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

and Applied Analysis 3 boundary conditions (2) and (3), where ‖ cDα t u(t)‖ = M‖u‖, we construct the following correction functional: u n+1 (x, y, z, t) = u n (x, y, z, t) + J α t [ c D α t u (x, y, z, t) − f ((x, y, z, t) , u n (x, y, z, t) , D n 1 x u (x, y, z, t) , D n 2 y u n (x, y, z, t) , D n 3 z u n (x, y, z, t))] (8) or u n+1 (x, y, z, t) = u n (x, y, z, t) + 1 Γ (α)


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][2][3][4][5][6][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][14][15][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][18][19][20][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.

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

Definition 1.
A real function ( ), > 0 is said to be in space , ∈ if there exists a real number > , such that ( ) = 1 ( ), where 1 ( ) ∈ (0, ∞), and it is said to be in the space if ∈ , ∈ .

Analysis of the Variational Iteration Method
To solve the fractional singular partial differential equations (4) by using the variational iteration method, with initial and Abstract and Applied Analysis 3 boundary conditions (2) and (3), where ‖ ( )‖ = ‖ ‖, we construct the following correction functional: 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], and̃( , ) are considered as restricted variation; that is,̃( , ) = 0.
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) +1 ( , , , ) That is, The initial approximation 0 can be chosen by the following manner which satisfies initial conditions: where 0 = 0 ( , , ) , 1 = 1 ( , , ) .

Abstract and Applied Analysis
We can obtain the following first-order approximation by substitution of (15) into (14) 1 ( , , , ) Finally, by substituting the constant values of 0 and 1 into (16), we have the results as the first approximate solutions of (4) with (2) and (3).
and satisfies Lipschitz condition with Lipschitz constant , such that Proof.
Abstract and Applied Analysis 5 We need to show that { } is a Cauchy sequence in this Banach space: Finally, we have where , , , Γ( ) are constants and From the triangle inequality, we have Since 0 < < 1, so 1 − − < 1, and then But ‖ 1 − 0 ‖ < ∞; then ‖ − ‖ → 0 as → ∞. We conclude that is a Cauchy sequence in [ ], so the sequence converges and the proof is complete.

Numerical Examples
Example 1. Consider the following fourth-order fractional singular partial differential equation: With initial conditions Suppose that an initial approximation has the following form which satisfies the initial conditions: ) .

Example 2.
Consider the following fourth-order fractional singular partial differential equation: the exact solution in special case = 2 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 . . .
(51) Table 3 shows the absolute error of VIM solution of example (37) (when = 1.5, = 0.1, and = 2), while Table 4 shows the maximum absolute truncated error of VIM solution (using Theorem 8) at different values of (when = 2). Example 3. Consider the following singular two-dimensional partial differential equation of fractional order: 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: Γ (2 + 2) ) . . .

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.