Solving the time-fractional diffusion equation via Sinc–Haar collocation method

doi:10.1016/j.amc.2014.12.110

Applied Mathematics and Computation

Volume 257, 15 April 2015, Pages 317-326

https://doi.org/10.1016/j.amc.2014.12.110 Get rights and content

Highlights

•
Expanding the approximation with the elements of Sinc and Haar functions.
•
An efficient algorithm with exponential convergence rate.
•
Computational speed is high due to using the Haar operational matrices.
•
To improve the accuracy by increasing the number of collocation points.

Abstract

The present study investigates the Sinc–Haar collocation method for the solution of the time-fractional diffusion equation. The advantages of this technique are that not only the convergence rate of Sinc approximation is exponential but the computational speed also is high due to the use of the Haar operational matrices. This technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of Sinc functions in space and Haar functions in time with unknown coefficients. The effectiveness of the proposed method is examined by comparing the numerical results with the exact solutions.

Introduction

In recent years, the field of fractional differential equations (FDEs) has attracted the attention of scientists in several areas and has also been used to solve many actual problems, modeled in mathematical physics, such as fluid and continuum mechanics, viscoelastic flows, biology, chemistry, acoustics and psychology [1], [2], [3], [4]. The time FDEs are obtained from the classical PDEs by replacing the integer-order time derivative with a fractional derivative of order $α$ with $0 < α$ . The advantages of time FDEs in comparison with integer-order are the capability of simulating natural physical process and dynamic system more accurately [5].

In this paper, we study the time-fractional diffusion equation which can be formulated as follows: $\frac{\partial^{α} y (x, t)}{\partial t^{α}} + a (x) \frac{\partial y (x, t)}{\partial x} + b (x) \frac{\partial^{2} y (x, t)}{\partial x^{2}} = f (x, t),$ $0 < x < 1, 0 < t ⩽ 1$ with initial conditions: $y (x, 0) = g (x), 0 < x < 1$ and boundary conditions: $y (0, t) = y (1, t) = 0, 0 < t ⩽ 1,$ where $a (x), b (x) \neq 0$ are continuous functions and $0 < α ⩽ 1$ . Here the time-fractional derivative is defined as the Caputo fractional derivatives. Time fractional diffusion equations are used when we are trying to explain transport processes with long memory in which the rate of diffusion is unreliable with the classical Brownian motion model [6].

Because of the lack of appropriate mathematical methods, most of the analytical solutions for time FDEs are challenging to acquire. Therefore, approximation and numerical techniques such as first-order finite difference method [7], Chebyshev spectral approximation [8], Walsh function method [9], variational iteration method [10], Adomian decomposition method [11], [12], Homotopy perturbation method [13] and generalized differential transform method [14] have been proposed. Uddin and Haq [15] used the radial basis functions (RBFs) approximation for the solution of Eq. (1) with constant coefficients. They replaced the first order time derivative with the Caputo fractional derivative of order $α \in (0, 1]$ and approximated the spatial derivatives by the derivative of interpolation in the Kansa method. Saadatmandi et. al. [16] solved this problem by expanding the approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. Jiang and Ma [17] developed the high-order finite difference method for solving time-fractional diffusion equation and proved an optimal convergence rate for the case of $a (x) = 0, b (x) = - 1$ . Chen et. al. [18] combined the orthogonal wavelet function with operational matrix and transformed the fractional diffusion equation into Sylvester equation which is easily to be solved.

The novelty of the present paper is that we investigate the behaviour of the combination of two different groups of orthogonal functions from two different intervals for solving the time fractional diffusion equation. The combination of piecewise orthogonal Haar functions obtained from wavelets transform, defined on $[0, 1)$ interval, with continuous orthogonal Sinc functions obtained from fourier transform, defined on $(- \infty, + \infty)$ interval. Based on the properties of orthogonal Sinc functions, it is apparent that the convergence rate of our method is exponential in space [19]. Another advantage of our method is that it transforms the problem into a system of algebraic equations, so that the computation becomes simple and computer oriented. In the new proposed algorithm, we extend the solution of the problem to the sum of basis functions and take good advantage of the orthogonality of Haar and Sinc functions to a set of equations for the coefficients of the solution.

The organization of the rest of the paper is as follows: In Section 2, we present a brief introduction to the some essential definitions from which are derived some tools for developing our method. In Section 3, the convergence rate analysis of the Sinc and Haar functions is given. In Section 4, we apply the method of Sinc–Haar collocation for solving the model equation. In Section 5, the proposed method is used in some types of time fractional diffusion equation and it is compared with the current analytic solutions revealed in different published works within the literature. The conclusion is presented in the final section.

Section snippets

Definitions of fractional derivatives and integrals [1,20]

Definition 1

For l to be the smallest integer that exceeds $α$ , Caputo’s time-fractional derivative operator of order $α > 0$ is defined as: ${}_{a}^{C}D_{t}^{α} y (x, t) = \{\begin{matrix} I_{t}^{l - α} D_{t}^{l} y (x, t), & l - 1 < α < l, \\ D_{t}^{l} y (x, t), & α = l, l \in N, \end{matrix}$ where, $I_{t}^{α} y (x, t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} y (x, τ) d τ, t, α > 0 .$

Some of the most essential properties of operator $I_{t}^{α}$ is as follows: $\begin{matrix} I_{t}^{α} (t^{β}) = \frac{Γ (β + 1)}{Γ (α + β + 1)} t^{α + β}, \\ I_{t}^{α} (x^{β} t^{γ}) = x^{β} I_{t}^{α} (t^{γ}) . \end{matrix}$ Caputo’s fractional differentiations are the linear operators given as follows: $D_{t}^{α} (κ y (x, t) + ρ s (x, t)) = κ D_{t}^{α} y (x, t) + ρ D_{t}^{α} s (x, t),$ where $κ$ and $ρ$ are constants.

Sinc functions

The Sinc

Sinc functions

The following theorem for which the proof can be found in [19] shows that the convergence rate of Sinc approximation is exponential.

Definition 2

Let $H^{2} (D_{E})$ be the class of functions f which are analytic in $D_{E}$ (the eye-shaped domain defined in Eq. (12)) satisfy: $\int_{ψ^{- 1} (p + T)} | f (z) | dz \to 0, x \to \pm \infty,$ where $T = \{iq : | q | < d ⩽ \frac{π}{2}\}$ and on the boundary of $D_{E}$ (denoted $\partial D$ ) satisfy: $N (f) = \int_{\partial D} | f (z) dz | < \infty .$

Theorem 1

Assume that $f ψ^{'} \in H^{2} (D_{E})$ then for all $z \in (0, 1)$ : $E (f, h) (z) = |f (z) - \sum_{k = - \infty}^{\infty} f (kh) S (k, h) \circ ψ (z)| ⩽ \frac{N (f ψ^{'})}{2 π d \sinh (π d / h)} ⩽ 2 \frac{N (f ψ^{'})}{π d} e^{- π d / h} .$

Moreover, if $| f (z) | ⩽ {Ce}^{- α}$

Sinc–Haar collocation method

A discrete approximation to the $\frac{\partial y (x, t)}{\partial t}$ can be expanded into $2 n + 1$ Sinc functions and k Haar functions as: $\frac{\partial y_{n, k} (x, t)}{\partial t} = \sum_{i = - n}^{n} \sum_{j = 0}^{k - 1} c_{ij} S_{i} (x) h_{j} (t) .$

Lemma 1

Suppose $0 < α < 1$ and $x_{m}$ be Sinc collocation points, given in Eq. (17). Then the following relations hold: $y_{n, k} (x_{m}, t) = g (x_{m}) + \sum_{j = 0}^{k - 1} \sum_{l = 0}^{k - 1} c_{ml} J_{lj} h_{j} (t),$ $\frac{\partial^{α} y_{n, k} (x_{m}, t)}{\partial t^{α}} = {(D_{t}^{α} g (x))}_{x_{m}} + \sum_{j = 0}^{k - 1} \sum_{l = 0}^{k - 1} c_{ml} J_{lj}^{(1 - α)} h_{j} (t),$ $\frac{\partial y_{n, k} (x_{m}, t)}{\partial x} = {(\frac{\partial g (x)}{\partial x})}_{x_{m}} + \sum_{i = - n}^{n} \sum_{j = 0}^{k - 1} \sum_{l = 0}^{k - 1} w_{im}^{(1)} c_{il} J_{lj} h_{j} (t),$ $\frac{\partial^{2} y_{n, k} (x_{m}, t)}{\partial x^{2}} = {(\frac{\partial^{2} g (x)}{\partial x^{2}})}_{x_{m}} + \sum_{i = - n}^{n} \sum_{j = 0}^{k - 1} \sum_{l = 0}^{n - 1} w_{im}^{(2)} c_{il} J_{lj} h_{j} (t),$ where, $\begin{matrix} J_{lj} = J_{k \times k} [l, j], \\ J_{lj} \end{matrix}$

Results and discussion

Example 1

To validate the effectiveness of the proposed method we consider the example given in [7]: $\begin{matrix} \frac{\partial^{α} y (x, t)}{\partial t^{α}} - \frac{\partial^{2} y (x, t)}{\partial x^{2}} = \frac{2}{Γ (3 - α)} t^{2 - α} \sin (2 π x) + 4 π^{2} t^{2} \sin (2 π x), 0 < x < 1, 0 < t ⩽ 1, \\ y (0, t) = y (1, t) = 0, 0 < t ⩽ 1, \\ y (x, 0) = 0, 0 < x < 1 . \end{matrix}$

The exact solution to this problem is $y (x, t) = t^{2} \sin (2 π x)$ . To solve the above problem with $α = 0.5$ by using the Sinc–Haar collocation method, we choose $h = \frac{π}{\sqrt{2 n}}$ . The $L^{2}$ -error and $L^{\infty}$ -error are used to explore the dependence of errors on the parameters $n, k$ . We have also calculated the convergence rate of the

Conclusion

In the present study, a numerical method for solving the time fractional diffusion equation based on the combination of two orthogonal Sinc and Haar functions was proposed. Also, Refs. [16], [30] have previously applied the combination of the Sinc functions with the other different groups of orthogonal Legendre and Chebyshev polynomials for solving the time fractional Diffusion equation. The effectiveness of the method was examined via comparing the obtained results with the other existed

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Solving the time-fractional diffusion equation via Sinc–Haar collocation method

Highlights

Abstract

Introduction

Section snippets

Definitions of fractional derivatives and integrals [1,20]

Sinc functions

Sinc functions

Sinc–Haar collocation method

Results and discussion

Conclusion

Chaos Solitons Fract.

Comput. Math. Appl.

J. Comput. Phy.

Commun. Nonlinear Sci. Numer. Simul.

Comput. Math. Appl.

Phys. Lett. A

Appl. Math. Comput.

Commun. Nonlinear Sci. Numer. Simul.

J. Comput. Appl. Math.

Commu. Nonlinear Sci. Numer. Simul.

Commun. Nonlinear Sci. Numer. Simul.

J. Comput. Appl. Math.

J. Comput. Sci.

Appl. Math. Comput.

Phys. Lett. A