New Iterative Method: An Application for Solving Fractional Physical Differential Equations

Abstract

The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which con…firm the power, accuracy, and simplicity of the given method compared with some of the other methods.

1. Introduction

Considerable attention has been devoted to the study of the fractional calculus during the past three decades and its numerous applications in the area of 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, chemical physics, optics, and signal processing can be successfully modelled by linear or nonlinear fractional differential equations.

So far there have been several fundamental works on the fractional derivative and fractional differential equations [1–3]. These works are to be considered as an introduction to the theory of fractional derivative and fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and uniqueness [4, 5]. Recently, many other researchers have paid attention to existence result of solution of the initial value problem and boundary problem for fractional differential equations [4–6].

Finding approximate or exact solutions of fractional differential equations 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 develop new methods for obtaining analytical solutions which reasonably approximate the exact solutions. Several such techniques have drawn special attention, such as Adomain’s decomposition method [7], homotopy perturbation method [8–10], homotopy analysis method [11, 12], variational iteration method [13–17], Chebyshev spectral method [18, 19], and new iterative method [20–22]. Among them, the new iterative method provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. The new iterative method is more superior than the other nonlinear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can find wide application in nonlinear problems without linearization or small perturbation.

The motivation of this paper is to extend the application of the new iterative method proposed by Daftardar-Gejji and Jafari [20–22] to solve linear and nonlinear ordinary and partial differential equations of fractional order. This motivation is based on the importance of these equations and their applications in various subjects in physical branches [10, 11, 14, 23–25].

There are several definitions of a fractional derivative of order 𝛼>0 [3, 26]. The two most commonly used definitions are Riemann-Liouville and Caputo. Each definition uses Riemann-Liouville fractional integration and derivative of whole order. The difference between the two definitions is in the order of evaluation. Riemann-Liouville fractional integration of order 𝛼 is defined as 𝐼^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝜉)^𝛼−1𝑓(𝜉)𝑑𝜉,𝛼>0,𝑥>0. (1) The next two equations define Riemann-Liouville and Caputo fractional derivatives of order 𝛼,  respectively, as𝐷^𝛼𝑓(𝑥)=𝑑^𝑚𝑑𝑥^𝑚𝐼^𝑚−𝛼𝑓(𝑥),𝐷^𝛼_∗𝑓(𝑥)=𝐼^𝑚−𝛼𝑑^𝑚𝑑𝑥^𝑚𝑓(𝑥), (2a)(2b)where 𝑚−1<𝛼<𝑚,  𝑚∈𝑁.

Caputo fractional derivative first computes an ordinary derivative followed by a fractional integral to achieve the desired order of fractional derivative. Riemann-Liouville fractional derivative is computed in the reverse order. Therefore, Caputo fractional derivative allows traditional initial and boundary conditions to be included in the formulation of the problem.

From properties of 𝐷^𝛼_∗  and 𝐼^𝛼,  it is important to note that𝐷^𝛼_∗𝑥^𝛽=Γ𝛽+1𝑥^𝛽−𝛼Γ𝛽+1−𝛼,𝛽≥𝛼, (3a) where 𝐷^𝛼_∗  is Caputo derivative operator of order 𝛼, 𝐼^𝛼𝑥^𝛽=Γ𝛽+1𝑥^𝛽+𝛼Γ.𝛽+1+𝛼 (3b)

2. Basic Idea of New Iterative Method

For the basic idea of the new iterative method, we consider the following general functional equation [20–22]: 𝑢(𝑥)=𝑓(𝑥)+𝑁(𝑢(𝑥)), (4) where 𝑁 is a nonlinear operator from a Banach space 𝐵→𝐵 and 𝑓  is a known function. We have been looking for a solution of (4) having the series form 𝑢(𝑥)=^∞_𝑖=0𝑢_𝑖(𝑥). (5) The nonlinear operator 𝑁 can be decomposed as 𝑁^∞_𝑖=0𝑢_𝑖=𝑁𝑢₀+^∞_𝑖=1𝑁^𝑖_𝑗=0𝑢_𝑗−𝑁^𝑖−1_𝑗=0𝑢_𝑗. (6) From (5) and (6), (4) is equivalent to ^∞_𝑖=0𝑢_𝑖=𝑓+𝑁𝑢₀+^∞_𝑖=1𝑁^𝑖_𝑗=0𝑢_𝑗−𝑁^𝑖−1_𝑗=0𝑢_𝑗. (7) We define the following recurrence relation: 𝑢₀𝑢=𝑓,₁=𝑁𝑢₀,𝑢_𝑛+1=𝑁𝑢₀+𝑢₁+⋯+𝑢_𝑛−𝑁𝑢₀+𝑢₁+⋯+𝑢_𝑛−1,𝑛=1,2,…. (8) Then, 𝑢₁+𝑢₂+⋯+𝑢_𝑛+1=𝑁𝑢₀+𝑢₁+⋯+𝑢_𝑛,𝑛=1,2,…,𝑢=^∞_𝑖=0𝑢_𝑖=𝑓+𝑁^∞_𝑖=0𝑢_𝑖. (9) If ‖𝑁(𝑥)−𝑁(𝑦)‖<𝑘‖𝑥−𝑦‖,  0<𝑘<1, then 𝑢_𝑛+1=𝑁𝑢₀+⋯+𝑢_𝑛−𝑁𝑢₀+⋯+𝑢_𝑛−1≤𝑘𝑢_𝑛≤⋯≤𝑘^𝑛+1𝑢₀,𝑛=0,1,2,…, (10) and the series ∑^∞_𝑖=0𝑢_𝑖 absolutely and uniformly converges to a solution of (4) [27], which is unique, in view of the Banach fixed point theorem [28]. The n-term approximate solution of (4) and (5) is given by 𝑢(𝑥)=∑^𝑛−1_𝑖=0𝑢_𝑖.

2.1. Convergence of the Method

Now we analyze the convergence of the new iterative method for solving any general functional equation (4). Let 𝑒=𝑢^∗−𝑢, where 𝑢^∗ is the exact solution, 𝑢 is the approximate solution, and 𝑒 is the error in the solution of (4); obviously 𝑒 satisfies (4), that is, 𝑒(𝑥)=𝑓(𝑥)+𝑁(𝑒(𝑥)) (11) and the recurrence relation (8) becomes 𝑒₀𝑒=𝑓,₁=𝑁𝑒₀,𝑒_𝑛+1=𝑁𝑒₀+𝑒₁+⋯+𝑒_𝑛−𝑁𝑒₀+𝑒₁+⋯+𝑒_𝑛−1,𝑛=1,2,…. (12) If ‖𝑁(𝑥)−𝑁(𝑦)‖≤𝑘‖𝑥−𝑦‖,  0<𝑘<1, then 𝑒₀𝑒=𝑓,₁=𝑁𝑒₀≤𝑘𝑒₀,𝑒₂=𝑁𝑒₀+𝑒₁−𝑁𝑒₀≤𝑘𝑒₁≤𝑘²𝑒₀,𝑒₃=𝑁𝑒₀+𝑒₁+𝑒₂−𝑁𝑒₀+𝑒₁≤𝑘𝑒₂≤𝑘³𝑒₀,⋮𝑒_𝑛+1=𝑁𝑒₀+⋯+𝑒_𝑛−𝑁𝑒₀+⋯+𝑒_𝑛−1≤𝑘𝑒_𝑛≤𝑘^𝑛+1𝑒₀,𝑛=0,1,2,…. (13) Thus 𝑒_𝑛+1→0 as 𝑛→∞, which proves the convergence of the new iterative method for solving the general functional equation (4). For more details, you can see [29].

3. Suitable Algorithm

In this section, we introduce a suitable algorithm for solving nonlinear partial differential equations using the new iterative method. Consider the following nonlinear partial differential equation of arbitrary order:𝐷^𝛼_∗𝑡𝑢()()(),𝜕𝑥,𝑡=𝐴𝑢,𝜕𝑢+𝐵𝑥,𝑡𝑚−1<𝛼<𝑚,𝑚∈𝑁,_𝑘𝜕𝑡^𝑘𝑢(𝑥,0)=ℎ_𝑘(𝑥),𝑘=0,1,…,𝑚−1, (14a)(14b)where 𝐴 is a nonlinear function of 𝑢 and 𝜕𝑢 (partial derivatives of 𝑢 with respect to 𝑥 and 𝑡) and 𝐵 is the source function. In view of the new iterative method, the initial value problem (14a) and (14b) is equivalent to the integral equation 𝑢(𝑥,𝑡)=^𝑚−1_𝑘=0ℎ_𝑘(𝑥)𝑡^𝑘𝑘!+𝐼^𝛼_𝑡𝐵+𝐼^𝛼_𝑡𝐴=𝑓+𝑁(𝑢), (15) where𝑓=^𝑚−1_𝑘=0ℎ_𝑘(𝑥)𝑡^𝑘𝑘!+𝐼^𝛼_𝑡𝑁(𝑢)𝐵,=𝐼^𝛼_𝑡𝐴. (16a)(16b)

Remark 1. When the general functional equation (4) is linear, the recurrence relation (8) can be simplified in the form 𝑢₀𝑢=𝑓,_𝑛+1=𝑁𝑢_𝑛,𝑛=0,1,2,…. (17)

Proof. From the properties of integration and by using (8) and (16b), we have 𝑢_𝑛+1=𝑁𝑢₀+⋯+𝑢_𝑛−1+𝑢_𝑛−𝑁𝑢₀+⋯+𝑢_𝑛−1=𝐼^𝛼_𝑡𝑢₀+⋯+𝑢_𝑛−1+𝑢_𝑛−𝐼^𝛼_𝑡𝑢₀+⋯+𝑢_𝑛−1=𝐼^𝛼_𝑡𝑢₀+⋯+𝐼^𝛼_𝑡𝑢_𝑛−1+𝐼^𝛼_𝑡𝑢_𝑛−𝐼^𝛼_𝑡𝑢₀−𝐼^𝛼_𝑡𝑢_𝑛−1=𝐼^𝛼_𝑡𝑢_𝑛=𝑁𝑢_𝑛,𝑛=0,1,2,…. (18) Therefore, we get the solution of (15) by employing the recurrence relation (8) or (17).

4. Applications

To illustrate the effectiveness of the proposed method, several test examples are carried out in this section.

Example 2. In this example, we consider the following initial value problem in the case of the inhomogeneous Bagely-Torvik equation [23, 24]: 𝐷²_∗𝑥𝑢(𝑥)+𝐷^1.5_∗𝑥𝑢(𝑥)(𝑥)(𝑥),𝑢(0)+𝑢=𝑔=1,𝑢^′(0)=1,𝑥∈[0,𝐿], (19) where 𝑔(𝑥)=1+𝑥. The exact solution of this problem is 𝑢(𝑥)=1+𝑥.

By applying the technique described in Sections 2 and 3, the initial value problem (19) is equivalent to the integral equation 𝑢(𝑥)𝑥=1+𝑥+²2+𝑥³6−𝐼²_𝑥𝐷^1.5_∗𝑥𝑢(𝑥)+𝑢(𝑥). (20)

Let 𝑁(𝑢)=−𝐼²_𝑥[𝐷^1.5_∗𝑥𝑢(𝑥)+𝑢(𝑥)]. In view of recurrence relation (17), we have the following first approximations: 𝑢₀(𝑥)𝑥=1+𝑥+²2+𝑥³6,𝑢₁(𝑥)=𝑁𝑢₀=−8𝑥^2.515√𝜋−16𝑥^3.5105√𝜋−𝑥²2−𝑥³6−𝑥⁴−𝑥24⁵,𝑢120₂(𝑥)=𝑁𝑢₁=8𝑥^2.515√𝜋+16𝑥^3.5105√𝜋+64𝑥^4.5945√𝜋+128𝑥^5.510395√𝜋+𝑥³6+𝑥⁴+𝑥12₅+𝑥120⁶+𝑥720⁷,5040 (21) and so on. In the same manner the rest of components can be obtained. The 6-term approximate solution for (19) is 𝑢(𝑥)=⁵_𝑖=0𝑢_𝑖𝑥=1+𝑥−⁶−1445𝑥⁷−𝑥5040⁹−𝑥36288₁₀−𝑥362880¹²−𝑥479001600¹³−622702080032𝑥^4.5945√𝜋−64𝑥^5.510395√𝜋−512𝑥^7.5405405√𝜋−1024𝑥^8.56891885√𝜋−512𝑥^10.5687465529√𝜋−128𝑥^11.51976463395√𝜋. (22)

Remark 3. In Example 2. we have used the recurrence relation (17). If we used the recurrence relation (8) in place of (17), we obtain the same result.

In Figure 1, we have plotted the 6-term approximate solution with the corresponding exact solution for (19). It is remarkable to note that the two solutions are almost equal.

Figure 1 

Plots of the approximate solution and the exact solution for (19).

Comparing these obtained results with those obtained by new Jacobi operational matrix in [23, 24], we can confirm the simplicity and accuracy of the given method.

Example 4. Consider the following fractional Riccati equation [10]: 𝐷^𝛼_∗𝑥𝑢(𝑥)+𝑢²(𝑥)=1,𝑢(0)=0,𝑥>0,0<𝛼≤1. (23) The exact solution when 𝛼=1 is 𝑢(𝑥)=(𝑒^2𝑥−1)/(𝑒^2𝑥+1).

By applying the technique described in Sections 2 and 3, the initial value problem (23) is equivalent to the integral equation 𝑢(𝑥)=𝑥^𝛼Γ(1+𝛼)−𝐼^𝛼_𝑥𝑢²(𝑥). (24)

Let 𝑁(𝑢)=−𝐼^𝛼_𝑥[𝑢²(𝑥)]. In view of recurrence relation (8), we have the following first approximations: 𝑢₀(𝑥)=𝑥^𝛼Γ(),𝑢1+𝛼₁(𝑥)=𝑁𝑢₀Γ()𝑥=−1+2𝛼^3𝛼Γ(1+3𝛼)Γ(1+𝛼)²,𝑢₂(𝑥)=𝑁𝑢₀+𝑢₁−𝑁𝑢₀=2Γ(1+2𝛼)Γ(1+4𝛼)𝑥^5𝛼Γ(1+3𝛼)Γ(1+5𝛼)Γ(1+𝛼)³−Γ(1+2𝛼)²Γ(1+6𝛼)𝑥^7𝛼Γ(1+3𝛼)²Γ(1+7𝛼)Γ(1+𝛼)⁴, (25) and so on. The 4-term approximate solution for (23) is 𝑢(𝑥)=³_𝑖=0𝑢_𝑖=𝑥^𝛼Γ()−Γ()𝑥1+𝛼1+2𝛼^3𝛼Γ(1+3𝛼)Γ(1+𝛼)²+2Γ(1+2𝛼)Γ(1+4𝛼)𝑥^5𝛼Γ(1+3𝛼)Γ(1+5𝛼)Γ(1+𝛼)³−Γ(1+2𝛼)²Γ(1+6𝛼)𝑥^7𝛼Γ(1+3𝛼)²Γ(1+7𝛼)Γ(1+𝛼)⁴−4Γ(1+2𝛼)Γ(1+4𝛼)Γ(1+6𝛼)𝑥^7𝛼Γ(1+3𝛼)Γ(1+5𝛼)Γ(1+7𝛼)Γ(1+𝛼)⁴+4Γ(1+2𝛼)²Γ(1+4𝛼)Γ(1+3𝛼)²Γ()⋅Γ()𝑥1+5𝛼1+8𝛼^9𝛼Γ(1+9𝛼)Γ(1+𝛼)⁵+2Γ(1+2𝛼)²Γ(1+6𝛼)Γ(1+8𝛼)𝑥^9𝛼Γ(1+3𝛼)²Γ(1+7𝛼)Γ(1+9𝛼)Γ(1+𝛼)⁵−4Γ(1+2𝛼)²Γ(1+4𝛼)²Γ(1+10𝛼)𝑥^11𝛼Γ(1+3𝛼)²Γ(1+5𝛼)²Γ(1+11𝛼)Γ(1+𝛼)⁶−2Γ(1+2𝛼)³Γ(1+3𝛼)³Γ()⋅Γ()Γ()𝑥1+7𝛼1+6𝛼1+10𝛼^11𝛼Γ(1+11𝛼)Γ(1+𝛼)⁶+4Γ(1+2𝛼)³Γ(1+4𝛼)Γ(1+6𝛼)Γ(1+3𝛼)³Γ()Γ()⋅Γ()𝑥1+5𝛼1+7𝛼1+12𝛼^13𝛼Γ(1+13𝛼)Γ(1+𝛼)⁷−Γ(1+2𝛼)⁴Γ(1+6𝛼)²Γ(1+14𝛼)𝑥^15𝛼Γ(1+3𝛼)⁴Γ(1+7𝛼)²Γ(1+15𝛼)Γ(1+𝛼)⁸. (26)

In Figure 2, we have plotted the 4-term approximate solution for (23) for different values of 𝛼 with the corresponding exact solution. It is remarkable to note that the approximate solution,  in case 𝛼=1,  and the exact solution are almost equal (continuous curve) whenever the approximate solution, in cases 𝛼=0.9,0.8, is of high agreement with the exact solution (dashed and dotted curves, resp.).

Figure 2 

Plots of the approximate solution for different values of 𝛼 and the exact solution for (23).

Comparing the obtained results with those obtained by homotopy analysis method, in case ℎ=−1, in [10], we can confirm the simplicity and accuracy of the given method.

Example 5. Consider the following initial value problem with fractional order [23, 24]: 𝐷³_∗𝑥𝑢(𝑥)+𝐷^2.5_∗𝑥𝑢(𝑥)+𝑢²(𝑥)=𝑥⁴,𝑢(0)=𝑢^′(0)=0,𝑢^′′(0)=2. (27)

The exact solution for this problem is 𝑢(𝑥)=𝑥².

As in Example 4, the initial value problem (27) is equivalent to the integral equation 𝑢(𝑥)=𝑥²+𝑥⁷210−𝐼³_𝑥𝐷^2.5_∗𝑥𝑢(𝑥)+𝑢²(𝑥). (28)

Let 𝑁(𝑢)=−𝐼³_𝑥[𝐷^2.5_∗𝑥𝑢(𝑥)+𝑢²(𝑥)]. In view of recurrence relation (8), we have the following first approximations: 𝑢₀(𝑥)=𝑥²+𝑥⁷,𝑢210₁(𝑥)=𝑁𝑢₀=−129024𝑥^7.542567525√𝜋−𝑥⁷−𝑥210₁₂−𝑥138600¹⁷,𝑢179928000₂(𝑥)=𝑁𝑢₀+𝑢₁−𝑁𝑢₀=1.71008⁻³𝑥^7.5+5.95252⁻⁴𝑥⁸+7.21501⁻⁶𝑥¹²+4.28636⁻⁶𝑥^12.5+9.09455⁻⁹𝑥¹⁷+1.31911⁻⁹𝑥^17.5−5.97301⁻¹⁰𝑥¹⁸+1.20298⁻¹²𝑥²²−2.48834⁻¹²𝑥^22.5−2.96617⁻¹⁵𝑥²⁷−1.02289⁻¹⁵𝑥^27.5−2.69485⁻¹⁸𝑥³²−6.62568⁻²⁸𝑥³⁷, (29) and so on. The 4-term approximate solution and the corresponding exact solution for (27) are plotted in Figure 3. It is remarkable to note that the two solutions are almost equal.

Figure 3 

Plots of the approximate solution and the exact solution for (27).

Comparing these obtained results with those obtained by new Jacobi operational matrix in [23, 24], we can confirm the simplicity and accuracy of the given method.

Example 6. Consider the following fractional order wave equation in 2-dimensional space [14]: 𝐷^𝛼_∗𝑡𝑢𝑥,𝑦,𝑡+𝑐𝑢_𝑥+𝑢_𝑦𝑢𝜋=0,𝑥,𝑦,0=sin𝑥+𝑦𝑙,0<𝛼≤1. (30)

The exact solution for this problem when 𝛼=1 is 𝑢𝑥,𝑦,𝑡=sin𝜋𝑥+𝑦−2𝑐𝑡𝑙. (31)

The initial value problem (30) is equivalent to the integral equation 𝑢𝑥,𝑦,𝑡=sin𝜋𝑥+𝑦𝑙−𝐼^𝛼_𝑡𝑐𝑢_𝑥+𝑢_𝑦. (32)

Let 𝑁(𝑢)=−𝐼^𝛼_𝑡[𝑐(𝑢_𝑥+𝑢_𝑦)].  In view of recurrence relation (17), we have the following first approximations: 𝑢₀𝑥,𝑦,𝑡=sin𝜋𝑥+𝑦𝑙,𝑢₁𝑥,𝑦,𝑡=−1Γ(1+𝛼)2𝑐𝜋𝑡^𝛼𝑙⋅cos𝜋𝑥+𝑦𝑙,𝑢₂𝑥,𝑦,𝑡=−1Γ(1+2𝛼)2𝑐𝜋𝑡^𝛼𝑙²⋅sin𝜋𝑥+𝑦𝑙,𝑢₃=1𝑥,𝑦,𝑡Γ(1+3𝛼)2𝑐𝜋𝑡^𝛼𝑙³⋅cos𝜋𝑥+𝑦𝑙, (33) and so on. The n-term approximate solution for (30) is 𝑢𝑥,𝑦,𝑡=^𝑛−1_𝑖=0𝑢_𝑖=sin𝜋𝑥+𝑦𝑙×11−Γ(1+2𝛼)2𝑐𝜋𝑡^𝛼𝑙²+1Γ(1+4𝛼)2𝑐𝜋𝑡^𝛼𝑙⁴−1Γ(1+6𝛼)2𝑐𝜋𝑡^𝛼𝑙⁶𝜋+⋯−cos𝑥+𝑦𝑙×1Γ(1+𝛼)2𝑐𝜋𝑡^𝛼𝑙−1Γ(1+3𝛼)2𝑐𝜋𝑡^𝛼𝑙³+1Γ(1+5𝛼)2𝑐𝜋𝑡^𝛼𝑙⁵−1Γ(1+7𝛼)2𝑐𝜋𝑡^𝛼𝑙⁷+⋯. (34) In closed form this gives: 𝑢𝑥,𝑦,𝑡=sin𝜋𝑥+𝑦𝑙⋅cos2𝑐𝜋𝑡^𝛼𝑙−cos𝜋𝑥+𝑦𝑙,sin2𝑐𝜋𝑡^𝛼𝑙=sin𝜋𝑥+𝑦−2𝑐𝑡^𝛼𝑙 (35) which is the exact solution for the given problem. When 𝛼=1, the above n-term approximate solution for (30) becomes 𝑢𝑥,𝑦,𝑡=^𝑛−1_𝑖=0𝑢_𝑖=sin𝜋𝑥+𝑦𝑙×11−2!2𝑐𝜋𝑡𝑙²+14!2𝑐𝜋𝑡𝑙⁴−16!2𝑐𝜋𝑡𝑙⁶𝜋+⋯−cos𝑥+𝑦𝑙×2𝑐𝜋𝑡𝑙−13!2𝑐𝜋𝑡𝑙³+15!2𝑐𝜋𝑡𝑙⁵−17!2𝑐𝜋𝑡𝑙⁷+⋯. (36) In closed form, this gives 𝑢𝑥,𝑦,𝑡=sin𝜋𝑥+𝑦𝑙⋅cos2𝑐𝜋𝑡𝑙−cos𝜋𝑥+𝑦𝑙,sin2𝑐𝜋𝑡𝑙=sin𝜋𝑥+𝑦−2𝑐𝑡𝑙, (37) which is the same result obtained by variational iteration method in [14].

Example 7. Consider the following fractional order heat equation in 2-dimensional space [11]: 𝐷^𝛽_∗𝑡𝑢𝑥,𝑦,𝑡=𝛼𝑢_𝑥𝑥+𝑢_𝑦𝑦,𝑢𝑥,𝑦,0=𝑐sin𝜋𝑥𝑙+sin𝜋𝑦𝑙,0<𝛽≤1. (38)

The exact solution for this problem when 𝛽=1 is 𝑢𝑥,𝑦,𝑡=𝑐𝑒^{−𝛼𝜋2𝑡/𝑙2}sin𝜋𝑥𝑙+sin𝜋𝑦𝑙. (39)

The initial value problem (38) is equivalent to the integral equation 𝑢𝑥,𝑦,𝑡=𝑐sin𝜋𝑥𝑙+sin𝜋𝑦𝑙+𝐼^𝛽_𝑡𝛼𝑢_𝑥𝑥+𝑢_𝑦𝑦. (40)

Let 𝑁(𝑢)=𝐼^𝛽_𝑡[𝛼(𝑢_𝑥𝑥+𝑢_𝑦𝑦)]. In view of recurrence relation (17), we have the following first approximations: 𝑢₀𝑥,𝑦,𝑡=𝑐sin𝜋𝑥𝑙+sin𝜋𝑦𝑙,𝑢₁𝑥,𝑦,𝑡=−𝑐Γ1+𝛽𝛼𝜋²𝑡^𝛽𝑙²×sin𝜋𝑥𝑙+sin𝜋𝑦𝑙,𝑢₂=𝑐𝑥,𝑦,𝑡Γ1+2𝛽𝛼𝜋²𝑡^𝛽𝑙²²×sin𝜋𝑥𝑙+sin𝜋𝑦𝑙,𝑢₃𝑥,𝑦,𝑡=−𝑐Γ1+3𝛽𝛼𝜋²𝑡^𝛽𝑙²³×sin𝜋𝑥𝑙+sin𝜋𝑦𝑙, (41) and so on. The n-term approximate solution for (38) is 𝑢𝑥,𝑦,𝑡=^𝑛−1_𝑖=0𝑢_𝑖=𝑐sin𝜋𝑥𝑙+sin𝜋𝑦𝑙×11−Γ1+𝛽𝛼𝜋²𝑡^𝛽𝑙²+1Γ1+2𝛽𝛼𝜋²𝑡^𝛽𝑙²²−1Γ1+3𝛽𝛼𝜋²𝑡^𝛽𝑙²³+⋯. (42) When 𝛽=1, The n-term approximate solution for (38) becomes 𝑢𝑥,𝑦,𝑡=^𝑛−1_𝑖=0𝑢_𝑖=𝑐sin𝜋𝑥𝑙+sin𝜋𝑦𝑙×11−1!𝛼𝜋²𝑡𝑙²+12!𝛼𝜋²𝑡𝑙²²−13!𝛼𝜋²𝑡𝑙²³+⋯. (43) In closed form, this gives 𝑢𝑥,𝑦,𝑡=𝑐𝑒^{−𝛼𝜋2𝑡/𝑙2}sin𝜋𝑥𝑙+sin𝜋𝑦𝑙 (44) which is the exact solution for the given problem.

The obtained results in this example are the same as these obtained in [11] by the homotopy perturbation method, in case 𝛽=1, but with the simplicity of the given method.

Example 8. In this last example, we consider the following fractional order nonlinear wave equation [25]: 𝐷^𝛼_∗𝑥𝑢(𝑥,𝑡)−𝑢𝑢_𝑡𝑡=𝑥^2−𝛼Γ()−𝑥3−𝛼²+𝑡²2,𝑢()=𝑡0,𝑡²2,𝑢_𝑥(0,𝑡)=0,1<𝛼≤2. (45)

The exact solution for this problem when 𝛼=2 is 𝑢(𝑥,𝑡)=(1/2)(𝑥²+𝑡²) where 0≤𝑥,  𝑡≤1.

The initial value problem (45) is equivalent to the integral equation 𝑢()=1𝑥,𝑡2𝑥²+𝑡²−𝑥^2+𝛼Γ()−𝑡3+𝛼²𝑥^𝛼2Γ(1+𝛼)+𝐼^𝛼_𝑥𝑢𝑢_𝑡𝑡. (46)

Let 𝑁(𝑢)=𝐼^𝛼_𝑥[𝑢𝑢_𝑡𝑡]. In view of recurrence relation (8), we have 𝑢₀()=1𝑥,𝑡2𝑥²+𝑡²−𝑥^2+𝛼Γ()−𝑡3+𝛼²𝑥^𝛼(),𝑢2Γ1+𝛼₁()=𝑥𝑥,𝑡^2+𝛼Γ()−𝑥3+𝛼Γ(^2+2𝛼)−Γ()𝑥3+2𝛼3+𝛼^2+2𝛼()Γ()+Γ()𝑥2Γ1+𝛼3+2𝛼3+2𝛼^2+3𝛼Γ(1+𝛼)Γ(3+𝛼)Γ(3+3𝛼)+𝑡²𝑥^𝛼()−𝑥2Γ1+𝛼Γ(^2𝛼)+Γ()𝑥1+2𝛼1+2𝛼^3𝛼2Γ(1+𝛼)²Γ(),1+3𝛼 (47) and so on. The 3-term approximate solution and the corresponding exact solution for (45) are plotted in Figure 4(a), in case 𝑡=1/2, for  𝛼=1.8,1.9,2., in Figure 4(b), in case 𝑡=1, for 𝛼=1.8,1.9,2., and in Figure 4(c), in case 𝛼=2. It is remarkable to note that in the first two figures all the solutions are almost equal.

(a)

(b)

(c)

Figure 4 

(a) Plots of the approximate solution for different values of 𝛼 and the exact solution, in case 𝑡=1/2; for (45). (b) Plots of the approximate solution for different values of 𝛼 and the exact solution, in case 𝑡=1; for (45). (c) Plots of the approximate solution, in case 𝛼=2 for (45).

Comparing these results with those obtained by the modification homotopy perturbation method in [25], we can confirm the accuracy and simplicity of the given method.

5. Conclusion

In this paper, the new iterative method with suitable algorithm is successfully used to solve linear and nonlinear ordinary and partial differential equations with fractional order. It is clear that the computations are easy and the solutions agree well with the corresponding exact solutions and more accurate than the solutions obtained by other methods. Moreover, the accuracy is high with little computed terms of the solution which confirm that this method with the given algorithm is a powerful method for handling fractional differential equations.