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This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

r-Modified Crank-Nicholson difference scheme for fractional parabolic PDE

Allaberen Ashyralyev12 and Zafer Cakir3*

Author Affiliations

1 Department of Mathematics, Fatih University, Istanbul, Turkey

2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan

3 Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey

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Boundary Value Problems 2014, 2014:76  doi:10.1186/1687-2770-2014-76

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/76


Received:2 December 2013
Accepted:18 March 2014
Published:31 March 2014

© 2014 Ashyralyev and Cakir; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

The second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation are presented using by r-modified Crank-Nicholson difference scheme. Stability estimate for the solution of this difference scheme is obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations. Numerical results for this scheme and the Crank-Nicholson scheme are compared in test examples.

1 Introduction

At present, there is a huge number of theoretical and applied works devoted to the study of fractional differential equations. Solutions of various problems for fractional differential equations can be found, for example, in the monographs of Podlubny [1], Kilbas, Srivastava, and Trujillo [2], Diethelm [3], and in [4-11]. These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations.

Many problems in fluid flow, dynamical and diffusion processes, control theory, mechanics, and other areas of physics can be reduced fractional partial differential equations.

In [12] the simple connection of fractional derivatives with fractional powers of first order differential operator was presented. This approach is important to obtain the formula for the fractional difference derivative. Presently, many mathematicians apply this approach and operator tools to investigate various problems for fractional partial differential equations which appear in applied problems (see, e.g., [13-20] and the references therein).

In previous paper [17] authors investigated stability estimates for Crank-Nicholson schemes for the Dirichlet problem for the fractional parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M1">View MathML</a>

(1.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M2">View MathML</a> is the standard Riemann-Liouville’s derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M3">View MathML</a> and Ω is the open cube in the m-dimensional Euclidean space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M4">View MathML</a>

with boundary S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M7">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M8">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M7">View MathML</a>) are given smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M12">View MathML</a>.

In [18] the authors investigated stability estimates for Crank-Nicholson schemes for the Neumann problem for the fractional parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M13">View MathML</a>

(1.2)

The role played by stability inequalities (well posedness) in the study of boundary-value problems for parabolic partial differential equations is well known (see, e.g., [21-26]).

In the present paper, we consider an r-modified Crank-Nicholson difference scheme of the above mentioned two problems (1.1), (1.2). This r-modified scheme is of the second order of accuracy in t and in space variables difference schemes for the approximate solution of problems. The stability estimate for the solution of this difference scheme is established. We use a procedure of a modified Gauss elimination method for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations.

2 Stability of difference scheme

Let us define the grid space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M14">View MathML</a>

We introduce the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M15">View MathML</a> of the grid function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M16">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M17">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M18">View MathML</a>

To the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M19">View MathML</a> generated by problem (1.1) or (1.2), respectively, we assign the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M21">View MathML</a>

(2.1)

acting in the space of grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M22">View MathML</a>, satisfying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M23">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M24">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M25">View MathML</a>). It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20">View MathML</a> is a self-adjoint positive definite operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M27">View MathML</a>. Here,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M28">View MathML</a>

With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M20">View MathML</a>, we arrive at the initial boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M30">View MathML</a>

(2.2)

for a finite system of ordinary fractional differential equations.

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M31">View MathML</a>

Applying the second order of the approximation formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M32">View MathML</a>

(2.3)

for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M33">View MathML</a>

(see [16]) and the Crank-Nicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to t and to x. Here,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M34">View MathML</a>

Now, we introduce the second order accuracy r-modified Crank-Nicholson difference scheme in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M35">View MathML</a>

(2.4)

for the approximate solution of problem (2.2).

Theorem 2.1Letτand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M36">View MathML</a>be sufficiently small positive numbers. Then the solutions of the difference scheme (2.4) satisfy the following stability estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M37">View MathML</a>

(2.5)

whereCdoes not depend onτ, h, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39">View MathML</a>.

Proof Consider the difference scheme (2.4). We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M40">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M41">View MathML</a>

We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M42">View MathML</a>

(2.7)

Let us write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M43">View MathML</a>. Using (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M44">View MathML</a>

Now, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39">View MathML</a>. From the triangle inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M47">View MathML</a>

(2.8)

Applying the triangle inequality and the estimates [24]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M48">View MathML</a>

(2.9)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M49">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M50">View MathML</a>

(2.11)

Hence, applying the difference analog of the integral inequality and inequalities (2.8), (2.10), and (2.11), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M51">View MathML</a>

(2.12)

The proof of estimate (2.5) for the solution of (2.4) follows from (2.6), (2.9), and (2.12). Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M53">View MathML</a> are independent from τ, h, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M39">View MathML</a>. Theorem 2.1 is proved. □

3 Numerical analysis

We consider two examples for numerical results.

Example 3.1 We consider the following initial boundary value problem with Dirichlet condition for the one-dimensional fractional parabolic partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M56">View MathML</a>

(3.1)

It is clear that the exact solution of problem (3.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M57">View MathML</a>

Applying the r-modified Crank-Nicholson difference scheme (2.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M59">View MathML</a> is defined by (2.3) for any n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M60">View MathML</a>. We can rewrite it in the system of equations with matrix coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M61">View MathML</a>

(3.2)

Here and in the sequel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M62">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M63">View MathML</a> zero matrix,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M64">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M65">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M66">View MathML</a>

(3.5)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M67">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M68">View MathML</a>

(3.6)

For solving (3.2) we use a modified Gauss elimination method [27]. Hence, we seek a solution of the matrix equation in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M69">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M71">View MathML</a>) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M72">View MathML</a> square matrices and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M73">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M71">View MathML</a>) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M75">View MathML</a> column matrices defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M76">View MathML</a>

(3.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M77">View MathML</a>

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M79">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M72">View MathML</a> zero matrix and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M81">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M82">View MathML</a> zero matrix and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M83">View MathML</a>.

Example 3.2 We consider the following initial boundary value problem with Neumann condition for the one-dimensional fractional parabolic partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M84">View MathML</a>

(3.10)

It is clear that the exact solution of problem (3.10) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M85">View MathML</a>

Applying the r-modified Crank-Nicholson difference scheme (2.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M86">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M59">View MathML</a> is defined by (2.3). We can rewrite it in the form of a system of equations with matrix coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M88">View MathML</a>

Here, A, B, C are defined by (3.3), (3.4), and (3.5):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M89">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M67">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M91">View MathML</a>

(3.11)

For solving this matrix equation we will use the same method as for Example 3.1. Namely, we use (3.7), (3.8), (3.9), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M92">View MathML</a>

Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of the modification parameter r, and discretization parameters N and M. Besides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M93">View MathML</a> represents the numerical solutions of these difference schemes at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M94">View MathML</a>. The error is computed by the following formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M95">View MathML</a>

Table 1 and Table 2 are constructed for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/76/mathml/M96">View MathML</a>, respectively. As can be seen from Table 1, the r-modified Crank-Nicholson difference scheme is more accurate than the Crank-Nicholson and Rothe difference schemes. Table 2 shows that the r-modified Crank-Nicholson difference scheme has the same order error as the Crank-Nicholson difference scheme.

Table 1. Error analysis for Dirichlet problem

Table 2. Error analysis for Neumann problem

4 Conclusion

In this study, the second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation is investigated. We have obtained a stability estimate for the solution of this difference scheme. The theoretical statements for the solution of this difference scheme for one-dimensional parabolic equations are supported by numerical examples obtained by computer.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZC proposed the main idea of this paper, obtained the theoretical and numerical results. AA designed the study and interpreted the results. All authors read and approved the final manuscript.

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