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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 518343, 11 pages

http://dx.doi.org/10.1155/2014/518343
Research Article

Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

1Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Malaysia

2Department of Mathematics, Faculty of Science, University Putra Malaysia, 4300 Serdang, Selangor, Malaysia

Received 5 March 2014; Revised 16 May 2014; Accepted 10 June 2014; Published 16 July 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 Asma Ali Elbeleze et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

1. Introduction

In recent years, considerable attention has been devoted to the study of the fractional calculus and its numerous applications in many areas such as physics and engineering. The applications of fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, and signal processing can be successfully modeled by linear or nonlinear FDEs [17]. Further, fractional partial differential equations appeared in many fields of engineering and science, including fractals theory, statistics, fluid flow, control theory, biology, chemistry, diffusion, probability, and potential theory [8, 9].

The singular partial differential equations of fractional order (FSPDEs), as generalizations of classical singular partial differential equations of integer order (SPDEs), are increasingly used to model problems in physics and engineering. Consequently, considerable attention has been given to the solution of singular partial differential equations of fractional order. Finding approximate or exact solutions of SPDEs is an important task. Except for a limited number of these equations, we have difficulty in finding their analytical solutions. Therefore, there have been attempts to find methods for obtaining approximate solutions. Several such techniques have drawn special attention, such as variational iteration method [10], homotopy analysis method [11], and homotopy iteration method [12].

The variational iteration method (VIM) was proposed by He [1316] due to its flexibility and convergence and efficiently works with different types of linear and nonlinear partial differential equations of fractional order and gives approximate analytical solution for all these types of equations without linearization or discretization; many author have been studying it; for example, see [1721]. In this paper, we discuss the VIM for solving FSPDEs and obtain the convergence results of this method. The contribution of this work can be summarized in three points.(1)Based on the sufficient condition that guarantees the existence of a unique solution to our problem (see Theorem 6) and using the series solution, convergence of VIM is discussed (see Theorem 7).(2)Using point one, the maximum absolute truncated error of series solution of VIM is estimated (see Theorem 8).(3)Some numerical examples are given. Consider fractional singular partial differential equations with variable coefficients where the variable coefficients subject to initial conditions and boundary conditions where is the fractional derivative in the Caputo sense, , and and are continuous. The , , and are linear bounded operator; that is, it is possible to find numbers such that ,  ,  . Equation (1) can be written as where .

2. Preliminaries

In this section, we give some basic definitions and properties of fractional calculus theory used in this paper.

Definition 1. A real function is said to be in space if there exists a real number , such that , where , and it is said to be in the space if , .

Definition 2. The Riemann-Liouville fractional integral operator of order of a function is defined as In particular .

For and , some properties of the operator are(1) ,(2) ,(3) .

Definition 3. The Caputo fractional derivative of , is defined as

Lemma 4. If , , then the following two properties hold: (1) ,(2) .

Lemma 5. Suppose that and their partial derivatives are continuous; then the fractional derivative, , is bonded.

Proof. We need to prove that it is possible to find number such that . From the definition of Caputo fractional derivative above we have where .

3. Analysis of the Variational Iteration Method

To solve the fractional singular partial differential equations (4) by using the variational iteration method, with initial and boundary conditions (2) and (3), where , we construct the following correction functional: or is the Riemann-Liouville fractional integral operator of order , with respect to variable , and is a general Lagrange multiplier which can be identified as optimally variational theory [22], and are considered as restricted variation; that is, .

Making the above correction functional stationary, the following condition can be obtained: and yields to Lagrange multiplier We obtain the following iteration formula by substitution of (11) in (9) That is, This yields the following iteration formula:

The initial approximation can be chosen by the following manner which satisfies initial conditions: where   .

We can obtain the following first-order approximation by substitution of (15) into (14)

Finally, by substituting the constant values of and into (16), we have the results as the first approximate solutions of (4) with (2) and (3).

3.1. Convergence Analysis
3.1.1. Existence and Uniqueness Theorem

Define contentious mapping, and the function exists with continuous and bounded derivatives, where is the Banach space , the space of all continuous functions on with the norm and satisfies Lipschitz condition with Lipschitz constant , such that

Theorem 6. Let satisfy the Lipschitz condition (18) then the problem (4) with (2) and (3) has unique solution , whenever .

Proof. (1) The existence of the solution. From equation (4) we have

The mapping is defined as

Let ; then where , then we get therefore the mapping is contraction, and there exists unique solution to problem (4). (2)The uniqueness of the solution (see [23]).

3.1.2. Proof of Convergence

Theorem 7. Suppose that is Banach space and satisfies condition (18). Then, the sequence (14) converges to the solution of (4) with (2) and (3).

Proof. Defined is the Banach space, the space of all continuous functions on with the norm We need to show that is a Cauchy sequence in this Banach space: where Finally, we have where are constants and

Let . Then

From the triangle inequality, we have

Since , so , and then

But ; then as . We conclude that is a Cauchy sequence in , so the sequence converges and the proof is complete.

3.1.3. Error Analysis

Theorem 8. The maximum absolute error of the approximate solution to problem (4)-(3) is estimated to be where

Proof. From Theorem (9) and inequality (30) we have as ; then and where , and thus, the maximum absolute error in the interval is This completes the proof.

4. Numerical Examples

Example 1. Consider the following fourth-order fractional singular partial differential equation: With initial conditions and boundary conditions the exact solution in special case is and we solve the problem (36) by variational iteration method. According to variational iteration method, formula (14) for (36) can be expressed in the following form:

Suppose that an initial approximation has the following form which satisfies the initial conditions: Now by iteration formula (16), we obtain the following approximations: The second approximation takes the following form:

Table 1 shows the absolute error of VIM solution of example (36) (when ,   , and ), while Table 2 shows the maximum absolute truncated error of VIM solution (using Theorem 8) at different values of (when ).

tab1
Table 1: Absolute error.
tab2
Table 2: Maximum absolute error.

Example 2. Consider the following fourth-order fractional singular partial differential equation: With initial conditions and boundary conditions the exact solution in special case is

According to variational iteration method, formula (14) for (44) can be expressed in the following form: Suppose that an initial approximation has the following form which satisfies the initial condition: Now by iteration formula (48), we obtain the first approximation and second approximation

Table 3 shows the absolute error of VIM solution of example (37) (when ,   , and ), while Table 4 shows the maximum absolute truncated error of VIM solution (using Theorem 8) at different values of (when ).

tab3
Table 3: Absolute error.
tab4
Table 4: Maximum absolute error.

Example 3. Consider the following singular two-dimensional partial differential equation of fractional order: With initial conditions and boundary conditions the exact solution in special case is

According to variational iteration method, formula (14) for (52) can be expressed in the following form: Suppose that an initial approximation has the following form which satisfies the initial conditions: Now by iteration formula (56), we obtain the following approximations:

The second approximation takes the following form:

Table 5 shows the absolute error of VIM solution of example (38) (when ,   , and ), while Table 6 shows the maximum absolute truncated error of VIM solution (using Theorem 8, resp.) at different values of (when ).

tab5
Table 5: Absolute error.
tab6
Table 6: Maximum absolute error.

5. Conclusion

The variational iteration method has been known as powerful tools for solving many equations in fractional calculus such as ordinary equations, partial differential equations, integrodifferential equations, and so many other equations. In this paper, this method has been analyzed with an aim to investigate the conditions which result in the convergence of generated series solutions of the singular partial differential equations of fractional order. The theorems outlined in the paper have proved that the approximate solutions successfully converge to the exact solution. We consider three examples to verify convergence hypothesis simplicity of the method. From the results we see that the exact error coincides with the approximate error obtained from using the theorems; for example, see Tables 1, 2, 3, and 4. Further, the high agreement of the numerical results so obtained between the variational iteration method and the exact solution in all examples reinforces the conclusion that the efficiency of this method and related phenomena give the method much wider applicability. Furthermore, the results obtained by proposed method confirm the robustness and efficiency of it. And we hope that the work in this paper is a step in this direction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors express their sincere thanks to the referees for the careful and noteworthy reading of the paper, and the very helpful suggestions that improved the paper substantially.

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