Numerical scheme with high order accuracy for solving a modified fractional diffusion equation

doi:10.1016/j.amc.2014.07.044

Applied Mathematics and Computation

Volume 244, 1 October 2014, Pages 772-782

https://doi.org/10.1016/j.amc.2014.07.044 Get rights and content

Abstract

In recent years, some researchers have developed various numerical schemes to solve the modified fractional diffusion equation. For the numerical solutions of the modified fractional diffusion equation, there are already some important progresses. In this paper, a numerical scheme with second order temporal accuracy and fourth order spatial accuracy is developed to solve a modified fractional diffusion equation; the convergence, stability and solvability of the numerical scheme are analyzed by Fourier analysis; the theoretical results extremely consistent with the numerical experiment.

Introduction

In recent years, the time or space or time–space fractional diffusion equations are widely used to describe anomalous diffusion processes in numerous physical and biological systems, many authors have developed various numerical schemes to solve these fractional diffusion equations. 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 has been proposed [3], [10], [11] $\frac{\partial p (x, t)}{\partial t} = (A \frac{\partial^{1 - α}}{\partial t^{1 - α}} + B \frac{\partial^{1 - β}}{\partial t^{1 - β}}) \frac{\partial^{2} p (x, t)}{\partial x^{2}},$ 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 [6]; Merdan et al. presented the numerical solution of time-fraction modified equal width wave equation, and show how an application of fractional two dimensional differential transformation method obtained approximate analytical solution of time-fraction modified equal width wave equation [9]; Liu et al. discussed the numerical method and analytical technique of the modified fractional diffusion equation with a nonlinear source term, they proved that this method has first order temporal accuracy and second order spatial accuracy using a new energy method [7]; Liu et al. researched the finite element approximation for a modified fractional diffusion equation, they proved that this approximation has first order temporal accuracy and m (here m is the degree of the piecewise polynomials) order spatial accuracy [8]; Zhang et al. studied the finite difference/element method for a two-dimensional modified fractional diffusion equation, they proved that this method has $(1 + \min \{α, β\})$ order temporal accuracy and m (here m is the degree of the piecewise polynomials) order spatial accuracy [12]; Chen researched the numerical method for solving a two-dimensional variable-order modified fractional diffusion equation, this method has first order temporal accuracy and fourth order spatial accuracy, further, a numerical method for improving temporal accuracy have also been developed [4]; apply the fourth-order compact scheme of the second-order space partial derivative and the Grünwald–Letnikov discretization of the Rieman–Liouille fractional time derivative, Abbaszadeh et al. proposed a high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, they proved that this scheme has first order temporal accuracy and fourth order spatial accuracy using the Fourier method [1], and they also presented a fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term, they proved that this solution has first order temporal accuracy and fourth order spatial accuracy using the Fourier method [2].

On the basis of the literature, for the cases of one-dimensional and multidimensional of the modified fractional diffusion equation Eq. (1), although the numerical schemes with ideal spatial accuracy can be constructed by the fourth-order compact scheme of the second-order space partial derivative or the finite element approximation, but unfortunately that numerical scheme with second-order temporal accuracy is still unable to achieve. Chen constructed a numerical method for improving temporal accuracy for solving a two-dimensional variable-order modified fractional diffusion equation, but it should be point out that it only is supported by the numerical experiment, whereas it is not supported by rigorous numerical analysis [4]. Therefore to date, numerical scheme with second order temporal accuracy for solving modified fractional diffusions supported by rigorous numerical analysis does not exist. So, how to improve temporal accuracy of numerical scheme for solving modified fractional diffusions is a exciting and meaningful work. Easy to understand that the key problem is that convergence and stability analysis of numerical scheme are very difficult. We expect that this article will change this situation.

In this paper, we study numerical scheme and perform related numerical analysis for solving the following modified fractional diffusion equation (MFDE) $\frac{\partial p (x, t)}{\partial t} = (\frac{\partial^{1 - α}}{\partial t^{1 - α}} + \frac{\partial^{1 - β}}{\partial t^{1 - β}}) \frac{\partial^{2} p (x, t)}{\partial x^{2}} + f (x, t),$ with the following initial and boundary conditions: $p (x, 0) = w (x), 0 ⩽ x ⩽ L,$ $p (0, t) = φ (t), p (L, t) = ψ (t), 0 ⩽ t ⩽ T,$ where $0 < α < β < 1$ , and $_{0} D_{t}^{1 - γ} p (x, t)$ is the Rieman–Liouille fractional partial derivative of order $1 - γ$ defined by $_{0} D_{t}^{1 - γ} p (x,, t) = \frac{1}{Γ (γ)} \frac{\partial}{\partial t} \int_{0}^{t} \frac{p (x, s)}{{(t - s)}^{1 - γ}} ds .$

In this paper, we always suppose that $p (x, t) \in P (Ω)$ is the exact solution of the problem (1), (2), (3) and $\frac{\partial^{2} f (x, t)}{\partial t^{2}} \in C (Ω)$ , where $Ω = \{(x, t) | 0 ⩽ x ⩽ L, 0 ⩽ t ⩽ T\},$ $P (Ω) = \{p (x, t) | \frac{\partial^{6} p (x, t)}{\partial x^{6}}, \frac{\partial^{4} p (x, y, t)}{\partial x^{2} \partial t^{2}} \in C (Ω)\} .$

Section snippets

A numerical scheme for solving MFDE

In this paper, we let $x_{j} = jh, j = 0, 1, \dots, J; t_{k} = k τ, k = 0, 1, \dots, K,$ where $h = L / J$ and $τ = T / K$ is space step and time step, respectively. And define $δ_{x}^{2} v_{j}^{i} = v_{j - 1}^{i} - 2 v_{j}^{i} + v_{j + 1}^{i} .$

Integrating both sides of Eq. (2) from $t_{k - 1}$ to $t_{k}$ and take $x = x_{j}$ , get that $p (x_{j}, t_{k}) - p (x_{j}, t_{k - 1}) = \frac{1}{Γ (α)} [\int_{0}^{t_{k}} \frac{\partial^{2} p (x_{j}, s)}{\partial x^{2}} \frac{ds}{{(t_{k} - s)}^{1 - α}} - \int_{0}^{t_{k - 1}} \frac{\partial^{2} p (x_{j}, s)}{\partial x^{2}} \frac{ds}{{(t_{k - 1} - s)}^{1 - α}}] + \frac{1}{Γ (β)} [\int_{0}^{t_{k}} \frac{\partial^{2} p (x_{j}, s)}{\partial x^{2}} \frac{ds}{{(t_{k} - s)}^{1 - β}} - \int_{0}^{t_{k - 1}} \frac{\partial^{2} p (x_{j}, s)}{\partial x^{2}} \frac{ds}{{(t_{k - 1} - s)}^{1 - β}}] + \int_{t_{k - 1}}^{t_{k}} f (x_{j}, s) ds$ from this we have that $(1 + \frac{1}{12} δ_{x}^{2}) p (x_{j}, t_{k}) - (1 + \frac{1}{12} δ_{x}^{2}) p (x_{j}, t_{k - 1}) = (1 + \frac{1}{12} δ_{x}^{2}) \frac{1}{Γ (α)} [\int_{0}^{t_{k}} \frac{\partial^{2} p (x_{j}, s)}{\partial x^{2}} \frac{ds}{{(t_{k} - s)}^{1 - α}} - \int_{0}^{t_{k - 1}} \partial^{2} p (]$

Convergence of the numerical scheme for solving MFDE

In this section, we study convergence of the numerical scheme (8), (9), (10). Subtracting (8) from (6), we obtain the following error equation $(1 + \frac{1}{12} δ_{x}^{2}) E_{j}^{k} - (1 + \frac{1}{12} δ_{x}^{2}) E_{j}^{k - 1} = μ_{α} \sum_{l = 0}^{k} λ_{α}^{(l)} δ_{x}^{2} E_{j}^{k - l} + μ_{β} \sum_{l = 0}^{k} λ_{β}^{(l)} δ_{x}^{2} E_{j}^{k - l} + R_{j}^{k},$ $k = 1, 2, \dots, K; j = 1, 2, \dots, J - 1,$ where $E_{j}^{k} = u (x_{j}, t_{k}) - u_{j}^{k}$ .

For $k = 0, 1, \dots, K$ , define the following grid functions: $E^{k} (x) = \{\begin{matrix} E_{j}^{k}, & when & x \in (x_{j - \frac{1}{2}}, x_{j + \frac{1}{2}}], j = 1, 2, \dots, J - 1; \\ 0, & when & x \in [0, \frac{h}{2}] ⋃ (L - \frac{h}{2}, L], \end{matrix}$ and $R^{k} (x) = \{\begin{matrix} R_{j}^{k}, & when & x \in (x_{j - \frac{1}{2}}, x_{j + \frac{1}{2}}], j = 1, 2, \dots, J - 1; \\ 0, & when & x \in [0, \frac{h}{2}] ⋃ (L - \frac{h}{2}, L], \end{matrix}$ respectively. Then, $E^{k} (x)$ and $R^{k} (x)$ can be developed Fourier

Stability of the numerical scheme for solving MFDE

In this subsection, we research stability of the numerical scheme (8), (9), (10), to this end consider the following difference equation $(1 + \frac{1}{12} δ_{x}^{2}) ρ_{j}^{k} - (1 + \frac{1}{12} δ_{x}^{2}) ρ_{j}^{k - 1} = μ_{α} \sum_{l = 0}^{k} λ_{α}^{(l)} δ_{x}^{2} ρ_{j}^{k - l} + μ_{β} \sum_{l = 0}^{k} λ_{β}^{(l)} δ_{x}^{2} ρ_{j}^{k - l},$ $k = 1, 2, \dots, K; j = 1, 2, \dots, J - 1,$ where $ρ_{j}^{k} = p_{j}^{k} - {\tilde{p}}_{j}^{k}$ , whereas ${\tilde{p}}_{j}^{k}$ is an approximation for $p_{j}^{k}$ .

For $k = 0, 1, \dots, K$ , define the following grid function: $ρ^{k} (x) = \{\begin{matrix} ρ_{j}^{k}, & when & x \in (x_{j - \frac{1}{2}}, x_{j + \frac{1}{2}}], j = 1, 2, \dots, J - 1; \\ 0, & when & x \in [0, \frac{h}{2}] ⋃ (L - \frac{h}{2}, L], \end{matrix}$ then $ρ^{k} (x)$ can be developed Fourier series $ρ^{k} (x) = \sum_{l = - \infty}^{\infty} ζ_{k} (l) e^{i 2 π lx / L}, k = 0, 1, \dots, K,$ where $ζ_{k} (l) = \frac{1}{L} \int_{0}^{L} u^{k} (x) e^{- i 2}$

Numerical experiment

In order to verify the theoretical analysis results, we now apply the numerical scheme (8), (9), (10) to solve the following initial-boundary value problem of a modified fractional diffusion equation $\frac{\partial p (x, t)}{\partial t} = (\frac{\partial^{1 - α}}{\partial t^{1 - α}} + \frac{\partial^{1 - β}}{\partial t^{1 - β}}) \frac{\partial^{2} p (x, t)}{\partial x^{2}} + f (x, t),$ $0 < t ⩽ 1, 0 < x < 1,$ $p (x, 0) = 0, 0 ⩽ x ⩽ 1 .$ $p (0, t) = t^{2}, p (1, t) = {et}^{2}, 0 ⩽ t ⩽ 1,$ where $f (x, t) = 2 e^{x} (t - \frac{t^{1 + α}}{Γ (2 + α)} - \frac{t^{1 + β}}{Γ (2 + β)})$ . The exact solution of the problem (29), (30), (31) is $p (x, t) = e^{x} t^{2} .$

Let that the maximum error of the numerical solutions $E_{\infty} = \max_{0 ⩽ k ⩽ K} \{‖ E^{k} ‖_{2}\} = O (τ^{a} + h^{b}) .$ Assume that

Conclusion

In this paper, a numerical scheme with second order temporal accuracy and fourth order spatial accuracy for solving a modified fractional diffusion equation has been developed; the convergence, stability and solvability of the numerical scheme have been analyzed by Fourier analysis; the numerical experiment extremely consistent with our theoretical results.

References (12)

A. Mohebbi et al.
A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term
J. Comput. Phys.
(2013)
M. Abbaszadeh et al.
A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term
Comput. Math. Appl.
(2013)
Chang-Ming Chen
Numerical methods for solving a two-dimensional variable-order modified diffusion equation
Appl. Math. Comput.
(2013)
T.A.M. Langlands
Solution of a modified fractional diffusion equation
Physica A
(2006)
F. Liu et al.
Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term
J. Comput. Appl. Math.
(2009)
Q. Liu et al.
Finite element approximation for a modified anomalous subdiffusion equation
Appl. Math. Model.
(2011)

There are more references available in the full text version of this article.

Cited by (2)

Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation
2018, Applied Mathematics and Computation
Citation Excerpt :
Compared with the numerical simulation method in [13], the numerical simulation method presented in this paper has the advantage of simple calculation. In particular, compared with the numerical simulation method in [6], the numerical simulation method presented in this paper does not require smoothness on the right end f(x, t), this will undoubtedly broaden the scope for application of the numerical simulation method, and without any condition, we can prove the solvability of the numerical simulation method by algebraic theory. By simple calculations, Lemma 2 is easy to get.
In this paper, based on the second order compact approximation formula of first order derivative, the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy is developed to simulate the modified fractional diffusion equation; the convergence, stability and solvability of the numerical simulation method are analyzed by Fourier analysis and algebraic theory, respectively; the theoretical results extremely consistent with the numerical experiment.
The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term
2017, Journal of Computational and Applied Mathematics
In this paper, the implicit midpoint method is used to solve the semi-discrete modified anomalous sub-diffusion equation with a nonlinear source term, and the weighted and shifted Grünwald–Letnikov difference operator and the compact difference operator are applied to approximate the Riemann–Liouville fractional derivative and space partial derivative respectively, then the new numerical scheme is constructed. The stability and the convergence of this method are analyzed. Numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results.

View full text

Numerical scheme with high order accuracy for solving a modified fractional diffusion equation

Abstract

Introduction

Section snippets

A numerical scheme for solving MFDE

Convergence of the numerical scheme for solving MFDE

Stability of the numerical scheme for solving MFDE

Numerical experiment

Conclusion

J. Comput. Phys.

Comput. Math. Appl.

Appl. Math. Comput.

Physica A

J. Comput. Appl. Math.

Appl. Math. Model.