Research Article  Open Access
Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order
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 [1–7]. 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 [13–16] 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 [17–21]. 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 RiemannLiouville 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 RiemannLiouville 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 firstorder 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 fourthorder 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 ).


Example 2. Consider the following fourthorder 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 ).


Example 3. Consider the following singular twodimensional 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 ).

