A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation

In this paper, a new Sumudu transform iterative method is established and successfully applied to find the approximate analytical solutions for time-fractional Cauchy reaction–diffusion equations. The approach is easy to implement and understand. The numerical results show that the proposed method is very simple and efficient.

and numerical solutions of the time-fractional Cauchy reaction-diffusion equations. Our iterative method is new and generalizes NIM due to Gejji and Jafari (2006). The advantage of this new method which we proposed is to make the calculation simple and highly accurate to approximate the exact solution.

Basic definition
In this section, we give some basic definitions and properties of fractional calculus and Sumudu transform, which we will use in this paper: Definition 1 A real function f (x), x > 0, is said to be in the space C µ , µ ∈ R if there exists a real number p, (p > µ), such that f (x) = x p f 1 (x), where f 1 (x) ∈ C[0, ∞), and it is said to be in the space C m µ if f (m) ∈ C µ , m ∈ N (Dimovski 1982).

The new Sumudu transform iterative method (NSTIM)
To illustrate the basic idea of the NSTIM for the fractional partial differential equation, we consider the following equation with the initial condition as where D nα t is the Caputo fractional derivative operator, D nα t = ∂ nα ∂t nα , L is a linear operator, R is general nonlinear operator, g(x, t) is continuous function.
Applying Sumudu transform on both sides of Eq. (6), we get Using the property of the Sumudu transform, we can obtain On simplifying (8), we have Operating the inverse Sumudu transform on both sides of Eq. (9), we get Next assume that Similarly, the convergence of the NSTIM, we can refer the paper Gejji and Jafari (2006 andBhalekar andGejji (2011).

Numerical examples
Example 1 Consider the following time-fractional Cauchy reaction-diffusion equation (Kumar 2013) Applying Sumudu transform on both sides of Eq. (20) and using the differential property of Sumudu transform, we obtain Using the inverse Sumudu transform on both sides of Eq. (21), we get namely According to the NSTIM, we have By iteration, the following results are obtained Therefore, we have the solution of the problem as follows The Eq. (24) is approximate to the form u(x, t) = e −x + xe −t for α = 1, which is the exact solution of Eq. (20) for α = 1. The result is same as HPTM (Kumar 2013).
Example 2 We consider the following time-fractional Cauchy reaction-diffusion equation (Kumar 2013) as follows Applying Sumudu transform on both sides of Eq. (25) and using the differential property of Sumudu transform, we obtain Using the inverse Sumudu transform on both sides of Eq. (26), we have According to the NSTIM, we can obtain By iteration, we get the following results as The solution of Eq. (25) is given as The series (28) approximate solutions of the linear fractional Cauchy reaction-diffusion equations at different values for α = 0.6, 0.8, 1 and the exact solutions for α = 1. It is very easy to find that the solution continuously depend on the values of time-fractional derivative. Remark 2 Figures 13, 14 and 15 show the absolute error between approximate solutions and exact solutions for α = 1. In Tables 1, 2 and 3, we compute the approximate solutions and the exact solutions at different points for α = 1. By comparison, we find that it is evident the accuracy and efficiency of this method can be dramatically enhanced by computing further terms. In this paper, we only use several terms. If we use more terms, the accuracy of the approximate solution will be greatly improved. Therefore, the proposed method is accurate and efficient for linear differential equation.   Operating the Sumudu transform on both sides of Eq. (33) and applying the property of Sumudu transform for fractional derivative, we get   By iteration, the following result is obtained . . . . . . u n = e x t nα Ŵ(nα + 1) .

Remark 4
In this example, we apply the NSTIM to solve the nonlinear Cauchy reaction-diffusion equation. In Table 4, we compute the different values between the 10thorder approximate solution and the exact solution of Eq. (33) for α = 1. Figs. 16, 17, 18 and 19 show 10th-order approximate solutions for α = 0.6, α = 0.8, α = 1, and the exact solution of Eq. (33). Figure 20 show the absolute error between approximate solution and exact solution for α = 1. By comparing Table 4 with Figs. 16, 17, 18 and 19, we can find the NSTIM is very accurate and efficient to solve the nonlinear Cauchy reaction-equation. The accuracy of this method depends on the number of terms. So, the NSTIM is a very efficient method to solve the nonlinear fractional differential equation.

Conclusion
In this paper, the new Sumudu transform iterative method has been successfully applied for finding the approximate solution for the time-fractional Cauchy reaction-diffusion equation. The advantage of the new Sumudu transform iterative method (NSTIM) is to combine new iterative method (NIM) and Sumudu transform for obtaining exact and approximate analytical solutions for the time-fractional Cauchy reaction-diffusion equations.The numerical results show that the Sumudu transform iterative method is highly efficient and accurate with less calculation than existing methods.