Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation

Abstract

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

1. Introduction

The topic of fractional calculus (theory of integration and differentiation of an arbitrary order) was started over 300 years ago. Recently, fractional differential equations have attracted many scientists and researchers due to the tremendous use in fluid mechanics, mathematical biology, electrochemistry, and physics. For example, differential equations with fractional order have recently proved to be suitable tools to modeling of many physical phenomena [1] and the fluid-dynamic traffic model with fractional derivative [2], and nonlinear oscillation of earthquake can be modeled with fractional derivatives [3].

There are several types of time fractional differential equations. (1) Fractional Klein-Gordon equations 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼−𝜕²𝑢(𝑥,𝑡)𝜕𝑥²+𝑎𝑢(𝑥,𝑡)+𝑏𝑢²+𝑐𝑢³=𝑓(𝑥,𝑡),𝑥∈𝑅. (1) This model is obtained by replacing the order time derivative with the fractional derivative of order 𝛼. The linear and nonlinear Klein-Gordon equations are used to modeling many problems in classical and quantum mechanics and condensed matter physics. For example, nonlinear sine Klein-Gordon equation models a Josephson junction [4, 5]. (2) Fractional Burger's equation 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝜕²𝑢(𝑥,𝑡)𝜕𝑥²+𝜕𝑢(𝑥,𝑡)()𝜕𝑥+𝑓𝑥,𝑡,𝑥∈𝑅. (2) In general, fractional Burger's model is derived from well-known Burger's equation model by replacing the ordinary time derivatives to fractional order time derivatives. Reference [6] has investigated unsteady flows of viscoelastic fluids with fractional Burger's model and fractional generalized Burger's model through channel (annulus) tube and solutions for velocity field.(3)Fractional Black-Scholes European option pricing equations In financial model the fractional Black-Scholes equation is obtained by replacing the order of derivative with a fractional derivative order [10]. 𝜕^𝛼𝑣𝜕𝑡^𝛼+𝜎𝑥²2𝜕²𝑣𝜕𝑥²+𝑟(𝑡)𝑥𝜕𝑣(𝑡)()𝜕𝑥−𝑟𝑣=0,𝑥,𝑡∈𝑅⁺×(0,𝑇),0<𝛼≤1, (3) where 𝑣(𝑥,𝑡) is the European call option price at asset price 𝑥 and at time 𝑡, 𝑇 is the maturity, 𝑟(𝑡) is the risk-free interest rate, and 𝜎(𝑥,𝑡) represents the volatility function of underlying asset. The payoff functions are 𝑣_𝑐()(),𝑣𝑥,𝑡=max𝑥−𝐸,0_𝑝(𝑥,𝑡)=max(𝐸−𝑥,0), (4) where 𝑣_𝑐(𝑥,𝑡) and 𝑣_𝑝(𝑥,𝑡) are the value of the European call and put options, respectively, 𝐸 denotes the expiration price for the option, and the function max(𝑥,0) gives the large value between 𝑥 and 0. The Black-Scholes equation is one of the most significant mathematical models for a financial market. This equation is used to submit a reasonable price for call or put options based on factors such as underlying stock volatility and days to expiration.

Formerly, [7] investigated approximate analytical solution of fractional nonlinear Klein-Gordon equation (1) when 0<𝛼≤1 by using HPM, while [8] solved this equation by using HAM also when 1≤𝛼<2. Reference [9] solved the coupled Klein-Gordon equation with time fractional derivative by ADM. References [10, 11] solved fractional Black-Scholes equations by using HPM using Sumudu and Laplace transforms, respectively. Reference [12] gave the exact solution of fractional Burgers equation, while [13] used DTM to find the approximate and exact solution of space- and time fractional Burgers equations. Reference [14] solved this equation by using VIM.

The variational iteration method [15–29] is one of approaches to provide an analytical approximation solutions to linear and nonlinear problems. The fractional variational iteration method with Riemann-Liouville derivative was proposed by Wu and Lee [30] and applied to solve time fractional and space fractional diffusion equations. Furthermore Wu [31] explained a possible use of the fractional variational iteration method as a fractal multiscale method. Recently fractional variational iteration method has been used to obtain approximate solutions of fractional Riccati differential equation [32].

The objective of this paper is to extend the application of the fractional variational iteration method to obtain analytical approximate solution for some fractional partial differential equations. These equations include fractional Klein-Gordon equation (1), Burgers equation (2), and fractional Black-Scholes equations (3).

Motivated and inspired by the ongoing research in this field, we will consider the following time fractional differential equation: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=𝑅[𝑥]𝑢(𝑥,𝑡)+𝑞(𝑥,𝑡),0<𝛼≤1,𝑥∈𝐑,𝑡>0, (5) with initial condition 𝑢(𝑥,0)=𝑓(𝑥), (6) where 𝜕^𝛼/𝜕𝑡^𝛼 is modified Riemann-Liouville derivative [33–35] of order 𝛼 defined in Section 2, 𝑓(𝑥) and 𝑞(𝑥,𝑡) are continuous functions, 𝑅[𝑥]𝑢(𝑥,𝑡) are linear and nonlinear operators, and 𝑢(𝑥,𝑡) is unknown function.

To solve the problem (1)-(2), we consider the FVIM in this work. This method is based on variational iteration method [19, 36] and modified Riemann-Liouville derivatives proposed by Jumarie.

This paper is organized as follows. in Section 2 some basic definitions of fractional calculus theory are given. In Section 3, the solution procedure of the fractional iteration method is given; we present the application of the FVIM for some fractional partial differential equations in Section 4. The conclusions are drawn in Section 5.

2. Fractional Calculus

2.1. Fractional Derivative via Fractional Difference

Definition 1. The left-sides Riemann-Liouville fractional integral operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is a defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡,𝛼>0,𝑥>0,𝐽⁰𝑓(𝑥)=𝑓(𝑥). (7)

Definition 2. The modified Riemann-Liouville derivative [34, 35] is defined as 𝐷^𝑥_𝛼𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥₀(𝑥−𝑡)^𝑛−𝛼𝑓(𝑡)−𝑓(0)𝑑𝑡, (8) where 𝑥∈[0,1], 𝑛−1≤𝛼<𝑛, and 𝑛≥1.

Definition 3. Let 𝑓∶𝑅→𝑅, 𝑥→𝑓(𝑥) denote a continuous (but not necessarily differentiable) function, and let ℎ>0 denote a constant discretization span. Define the forward operator FW(ℎ) by the equality FW(ℎ)𝑓(𝑥)∶=𝑓(𝑥+ℎ). (9) Then the fractional difference of order 𝛼, 0<𝛼<1, of 𝑓(𝑥) is defined by the expression Δ^(𝛼)𝑓(𝑥)∶=(FW−1)^𝛼𝑓(𝑥)=^∞_𝑘=0(−1)^𝑘𝛼𝑘𝑓[𝑥+(𝑎−𝑘)ℎ], (10) and its fractional derivative of order 𝛼 is defined by the limit 𝑓^𝛼(𝑥)=lim_𝑥→0Δ^(𝛼)𝑓(𝑥)−𝑓(0)ℎ^𝛼. (11)

Equation (11) is defined as Jumarie fractional derivative of order 𝛼 which is equivalent to (8). For more details we refer the reader to [35].

For 0<𝛼≤1, some properties of the fractional modified Riemann-Liouville derivative.

Fractional Leibnitz product law: ₀𝐷^𝛼_𝑥(𝑢𝑣)=𝑢^(𝛼)𝑣+𝑢𝑣^(𝛼), (12) fractional Leibnitz formulation: ₀𝐼^𝛼_𝑥𝐷^𝛼_𝑥(𝑢𝑣)=𝑓(𝑥)−𝑓(0), (13) The fractional integration by parts formula: _𝑎𝐼^𝛼_𝑏𝑢^(𝛼)𝑣=(𝑢𝑣)|^𝑏_𝑎−_𝑎𝐼^𝛼_𝑏𝑢𝑣^(𝛼). (14)

Definition 4. Fractional derivative of compounded function [34, 35] is defined as 𝑑^𝛼𝑓≅Γ(1+𝛼)𝑑𝑓,0<𝛼<1. (15)

Definition 5 (see [34, 35]). The integral with respect to (𝑑𝑡)^𝛼 is defined as the solution of the fractional differential equation 𝑑𝑥≅𝑓(𝑥)(𝑑𝑡)^𝛼,𝑡≥0,𝑥(0)=0,0<𝛼<1. (16)

Lemma 6 (see [34, 35]). Let 𝑓(𝑥) denote a continuous function; then the solution of (2) is defined as 𝑦=^𝑥₀𝑓(𝜏)(𝑑𝜏)^𝛼=𝛼^𝑥₀(𝑥−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,0<𝛼<1, (17) that is, 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝜏)^𝛼−1𝑓(𝜏)=1𝑑𝜏(Γ(𝛼+1))^𝑥₀𝑓(𝜏)(𝑑𝜏)^𝛼. (18)

For example, with 𝑓(𝑥)=𝑥^𝛽 in (7), one obtains ^𝑥₀𝑡^𝛽(𝑑𝑡)^𝛼=Γ𝛽+1Γ(𝛼+1)Γ𝑥𝛼+𝛽+1^𝛼+𝛽,0<𝛼<1. (19)

Definition 7. The Mittag-Leffler function 𝐸_𝛼(𝑧) with 𝛼>0 is defined by the following series representation, valid in the whole complex plane [37]: 𝐸_𝛼(𝑧)=^∞₀𝑧^𝑛Γ().𝛼𝑛+1 (20)

3. Fractional Variational Iteration Method

To describe the solution procedure of fractional variational iteration method, we consider the time-fractional differential equations (1)–(3).

According to variational iteration method we construct the following correction function: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+𝐽^𝛼_𝑡𝜇𝜕^𝛼𝑢(𝑥,𝑠)𝜕𝑠^𝛼−𝑅[𝑥]𝑢(𝑥,𝑠)−𝑞(𝑥,𝑠)=𝑢_𝑛()+1𝑥,𝑡Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝜇(𝑠)𝜕^𝛼𝑢(𝑥,𝑠)𝜕𝑠^𝛼−𝑅[𝑥]𝑢(𝑥,𝑠)−𝑞(𝑥,𝑠)𝑑𝑠, (21) where 𝜇 is the general Lagrange multiplier which can be defined optimally via variational theory [22] and 𝑢(𝑥,𝑡) is the restricted variation, that is, 𝛿𝑢(𝑥,𝑡)=0.

By using (7), we obtain a new correction functional 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()+1𝑥,𝑡Γ(𝛼+1)×^𝑡₀𝜇(𝑠)𝜕^𝛼𝑢(𝑥,𝑠)𝜕𝑠^𝛼[𝑥]𝑢()()−𝑅𝑥,𝑠−𝑞𝑥,𝑠𝑑𝑠^𝛼. (22) Making the above functional stationary the following conditions can be obtained: 𝛿𝑢_𝑛+1(𝑥,𝑡)=𝛿𝑢_𝑛()+𝛿𝑥,𝑡Γ()×𝛼+1_𝑡0𝜇(𝑠)𝜕^𝛼𝑢(𝑥,𝑠)𝜕𝑠^𝛼[𝑥]𝑢()()−𝑅𝑥,𝑠−𝑞𝑥,𝑠𝑑𝑠^𝛼. (23) Now, we can get the coefficients of 𝛿𝑢 to zero: (𝑠)𝜕1+𝜇=0,^𝛼𝜇(𝑠)𝜕𝑠^𝛼=0. (24) So, the generalized Lagrange multiplier can be identified as 𝜇=−1. (25) Then we obtain the following iteration formula by substituting (25) in (23): 𝛿𝑢_𝑛+1(𝑥,𝑡)=𝛿𝑢_𝑛()−𝛿𝑥,𝑡Γ()×𝛼+1_𝑡0𝜇(𝑠)𝜕^𝛼𝑢(𝑥,𝑠)𝜕𝑠^𝛼[𝑥]𝑢()()−𝑅𝑥,𝑠−𝑞𝑥,𝑠𝑑𝑠^𝛼, (26) where 0<𝛼≤1 and 𝑢₀(𝑥,𝑡) is an initial approximation which can be freely chosen if it satisfies the initial and boundary conditions of the problem.

4. Applications

In this section, we have applied fractional variational iteration method (FVIM) to fractional partial differential equations.

Example 8. In this example we consider the following fractional nonlinear Klein-Gordon differential equation: 𝜕^𝛼𝑢𝜕𝑡^𝛼−𝜕²𝑢𝜕𝑥²+𝑢²=0,𝑡≥0,0<𝛼≤1, (27) subject to initial condition 𝑦(𝑥,0)=1+sin(𝑥). (28) Substituting (𝑎=0, 𝑏=0 and 𝑐=1) in (1). Construction the iteration formula as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢_𝑛𝜕𝑠^𝛼−𝜕²𝑢_𝑛𝜕𝑥²+𝑢²_𝑛(𝑑𝑠)^𝛼. (29) Taking the initial value 𝑢₀(𝑥,𝑡)=1+sin(𝑥) we can derive the first approximate 𝑢₁(𝑥,𝑡) as follows: 𝑢₁(𝑥,𝑡)=𝑢₀()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₀𝜕𝑠^𝛼−𝜕²𝑢₀𝜕𝑥²+𝑢²₀(𝑑𝑠)^𝛼(𝑥)−𝑡=1+sin^𝛼+1Γ()×(𝑥)𝛼+11+3sin+sin²(𝑥,𝑢₂)(𝑥,𝑡)=𝑢₁()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₁𝜕𝑠^𝛼−𝜕²𝑢₁𝜕𝑥²+𝑢²₁(𝑑𝑠)^𝛼(𝑥)−𝑡=1+sin^𝛼+1Γ()(𝑥)𝛼+11+3sin+sin²(𝑥)+𝑡^2𝛼+1Γ()(𝑥)2𝛼+111sin+12sin²(𝑥)+2sin³(𝑥),𝑢₃(𝑥,𝑡)=𝑢₂()−1𝑥,𝑡Γ(𝛼+1)×^𝑡₀𝜕^𝛼𝑢₂𝜕𝑠^𝛼−𝜕²𝑢₂𝜕𝑥²+𝑢²₂(𝑑𝑠)^𝛼(𝑥)−𝑡=1+sin^𝛼Γ()(𝑥)𝛼+11+3sin+sin²(𝑥)+𝑡^2𝛼Γ()(𝑥)2𝛼+111sin+12sin²(𝑥)+2sin³(𝑥)+𝑡^3𝛼Γ()(𝑥)3𝛼+118−57sin−160sin²(𝑥)−82sin³(𝑥)−10sin⁴(𝑥). (30) Thus, the approximate solution is 𝑢()(𝑥)−𝑡𝑥,𝑡=1+sin^𝛼Γ()×(𝑥)𝛼+11+3sin+sin²(𝑥)+𝑡^2𝛼Γ()×(𝑥)2𝛼+111sin+12sin²(𝑥)+2sin³(𝑥)+𝑡^3𝛼Γ()(𝑥)3𝛼+118−57sin−160sin²(𝑥)−82sin³(𝑥)−10sin⁴(𝑥)+⋯. (31)
In Figures 1 and 2 we have shown the surface of 𝑢(𝑥,𝑡) corresponding to the values 𝛼=0.01,0.5,1 for FVIM and HPM; the two figures indicate that the differences among VIM and HPM, and the exact solution in Example 8 are negligible when 𝛼=0.5,1 while when 𝛼=0.01 the results of VIM and HPM somewhat diverge from the exact solution.

(a)

(b)

(c)

(d)

Figure 1 

The surface shows the solution 𝑢(𝑥,𝑡) for (27) with initial condition (28): FVIM results are, respectively, (a) 𝛼=0.01 and (c) 𝛼=0.5; HPM [7] results are, respectively, (b) 𝛼=0.01 and (d) 𝛼=0.5.

(a)

(b)

(c)

Figure 2 

The surface shows the solution 𝑢(𝑥,𝑡) for (27) with initial condition (28): (a) FVIM when 𝛼=1, (b) HPM [7] when 𝛼=1, and (c) exact solution.

Example 9. We consider the one-dimensional linear inhomogeneous fractional Burger equation 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝜕𝑢−𝜕𝜕𝑥²𝑢𝜕𝑥²=2𝑡^2−𝛼Γ(3−𝛼)+2𝑥−2,𝑡>0,𝑥∈𝑅,0<𝛼≤1, (32) subject to initial condition 𝑢(𝑥,0)=𝑥². (33) By construction the iteration formula as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢_𝑛𝜕𝑡^𝛼+𝜕𝑢_𝑛−𝜕𝜕𝑥²𝑢_𝑛𝜕𝑥²−2𝑡^2−𝛼Γ())3−𝛼−2𝑥+2(𝑑𝑠^𝛼. (34) Taking the initial value 𝑢₀(𝑥,𝑡)=0 we can derive the first approximate 𝑢₁(𝑥,𝑡) as follows: 𝑢₁(𝑥,𝑡)=𝑢₀()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₀𝜕𝑡^𝛼+𝜕𝑢₀−𝜕𝜕𝑥²𝑢₀𝜕𝑥²−𝑡^2−𝛼Γ())3−𝛼−2𝑥+2(𝑑𝑠^𝛼=𝑥²+𝑡²,𝑢₂(𝑥,𝑡)=𝑢₁()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₁𝜕𝑡^𝛼+𝜕𝑢₁−𝜕𝜕𝑥²𝑢₀𝜕𝑥²−𝑡^2−𝛼Γ())3−𝛼−2𝑥+2(𝑑𝑠^𝛼=𝑥²+𝑡²⋮𝑢_𝑛(𝑥,𝑡)=𝑥²+𝑡². (35) So, the exact solution 𝑢(𝑥,𝑡)=𝑥²+𝑡² follows immediately. The exact solution is obtained by using two iterations and this is dependent on proper selection of initial guess 𝑢₀(𝑥,𝑡).

Example 10. We consider the following fractional Black-Scholes option pricing equation [38] as follows: 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕²𝑢𝜕𝑥²+(𝑘−1)𝜕𝑢𝜕𝑥−𝑘𝑢,0<𝛼≤1, (36) where 𝑘 is the risk-free interest rate subject to initial condition 𝑢(𝑥,0)=max𝑒^𝑥−1,0. (37) The exact solution for special case 𝛼=1 is given by 𝑢(𝑥,𝑡)=max𝑒^𝑥−1,0𝑒^−𝑘𝑡+max𝑒^𝑥,01−𝑒^−𝑘𝑡. (38) By construction the iteration formula as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢_𝑛𝜕𝑠^𝛼−𝜕²𝑢_𝑛𝜕𝑥²+(𝑘−1)𝜕𝑢_𝑛𝜕𝑥−𝑘𝑢_𝑛(𝑑𝑠)^𝛼. (39) Taking the initial value 𝑢₀(𝑥,𝑡)=max(𝑒^𝑥−1,0) we can derive the first approximate 𝑢₁(𝑥,𝑡) as follows: 𝑢₁(𝑥,𝑡)=𝑢₀()−1𝑥,𝑡Γ(𝛼+1)×^𝑡₀𝜕^𝛼𝑢₀𝜕𝑠^𝛼−𝜕²𝑢₀𝜕𝑥²+(𝑘−1)𝜕𝑢₀𝜕𝑥−𝑘𝑢₀(𝑑𝑠)^𝛼=max𝑒^𝑥−1,0−max𝑒^𝑥,0−𝑘𝑡^𝛼Γ()𝑒𝛼+1+max^𝑥−1,0−𝑘𝑡^𝛼Γ(),𝑢𝛼+1₂(𝑥,𝑡)=𝑢₁()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₁𝜕𝑠^𝛼−𝜕²𝑢₁𝜕𝑥²+(𝑘−1)𝜕𝑢₁𝜕𝑥−𝑘𝑢₁(𝑑𝑠)^𝛼=max𝑒^𝑥−1,0−max𝑒^𝑥×,0−𝑘𝑡^𝛼Γ()+𝛼+1−𝑘𝑡^𝛼2Γ()𝑒2𝛼+1+max^𝑥−1,0−𝑘𝑡^𝛼Γ()+𝛼+1−𝑘𝑡^𝛼2Γ()⋮𝑢2𝛼+1₃(𝑥,𝑡)=𝑢₂()−1𝑥,𝑡Γ()×𝛼+1_𝑡0𝜕^𝛼𝑢₂𝜕𝑠^𝛼−𝜕²𝑢₂𝜕𝑥²+(𝑘−1)𝜕𝑢₂𝜕𝑥−𝑘𝑢₂(𝑑𝑠)^𝛼=max𝑒^𝑥−1,0−max𝑒^𝑥×,0−𝑘𝑡^𝛼Γ()+𝛼+1−𝑘𝑡^𝛼2Γ()+2𝛼+1−𝑘𝑡^𝛼3Γ()𝑒3𝛼+1+max^𝑥×−1,0−𝑘𝑡^𝛼Γ()+𝛼+1−𝑘𝑡^𝛼2Γ()+2𝛼+1−𝑘𝑡^𝛼3Γ()⋮𝑢3𝛼+1_𝑛(𝑥,𝑡)=max𝑒^𝑥−1,0𝐸_𝛼−𝑘𝑡^𝛼+max𝑒^𝑥,01−𝐸_𝛼−𝑘𝑡^𝛼, (40) so that the solution 𝑢(𝑥;𝑡) of the problem is given by 𝑢_𝑛(𝑥,𝑡)=max𝑒^𝑥−1,0𝐸_𝛼−𝑘𝑡^𝛼+max𝑒^𝑥,01−𝐸_𝛼−𝑘𝑡^𝛼, (41) where 𝐸_𝛼(z) is Mittag-Leffler function in one parameter. Equation (41) represents the closed form solution of the fractional Black-Scholes equation (36). Now for the standard case 𝛼=1, this series has the closed form of the solution 𝑢(𝑥;𝑡)=max(𝑒^𝑥−1,0)𝑒^−𝑘𝑡+max(𝑒^𝑥,0)(1−𝑒^−𝑘𝑡), which is an exact solution of the given Black-Scholes equation (36) for 𝛼=1.

In Figure 3 we have shown the surface of 𝑢(𝑥,𝑡) corresponding to the value (𝛼=1 for FVIM&HPM and for FVIM 𝛼=0.01).

(a)

(b)

(c)

Figure 3 

The surface shows the solution 𝑢(𝑥,𝑡) for (36) with initial condition (37): (a) FVIM (𝛼=1), (b) HPM [10] (𝛼=1), and (c) FVIM (𝛼=0.01).

5. Conclusion

Variational iteration method has been known as a powerful method for solving many fractional equations such as partial differential equations, integrodifferential equations, and so many other equations. In this paper, based on the variational iteration method and modified Riemann-Liouville derivative, we have presented a general framework of fractional variational iteration method for analytical and numerical treatment of fractional partial differential equations in fluid mechanics and in financial models. All of the examples concluded that the fractional variational iteration method is powerful and efficient in finding analytical approximate solutions as well as numerical solutions. For example, the results of Examples 8 and 10 illustrate that the present method is in excellent agreement with those of HPM and exact solution, where the obtained solution is shown graphically. Further, in Example 9 we got the exact solution in two iterations. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus. Maple has been used for presenting graph of solution in the present paper.

Acknowledgments

The second author gratefully acknowledges that this research partially supported by Ministry of Higher Education (MOHE), Malaysia under the ERGS Grant 5527068.