On Efficient Method for System of Fractional Differential Equations

Abstract

The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations. In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation. The new approximate analytical procedure depends only on two components. Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate.

1. Introduction

Fractional differential equations have received considerable interest in recent years and have been extensively investigated and applied for many real problems which are modeled in different areas. One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional derivatives [1]. Another difficulty is that fractional derivatives have no evident geometrical interpretation because of their nonlocal character. However, during the last 12 years fractional calculus starts to attract much more attention of scientists. It was found that various, especially interdisciplinary, applications [2–6] can be elegantly modeled with the help of the fractional derivatives.

The homotopy perturbation method is a powerful devise for solving nonlinear problems. This method was introduced by He [7–9] in the year 1998. In this method, the solution is considered as the summation of an infinite series that converges rapidly. This technique is used for solving nonlinear chemical engineering equations [10], time-fractional Swift-Hohenberg (S-H) equation [11], viscous fluid flow equation [12], Fourth-Order Integro-Differential equations [13], nonlinear dispersive equations [14], Long Porous Slider equation [15], and Navier-Stokes equations [16]. It can be said that He's homotopy perturbation method is a universal one, which is able to solve various kinds of nonlinear equations. The new homotopy perturbation method (NHPM) was applied to linear and nonlinear ODEs [17].

In this paper, we construct the solution of system of fractional-order differential equations by extending the idea of [17, 18]. This method leads to computable and efficient solutions to linear and nonlinear operator equations. The corresponding solutions of the integer-order equations are found to follow as special cases of those of fractional-order equations.

We consider the system of fractional-order equations of the form

(1.1)

2. Basic Definitions

We give some basic definitions, notations, and properties of the fractional calculus theory used in this work.

Definition 2.1.

The Riemann-Liouville fractional integral operator of order on the usual Lebesgue space is given by

(2.1)

It has the following properties:

(i) exists for any ,

(ii),

(iii),

(iv),

(v),

where , and .

Definition 2.2.

The Caputo definition of fractal derivative operator is given by

(2.2)

where . It has the following two basic properties for and :

(2.3)

3. Analysis of New Homotopy Perturbation Method

Let us consider the system of nonlinear differential equations

(3.1)

where are the operators, are known functions and are sought functions. Assume that operators can be written as

(3.2)

where are the linear operators and are the nonlinear operators. Hence, (3.1) can be rewritten as follows:

(3.3)

We define the operators as

(3.4)

where is an embedding or homotopy parameter, and are the initial approximation of solution of the problem in (3.3) can be written as

(3.5)

Clearly, the operator equations and are equivalent to the equations and , respectively. Thus, a monotonous change of parameter from zero to one corresponds to a continuous change of the trivial problem to the original problem. Operator is called a homotopy map. Next, we assume that the solution of equation can be written as a power series in embedding parameter , as follows:

(3.6)

Now, let us write (3.5) in the following form:

(3.7)

By applying the inverse operator, to both sides of (3.7), we have

(3.8)

Suppose that the initial approximation of (3.3) has the form

(3.9)

where , are unknown coefficients and , are specific functions on the problem. By substituting (3.6) and (3.9) into (3.8), we get

(3.10)

Equating the coefficients of like powers of , we get the following set of equations:

(3.11)

Now, we solve these equations in such a way that . Therefore, the approximate solution may be obtained as

(3.12)

4. Applications

Application 1

Consider the following linear fractional-order 2-by-2 stiff system:

(4.1)

with the initial conditions

(4.2)

where and are constants. To obtain the solution of (4.1) by NHPM, we construct the following homotopy:

(4.3)

Applying the inverse operator, of both sides of the above equation, we obtain

(4.4)

The solution of (4.1) to has the following form:

(4.5)

Substituting (4.5) in (4.4) and equating the coefficients of like powers of , we get the following set of equations:

(4.6)

Assuming , , , , and and solving the above equation for and lead to the result

(4.7)

Vanishing and lets the coefficients by taking the following values:

(4.8)

Therefore, we obtain the solutions of (4.1) as

(4.9)

Our aim is to study the mathematical behavior of the solution and for different values of . This goal can be achieved by forming Pade' approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about and . It is well known that Pade' approximants will converge on the entire real axis, if and are free of singularities on the real axis. It is of interest to note that Pade' approximants give results with no greater error bounds than approximation by polynomials. To consider the behavior of solution for different values of , we will take advantage of the explicit formula (4.9) available for and consider the following two special cases.

Case 1.

Setting in (4.9), we obtain the approximate solution in a series form as

(4.10)

Case 2.

In this case, we will examine the linear fractional stiff equation (4.1). Setting in (4.9) gives

(4.11)

For simplicity, let , then

(4.12)

Calculating the [10/11] Pade' approximants and recalling that , we get

(4.13)

Application 2

Consider the following nonlinear fractional-order 2-by-2 stiff system:

(4.14)

with the initial conditions

(4.15)

To obtain the solution of (4.14) by NHPM, we construct the following homotopy:

(4.16)

Applying the inverse operator, of both sides of the above equation, we obtain

(4.17)

The solution of (4.14) to have the following form:

(4.18)

Substituting (4.18) in (4.17) and equating the coefficients of like powers of , we get the following set of equations:

(4.19)

Assuming , , , , and and solving the above equation for and lead to the result

(4.20)

Vanishing and lets the coefficients to take the following values:

(4.21)

Therefore, we obtain the solution of (4.14) as

(4.22)

The exact solution of (4.14) for is .

Application 3

Consider the following nonlinear Genesio system with fractional derivative:

(4.23)

with the initial conditions

(4.24)

where , , and are constants. To obtain the solution of (4.23) by NHPM, we construct the following homotopy:

(4.25)

Applying the inverse operator, of both sides of the above equation, we obtain

(4.26)

The solution of (4.23) to have the following form:

(4.27)

Substituting (4.27) in (4.26) and equating the coefficients of like powers of , we get the following set of equations:

(4.28)

Assuming , , , , , , , , and , and solving the above equation for and lead to the result

(4.29)

Vanishing , and lets the coefficients to take the following values:

(4.30)

Therefore, we obtain the solutions of (4.23) as

(4.31)

Application 4

Finally, we consider the following nonlinear matrix Riccati differential equation with fractional derivative:

(4.32)

where . To find the solution of this equation by NHPM, we will treat the matrix equation as a system of fractional-order differential equations

(4.33)

with the initial conditions

(4.34)

Therefore, we obtain the solution of (4.33) as

(4.35)

5. Concluding Remarks

The NHPM for solving system of fractional-order differential equations are based on two component procedure and polynomial initial condition. The NHPM applied on fractional-order Stiff equation, fractional Genesio equation, and the matrix Riccati-type differential equation. The Applications in problems 1–4 are plotted in Figures 1, 2, 3, and 4, which show the accuracy of NHPM. The computations associated with the applications discussed above, were performed by MATHEMATICA. The NHPM is very simple in application and is less computational more accurate in comparison with other mentioned methods. By using this method, the solution can be obtained in bigger interval. Unlike the ADM [19], the NHPM is free from the need to use Adomian polynomials. In this method, we do not need the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals, which eliminate the complications that exist in the VIM [20]. In contrast to the HPM and HAM, in this method, it is not required to solve the functional equations in each iteration. The efficiency of HAM is very much depended on choosing auxiliary parameter . All the applications are taken from [20] with fractional derivatives.

Figure 1

Solutions of linear stiff system for , (a) Exact, (b) Numerical, (c) NHPM-Pade [10/11], (d) NHPM-Pade [10/11], (color figure can be viewed in the online issue).

Figure 2

Solutions of nonlinear stiff system for , (a) Exact, (b) Numerical, (c) NHPM, and (c) NHPM, (color figure can be viewed in the online issue).

Figure 3

Solutions of nonlinear Genesio system for (a) Numerical, (b) NHPM , (c) NHPM, , (d) NHPM, (color figure can be viewed in the online issue).

Figure 4

Solutions of matrix Riccati equations , for (a) Numerical, (b) NHPM-Pade [9/11], (c) NHPM-Pade [9/11], (color figure can be viewed in the online issue).