A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation
- Kangle Wang†^{1}Email author and
- Sanyang Liu†^{1}
Received: 8 November 2015
Accepted: 26 May 2016
Published: 24 June 2016
Abstract
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.
Keywords
Sumudu transform Caputo fractional derivative Diffusion equationBackground
The fractional differential equations have gained a lot of attention of physicists, mathematicians and engineers in the past two decades (Oldham and Spanier 1974; Hilfer 2000; Kilbas et al. 1993; Podlubny 1999; Debnath 1997; Yang and Srivastava 2015; Yang et al. 2015b, c, d; Wang et al. 2014, 2015a, b, c; Jiwari and Mittal 2011). All kinds of interdisciplinary problems can be modeled with the help of fractional derivatives. However, it is very difficult for us to find their exact solutions to most fractional differential equations, so numerical and approximation methods have to be used. So far, many methods have been used to solve linear and nonlinear fractional differential equations. For example, the Adomain decomposition method (ADM) (Wazwaz 1999), the homotopy perturbation method (HPM) (He 1999), the variational iteration method (VIM) (Safari et al. 2009), homotopy analysis method (HAM) (Liao 1992, 2004) and differential quadrature method (Jiwari et al. 2012). The time-fractional Cauchy reaction–diffusion equation (Verma et al. 2014; Jiwari et al. 2014; Mittal and Jiwari 2011) is one of all the important fractional partial differential equations. The time-fractional Cauchy reaction–diffusion equations can be used to describe many kinds of linear and nonlinear systems in chemistry, physics, ecology, biology and engineering (Britton 1998; Grindrod 1996). Kumar (2013) have obtained the approximate solutions of time-fractional Cauchy reaction–diffusion equations by using the homotopy perturbation transform method with the help of Laplace transform. In Gejji and Jafari (2006), proposed NIM for solving linear and nonlinear integral and differential equation. The NIM is very easy to understand and implement and obtain better result than existing methods.
In this paper, we establish a new Sumudu transform iterative method (NSTIM) with the help of the Sumudu transform (Chaurasia and Singh 2010) for obtaining analytical 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 \(C_{\mu }\), \(\mu \in {R}\) if there exists a real number \(p,\,(p>\mu )\), such that \(f(x)=x^{p}f_{1}(x)\), where \(f_{1}(x)\in {C[0,\infty )}\), and it is said to be in the space \(C_{\mu }^{m}\) if \(f^{(m)}\in {C_{\mu }}\), \(m\in {N}\) (Dimovski 1982).
Definition 2
Properties of the operator \(I^{\alpha }\), which we will use here, are as follows
Definition 3
Definition 4
Definition 5
Definition 6
The new Sumudu transform iterative method (NSTIM)
Numerical examples
Example 1
Example 2
The series (28) is approximate to the form \(u(x,t)=e^{x^{2}+t}\) for \(\alpha =1\), which is the exact solution of Eq. (25) for \(\alpha =1\). The result is complete agreement with HPTM (Kumar 2013).
Example 3
The exact solution of Eq. (29) is \(u(x,t)=e^{x^{2}+t^{2}}\) for \(\alpha =1\).
Remark 1
Remark 2
Comparison between the 10th-order approximate solution of Eq. (20) and the exact solution for \(\alpha =1\)
x | t | \(u_{exa}\) | \(u_{NSTIM}\) | \(|u_{exa}-u_{10app}|\) |
---|---|---|---|---|
\(\alpha =1\) | ||||
0.2 | 0.3 | 0.9668943972 | 0.9668943972 | \(0.0\times 10^{-10}\) |
0.4 | 0.3 | 0.9666473343 | 0.9666473342 | \(1.0\times 10^{-10}\) |
0.5 | 0.6 | 0.8809364777 | 0.8809364778 | \(1.0\times 10^{-10}\) |
0.7 | 0.8 | 0.8111155787 | 0.8111155801 | \(1.4\times 10^{-9}\) |
Comparison between the 10th-order approximate solution of Eq. (25) and the exact solution for \(\alpha =1\)
x | t | \(u_{exa}\) | \(u_{NSTIM}\) | \(|u_{exa}-u_{10app}|\) |
---|---|---|---|---|
\(\alpha =1\) | ||||
0.3 | 0.4 | 1.632316221 | 1.632316221 | \(0.0\times 10^{-9}\) |
0.5 | 0.6 | 2.339646852 | 2.339646853 | \(1.0\times 10^{-9}\) |
0.6 | 0.7 | 2.886370989 | 2.886370989 | \(0.0\times 10^{-9}\) |
0.7 | 0.8 | 3.632786555 | 3.632786553 | \(2.0\times 10^{-9}\) |
Comparison between the 5th-order approximate solution of Eq. (29) and the exact solution for \(\alpha =1\)
x | t | \(u_{exa}\) | \(u_{NSTIM}\) | \(|u_{exa}-u_{5app}|\) |
---|---|---|---|---|
\(\alpha =1\) | ||||
0.2 | 0.3 | 1.138828383 | 1.138828331 | 0.000000052 |
0.4 | 0.5 | 1.506817785 | 1.506807822 | 0.000009963 |
0.3 | 0.6 | 1.568312186 | 1.568253566 | 0.000058620 |
0.7 | 0.8 | 3.095656500 | 3.094024665 | 0.001631835 |
Example 4
Remark 4
Comparison between the 10th-order approximate solution of Eq. (33) and the exact solution for \(\alpha =1\)
x | t | \(u_{exa}\) | \(u_{NSTIM}\) | \(|u_{exa}-u_{10app}|\) |
---|---|---|---|---|
\(\alpha =1\) | ||||
0.2 | 0.2 | 1.491824698 | 1.491824698 | \(0.0\times 10^{-9}\) |
0.4 | 0.3 | 2.013752707 | 2.013752707 | \(0.0\times 10^{-9}\) |
0.6 | 0.4 | 2.718281828 | 2.718281129 | \(1.0\times 10^{-9}\) |
0.7 | 0.8 | 4.481689070 | 4.481689066 | \(4.0\times 10^{-9}\) |
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.
Notes
Declarations
Authors' contributions
KW and SL contributed substantially to this paper, participated in drafting and checking the manuscript and have approved the version to be published. Both authors read and approved the final manuscript.
Acknowlegements
The authors are very grateful to the editor and referees for their insightful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 61373174).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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