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FDM for fractional parabolic equations with the Neumann condition

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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Advances in Difference Equations 2013, 2013:120  doi:10.1186/1687-1847-2013-120


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/120


Received:2 January 2013
Accepted:11 April 2013
Published:25 April 2013

© 2013 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 cited.

Abstract

In the present study, the first and second order of accuracy stable difference schemes for the numerical solution of the initial boundary value problem for the fractional parabolic equation with the Neumann boundary condition are presented. Almost coercive stability estimates for the solution of these difference schemes are obtained. The method is illustrated by numerical examples.

MSC: 34K37, 35R11, 35B35, 39A14, 47B48.

Keywords:
fractional parabolic equations; Neumann condition; difference schemes; stability

1 Introduction

Mathematical modeling of fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Numerical methods and theory of solutions of the problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1-31] and the references given therein).

The method of operators as a tool for investigation of the well-posedness of boundary value problems for parabolic partial differential equations is well known (see, e.g., [32-41]). In paper [42], the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M1">View MathML</a>

(1.1)

for the fractional differential equation in a Banach space E with the strongly positive operator A was investigated. This fractional differential equation corresponds to the Basset problem [43]. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M2">View MathML</a> is the standard Riemann-Liouville’s derivative of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M3">View MathML</a>.

The well-posedness of (1.1) in spaces of smooth functions was established. The coercive stability estimates for the solution of the 2mth order multidimensional fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable were obtained.

In paper [44], the stable first order of accuracy difference scheme for the approximate solution of initial value problem (1.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M4">View MathML</a>

(1.2)

was presented. Here (see, [45]),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M5">View MathML</a>

(1.3)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M6">View MathML</a> be the linear space of mesh functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M7">View MathML</a> with values in the Banach space E. Next, on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M6">View MathML</a> we introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M9">View MathML</a> with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M10">View MathML</a>

The well-posedness of (1.2) in difference analogues of spaces of smooth functions was established. Namely, we have the following theorems.

Theorem 1.1LetAbe a strongly positive operator in a Banach spaceE. Then, for the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12">View MathML</a>of initial value problem (1.2) the stability inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M13">View MathML</a>

(1.4)

Theorem 1.2LetAbe a strongly positive operator in a Banach spaceE. Then, for the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12">View MathML</a>of initial value problem (1.2) the almost coercive stability inequality is valid:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M16">View MathML</a>

(1.5)

Here, and in future, positive constants, which can differ in time (hence: not a subject of precision) will be indicated with an M. On the other hand <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M17">View MathML</a> is used to focus on the fact that the constant depends only on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M18">View MathML</a> .

Finally, the coercive stability and almost coercive stability estimates for the solution of difference schemes the first order of approximation in t for the 2mth order multidimensional fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable were obtained.

In the present paper, applying the second order of approximation formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M19">View MathML</a>

(1.6)

for

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M20">View MathML</a>

and using the Crank-Nicholson difference scheme for parabolic equations, we present the second order of accuracy difference scheme

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M21">View MathML</a>

(1.7)

for the approximate solution of initial value problem (1.1). Here,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M22">View MathML</a>

The well-posedness of (1.7) in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12">View MathML</a> is established. In applications, the initial boundary value problem for the fractional parabolic equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M24">View MathML</a>

(1.8)

is considered. Here, Ω is the open cube in the m-dimensional Euclidean space

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M25">View MathML</a>

with boundary S, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M26">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M27">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M28">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M29">View MathML</a>) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M30">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M29">View MathML</a>) are given smooth functions and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M33">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M35">View MathML</a> is the normal vector to S.

The first and second order of accuracy difference schemes for the approximate solution of problem (1.8) are presented. The almost coercive stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes for one-dimensional fractional parabolic equations are supported by numerical examples.

2 The well-posedness of difference scheme

It is clear that the following representation formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M36">View MathML</a>

(2.1)

holds for the solution of problem (1.7). Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M37">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M38">View MathML</a>.

Theorem 2.1LetAbe a strongly positive operator in a Banach spaceE. Then, for the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12">View MathML</a>of initial value problem (1.2) the following stability inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M41">View MathML</a>

(2.2)

Proof Using formulas (2.1) and (1.6), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M42">View MathML</a>

(2.3)

Now, let us first estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M43">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M44">View MathML</a>. Using formula (2.3) and the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M45">View MathML</a>

(2.4)

we get

(2.5)

(2.6)

Now we consider the case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M48">View MathML</a>. Applying formula (2.3), the triangle inequality and estimates [46]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M49">View MathML</a>

(2.7)

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M50">View MathML</a>

(2.8)

Applying the difference analogue of the integral inequality and inequalities (2.5), (2.6) and (2.8), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M51">View MathML</a>

(2.9)

Using the triangle inequality and equation (1.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M52">View MathML</a>

(2.10)

Estimate (2.2) follows from estimates (2.9) and (2.10). Theorem 2.1 is proved. □

Theorem 2.2LetAbe a strongly positive operator in a Banach spaceE. Then, for the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M11">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M12">View MathML</a>of initial value problem (1.2) the almost coercive stability inequality is valid:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M55">View MathML</a>

(2.11)

Proof Using formula (2.1), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M56">View MathML</a>

(2.12)

The proof of estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M57">View MathML</a>

(2.13)

for the solution of initial value problem (1.2) is based on formula (2.12) and estimate (2.2) and the following estimates [46]:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M58">View MathML</a>

Using these estimates, the triangle inequality and equation (1.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M59">View MathML</a>

(2.14)

Estimate (2.11) follows from estimates (2.13) and (2.14). Theorem 2.2 is proved. □

3 Applications

Now, we consider the applications of Theorems 2.1 and 2.2 to initial boundary value problem (1.8). The discretization of problem (1.8) is carried out in two steps. In the first step, let us define the grid space

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M60">View MathML</a>

We introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M61">View MathML</a> of the grid function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M62">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M63">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M64">View MathML</a>

To the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M65">View MathML</a> generated by problem (1.8), we assign the difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66">View MathML</a> by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M67">View MathML</a>

acting in the space of grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M68">View MathML</a>, satisfying the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M69">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M70">View MathML</a>. Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M71">View MathML</a> is the first or second order of approximation of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M72">View MathML</a>. It is known that (see, [47,48]) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66">View MathML</a> is a strongly positive definite operator in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M74">View MathML</a>. With the help of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66">View MathML</a> we arrive at the initial boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M76">View MathML</a>

(3.1)

for a finite system of ordinary fractional differential equations.

In the second step, applying the first order of approximation formula defined by (1.3) for

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M77">View MathML</a>

and using the first order of accuracy stable difference scheme for parabolic equations, we can present the first order of accuracy difference scheme

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M78">View MathML</a>

(3.2)

for the approximate solution of problem (1.8).

Moreover, applying the second order of approximation formula defined by (1.6) for

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M79">View MathML</a>

and using the Crank-Nicholson difference scheme for parabolic equations, we can present the second order of accuracy difference scheme

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M80">View MathML</a>

(3.3)

for the approximate solution of problem (1.8).

Theorem 3.1Letτand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M81">View MathML</a>be sufficiently small numbers. Then the solutions of difference scheme (3.2) satisfy the following almost coercive stability estimates:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M82">View MathML</a>

The proof of Theorem 3.1 is based on the abstract Theorem 1.2 and on the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M83">View MathML</a>

(3.4)

as well as on the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85">View MathML</a>[47,48], along with the following theorem on the almost coercivity inequality for the solution of the elliptic difference equation in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85">View MathML</a>.

Theorem 3.2[49]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M81">View MathML</a>be sufficiently small number. Then, for the solutions of the elliptic difference equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M88">View MathML</a>

(3.5)

the following almost coercivity inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M89">View MathML</a>

is valid.

Theorem 3.3Letτand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M90">View MathML</a>be sufficiently small numbers. Then the solutions of difference scheme (3.3) satisfy the following almost coercive stability estimates:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M91">View MathML</a>

The proof of Theorem 3.3 is based on the abstract Theorem 2.2 and on estimate (3.4) and on the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M66">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85">View MathML</a> and on Theorem 3.2 on the almost coercivity inequality for the solution of the elliptic difference equation in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M85">View MathML</a>.

Note that one has not been able to get a sharp estimate for the constants figuring in the almost coercive stability estimates of Theorems 3.1 and 3.3. Therefore, our interest in the present paper is studying the difference schemes (3.2) and (3.3) by numerical experiments. Applying these difference schemes, the numerical methods are proposed in the following section for solving the one-dimensional fractional parabolic partial differential equation. The method is illustrated by numerical experiments.

4 Numerical results

For the numerical result, the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M95">View MathML</a>

(4.1)

for the one-dimensional fractional parabolic partial differential equation is considered. The exact solution of problem (4.1) is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M96">View MathML</a>.

4.1 First order of accuracy difference scheme

Applying difference scheme (3.2), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M97">View MathML</a>

It can be rewritten in the matrix form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M98">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M99">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M100">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M101">View MathML</a>

So, we have the second order difference equation with respect to n matrix coefficients. This type system was developed by Samarskii and Nikolaev [50]. To solve this difference equation, we have applied a procedure for difference equation with respect to k matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M102">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M103">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M104">View MathML</a>) are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M105">View MathML</a> square matrices and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M106">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M104">View MathML</a>) are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M108">View MathML</a> column matrices defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M109">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M111">View MathML</a> is the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M112">View MathML</a> identity matrix and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M113">View MathML</a> is the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M114">View MathML</a> zero matrix.

4.2 Second order of accuracy difference scheme

Applying the formulas

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M115">View MathML</a>

and using difference scheme (3.3), we obtain the second order of accuracy difference scheme in t and in x

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M116">View MathML</a>

Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M117">View MathML</a> is the fractional difference derivative defined by the formula (1.6). It can be rewritten in the matrix form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M118">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M119">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M120">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M121">View MathML</a>

For the solution of the matrix equation (4.2), we use the same algorithm as in the first order of accuracy difference scheme, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M122">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M120">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M124">View MathML</a>

4.3 Error analysis

Finally, we give the results of the numerical analysis. The error is computed by the following formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M125">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M126">View MathML</a> represents the exact solution and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M127">View MathML</a> represents the numerical solutions of these difference schemes at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M128">View MathML</a>.

The numerical solutions are recorded for different values of N and M. Table 1 is constructed for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M129">View MathML</a>, respectively.

Table 1. Comparison of errors

Thus, the results show that, by using the Crank-Nicholson difference scheme increases faster then the first order of accuracy difference scheme.

5 Conclusion

In the present study, the second order of accuracy difference scheme for the approximate solution of initial value problem (1.1) is presented. A theorem on almost coercivity of this difference scheme in maximum norm is established. Almost coercive stability estimates for the solution of the first and second order of accuracy stable difference schemes for the numerical solution of the initial boundary value problem for the fractional parabolic equation with the Neumann boundary condition are obtained. Of course, stability estimates permits us to obtain the convergence of difference schemes for the numerical solution of the initial boundary value problem for the fractional parabolic equation with the Neumann boundary condition. Moreover, the Banach fixed-point theorem and method of the present paper enables us to obtain the estimate of convergence of difference schemes of the first and second order of accuracy for approximate solutions of the initial-boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M130">View MathML</a>

for semilinear fractional parabolic partial differential equations with smooth <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M131">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/120/mathml/M132">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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