœ™î…žî€·ğ‘¥ğ‘î€¸îƒ­â‰ğ‘Ÿğ‘š(2).(4.10) Equation (4.10) can be written as follows: ℎ0𝑥0𝑣𝑚0+â„Žî…žğ‘î€·ğ‘¥0𝑣𝑚𝑁=𝑟𝑚(1)−𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘›î€¸,ℎ0𝑥𝑁𝑣𝑚0+â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸ğ‘£ğ‘šğ‘=𝑟𝑚(2)−𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘›î€¸.(4.11) Solving (4.11) for 𝑣𝑚0 and 𝑣𝑚𝑁, we get 𝑣𝑚0=â„Žî…žğ‘î€·ğ‘¥0𝑟𝑚(2)−∑𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸î‚âˆ’â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸î‚€ğ‘Ÿğ‘š(1)−∑𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥0î€¸î‚â„Žî…žğ‘î€·ğ‘¥0ℎ0î€·ğ‘¥ğ‘î€¸âˆ’â„Žî…ž0𝑥0î€¸â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸,𝑣𝑚𝑁=ℎ0𝑥𝑁𝑟𝑚(1)−∑𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥0−ℎ0𝑥0𝑟𝑚(2)−∑𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸î‚â„Žî…žğ‘î€·ğ‘¥0ℎ0î€·ğ‘¥ğ‘î€¸âˆ’â„Žî…ž0𝑥0î€¸â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸.(4.12)

Then the last right term of (4.1) can be written as follows: 𝑀𝑚=0𝑑𝑖0𝑣𝑚0+ğ‘‘ğ‘–ğ‘ğ‘£ğ‘šğ‘î€¸ğ‘Žğ‘—ğ‘š=𝑀𝑚=0𝑐𝑖(1)𝑟𝑚(1)−𝑐𝑖(2)𝑟𝑚(2)î‚ğ‘Žğ‘—ğ‘š+𝑀𝑚=0𝑐𝑖(2)𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸âˆ’ğ‘ğ‘–(1)𝑁−1𝑛=1ğ‘£ğ‘šğ‘›â„Žî…žğ‘›î€·ğ‘¥0î€¸îƒªğ‘Žğ‘—ğ‘š=𝑀𝑚=0𝑐𝑖(1)𝑟𝑚(1)−𝑐𝑖(2)𝑟𝑚(2)î‚ğ‘Žğ‘—ğ‘š+𝑀𝑚=0𝑁−1𝑛=1𝑣𝑚𝑛𝑐𝑖(2)â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸âˆ’ğ‘ğ‘–(1)â„Žî…žğ‘›î€·ğ‘¥0î€¸î‚ğ‘Žğ‘—ğ‘š,(4.13) where 𝑐𝑖(1)=ℎ0î€·ğ‘¥ğ‘î€¸ğ‘‘ğ‘–ğ‘âˆ’â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸ğ‘‘ğ‘–0â„Žî…žğ‘î€·ğ‘¥0ℎ0î€·ğ‘¥ğ‘î€¸âˆ’â„Žî…ž0𝑥0î€¸â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸,𝑐𝑖(2)=ℎ0𝑥0î€¸ğ‘‘ğ‘–ğ‘âˆ’â„Žî…žğ‘î€·ğ‘¥0𝑑𝑖0â„Žî…žğ‘î€·ğ‘¥0ℎ0î€·ğ‘¥ğ‘î€¸âˆ’â„Žî…ž0𝑥0î€¸â„Žî…žğ‘î€·ğ‘¥ğ‘î€¸.(4.14)

Let 𝐆𝐌𝐍=̃𝑔01,̃𝑔02,…,̃𝑔0𝑁−1,̃𝑔11,…,̃𝑔1𝑁−1,…,̃𝑔𝑀𝑁−1𝑇,(4.15) where ̃𝑔𝑗𝑖=𝑀𝑚=0𝑐𝑖(1)𝑟𝑚(1)−𝑐𝑖(2)𝑟𝑚(2)î‚ğ‘Žğ‘—ğ‘š,(4.16)𝑑𝑖𝑛=𝑑𝑖𝑛+𝑐𝑖(2)â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸âˆ’ğ‘ğ‘–(1)â„Žî…žğ‘›î€·ğ‘¥0=−𝜆2â„Žğ‘›î€·ğ‘¥ğ‘–î€¸âˆ’ğœˆâ„Žî…žğ‘›î€·ğ‘¥ğ‘–î€¸+ğ·â„Žğ‘›î…žî…žî€·ğ‘¥ğ‘–î€¸+𝑐𝑖(2)â„Žî…žğ‘›î€·ğ‘¥ğ‘î€¸âˆ’ğ‘ğ‘–(1)â„Žî…žğ‘›î€·ğ‘¥0,𝑑𝐃=𝑖𝑗isamatrixof(𝑁−1)by(𝑁−1).(4.17)

Then the numerical scheme (4.1) leads to a system of equation of the form 𝐕𝐌𝐍=𝐅𝐌𝐍+𝐂𝐕𝐌𝐍,(4.18) where 𝐅𝐌𝐍=𝐆𝐌𝐍+𝐆𝐌𝐍, 𝐂=[ğ‘Žğ‘–ğ‘—î‚ğƒ] is a matrix of (𝑁−1)×(𝑀+1) by (𝑁−1)×(𝑀+1).

Remark 4.1. The implementation for the mixed boundary conditions also can be derived by (4.6) and (4.12).

4.3. Numerical Experiments

In this subsection, the proposed numerical scheme is applied to several test problems to show the efficiency and spectral accuracy.

In each example, we have calculated 𝐿2 errors and ğ¿âˆž errors given by the following formulas: 𝐿2⎷error=𝑁𝑀𝑖=0𝑗=0𝑢𝑗𝑖𝑥−𝑢𝑖,𝑡𝑗2𝑤𝑖𝑤𝑗;ğ¿âˆžerror=max𝑖,𝑗||𝑢𝑗𝑖𝑥−𝑢𝑖,𝑡𝑗||,(4.19) where 𝑢𝑗𝑖 is the numerical approximation solutions of the exact solutions 𝑢(𝑥𝑖,𝑡𝑗).

Example 4.2 (Dirichlet Boundary Conditions). In this example, we consider the following time fractional diffusion equations: 𝐷𝜇𝑡𝜕𝑢(𝑥,𝑡)=2𝑢(𝑥,𝑡)𝜕𝑥2+𝑓(𝑥,𝑡),𝑥∈(−1,1),𝑡∈(0,𝑇)𝑢(𝑥,0)=0,𝑥∈(ğ‘Ž,𝑏)𝑢(−1,𝑡)=𝑢(1,𝑡)=0,𝑡∈(0,𝑇),(4.20) where 𝑓(𝑥,𝑡)=Γ(1+𝛽)𝑡Γ(1+𝛽−𝜇)𝛽−𝜇sin(2𝜋𝑥)+4𝜋2𝑡𝛽sin(2𝜋𝑥).(4.21) The exact solution is given by 𝑢(𝑥,𝑡)=𝑡𝛽sin(2𝜋𝑥),𝛽>−1.(4.22)

In Figure 1, we plot the exact and numerical solutions, and, in Figure 2, we represent the associated errors field. Here we set 𝑁=𝑀=22, 𝜇=0.8, and 𝛽=−0.2. The results in Figures 1 and 2 denote that the numerical solution using the proposed collocation spectral method is excellent in agreement with the exact solution at the whole domain. The efficiency of this collocation spectral method can be further confirmed by Figures 3–4, which are the comparison of the exact solution and numerical solution when the space variable 𝑥 or time variable 𝑡 is fixed for various 𝜇 and 𝛽; here 𝑁=𝑀=22.

The main purpose of the numerical test is to check the convergence behavior of numerical solutions with respect to the polynomial degrees 𝑀 and 𝑁 for several 𝜇, especially the convergence in time because of the fractional derivative in time. In order to investigate the spatial accuracy when 𝑁 increases, we take 𝑀 larger enough so that the time discretization errors are negligible compared with the spatial discretization errors. Similary, for the temporal accuracy, we must keep 𝑁 large enough to preclude spatial errors. 𝐿2 errors and ğ¿âˆž errors in semilog scale varying with the polynomial degree 𝑁 (see (a)) or 𝑀 (see (b)) are shown in Figures 5 and 6 for 𝜇=0.05,0.75 with 𝛽=2.2, and Figure 7 for 𝜇=1 with 𝛽=1.2. It is clear that the spectral convergence is achieved both of spatial and temporal errors. This indicates that the convergence in space and time of the time-space collocation spectral method is exponential, even though for the classical diffusion equation (𝜇=1).

Example 4.3 (Neumann Boundary Conditions). Consider the time fractional advection-diffusion equation with Neumann boundary conditions 𝐷𝜇𝑡𝑢(𝑥,𝑡)=−𝜈𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑥+𝐷2𝑢(𝑥,𝑡)𝜕𝑥2𝜕+𝑓(𝑥,𝑡),𝑥∈(3,5),𝑡>0,𝑢(𝑥,0)=0,𝑥∈(3,5),𝑥𝑢(3,𝑡)=𝜋𝑡22,𝜕𝑥𝑢(5,𝑡)=−𝜋𝑡22,𝑡>0.(4.23) We choose the suitable source term 𝑓(𝑥,𝑡) to obtain the exact solution 𝑢(𝑥,𝑡)=𝑡2cos𝜋𝑥2.(4.24)

In Table 1, 𝐿2 errors and ğ¿âˆž errors varying with (𝑁,𝑀) and 𝜇 are given, and it indicates the efficiency of the present technique.


( 𝑁 , 𝑀 ) 𝜇 = 0 . 1 𝜇 = 0 . 6 𝜇 = 0 . 9 9
𝐿 2 error 𝐿 ∞ error 𝐿 2 error 𝐿 ∞ error 𝐿 2 error 𝐿 ∞ error

(6, 22) 8 . 3 8 2 2 𝑒 − 0 0 4 2 . 7 3 9 3 𝑒 − 0 0 4 1 . 8 3 0 2 𝑒 − 0 0 4 1 . 7 7 9 3 𝑒 − 0 0 4 8 . 4 5 3 9 𝑒 − 0 0 5 1 . 2 5 6 0 𝑒 − 0 0 4
(8, 22) 8 . 8 8 7 3 𝑒 − 0 0 6 2 . 3 8 8 3 𝑒 − 0 0 6 1 . 6 2 0 7 𝑒 − 0 0 6 1 . 5 8 4 3 𝑒 − 0 0 6 7 . 6 2 7 7 𝑒 − 0 0 7 1 . 1 4 3 9 𝑒 − 0 0 6
(10, 22) 5 . 1 2 2 9 𝑒 − 0 0 6 1 . 8 1 7 9 𝑒 − 0 0 6 4 . 4 0 8 1 𝑒 − 0 0 8 1 . 2 0 3 2 𝑒 − 0 0 7 4 . 3 5 0 9 𝑒 − 0 0 9 6 . 6 0 3 4 𝑒 − 0 0 9
(22, 6) 0.0013 3 . 8 7 7 1 𝑒 − 0 0 4 5 . 0 2 5 1 𝑒 − 0 0 5 5 . 6 2 3 0 𝑒 − 0 0 5 2 . 3 4 7 0 𝑒 − 0 0 7 2 . 9 2 6 1 𝑒 − 0 0 7
(22, 8) 3 . 7 5 1 6 𝑒 − 0 0 4 1 . 1 6 3 1 𝑒 − 0 0 4 1 . 0 9 1 6 𝑒 − 0 0 5 1 . 4 8 3 2 𝑒 − 0 0 5 4 . 5 7 0 7 𝑒 − 0 0 8 6 . 9 2 2 1 𝑒 − 0 0 8
(22, 10) 1 . 4 5 9 1 𝑒 − 0 0 4 4 . 6 3 0 9 𝑒 − 0 0 5 3 . 2 9 7 6 𝑒 − 0 0 6 5 . 2 2 9 7 𝑒 − 0 0 6 1 . 2 4 5 3 𝑒 − 0 0 8 2 . 1 8 3 8 𝑒 − 0 0 8

In Figure 8, we plot 𝐿2 errors and ğ¿âˆž errors in semilog scale. Figures 8(a) and 8(b) are concerned with the spatial errors with 𝑀=16 and the temporal errors with 𝑁=16, respectively. As expected, the spatial and temporal spectral convergence is achieved. Our greatest interest is to check the convergence in time because of the fractional derivative in time. So we plot the 𝐿2 errors and ğ¿âˆž errors as functions of 𝑀 for several more values 𝜇 in Figure 9, where 𝑁=16. This graph shows that the spectral convergence in time can be reached, and that the time-space collocation spectral method also works well for the classical advection-diffusion equation.

Example 4.4 (Mixed Boundary Conditions). Consider the time fractional reaction-subdiffusion equation 𝐷𝜇𝑡3𝑢(𝑥,𝑡)=−4𝜕𝑢(𝑥,𝑡)+2𝑢(𝑥,𝑡)𝜕𝑥2𝑥,𝑥∈(0,2𝜋),𝑡>0,𝑢(𝑥,0)=sin2,𝑥∈(0,2𝜋);𝑢(0,𝑡)=0,𝜕𝑥1𝑢(2𝜋,𝑡)=−2𝐸𝜇(−𝑡𝜇).(4.25) The exact solution is given by 𝑢(𝑥,𝑡)=𝐸𝜇(−𝑡𝜇𝑥)sin2.(4.26)

In Figure 10, we plot the visual fields of the exact and numerical solutions. Comparing Figures 10(a) and 10(c), and Figures 10(b) and 10(d), we see that the numerical solution is in good agreement with the exact solution; the shapes on the space variable are similar to Sine function. In addition, we can see that the solutions decay over time from the initial state. Comparing Figures 10(a) and 10(b) with Figures 10(c) and 10(d), (a) and (b) decay faster in the beginning and the trend is the opposite of (c) and (d). This is consistent with the behavior of the function 𝐸𝛼(𝑧). For 0<𝑡<0.77, 𝐸1/2(−𝑡1/2) decays faster than 𝐸1(−𝑡), whereas for 𝑡>0.77, the trend is just the opposite.

We plot in Figure 11 the 𝐿2 errors and ğ¿âˆž errors versus 𝑁 with 𝑀=16 in 11(a) and versus 𝑀 with 𝑁=16 in 11(b). As can be seen in Figure 11, the proposed method provides spatial and temporal spectral convergence for 𝐿2 errors and ğ¿âˆž errors.

5. Conclusion

This paper proposes a time-space collocation spectral method for a class of time fractional differential equations with Caputo derivatives. The proposed method also works well for the corresponding classical integer-order partial differential equations (𝜇=1), and it differs from (and is simpler than) the existing time-space spectral methods which are based on the Petrov-Galerkin or Dual-Petrov-Galerkin formulation. The main advantage of the present scheme is that it gives very accurate convergency by choosing less number of grid points and the problem can be solved up to big time, and the storage requirement due to the time memory effect can be considerably reduced. At the same time, the technique is also simpler and easier to apply to multidimensional problems than the existing Galerkin spectral method such as the methods in [19, 20].

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Copyright © 2012 Fenghui Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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