A wavelet iterative method based on a numerical integration by using the Coiflets orthogonal wavelets for a nonlinear fractional differential equation is proposed. With the help of Laplace transform, the fractional differential equation was converted into equivalent integral equation of convolution type. By using the wavelet approximate scheme of a function, the undesired jump or wiggle phenomenon near the boundary points was avoided and the expansion constants in the approximation of arbitrary nonlinear term of the unknown function can be explicitly expressed in finite terms of the expansion ones of the approximation of the unknown function. Then a numerical integration method for the convolution is presented. As an example, an iterative method which can solve the singular nonlinear fractional Riccati equations is proposed. Numerical results are performed to show the efficiency of the method proposed.
In the recent years, fractional differential equations have been found to be effective to describe some physical phenomena such as rheology, damping laws, fractional random walk, and fluid flow [
In this paper, we will introduce a modified fractional differential operator
Despite the progresses outlined above, the literatures on high accuracy and easy to implement numerical techniques are suitable for solving nonlinear Riccati differential equations. The wavelet method, which is a general analytical method and has been widely used by many mathematicians and engineers to solve various functional equations, was applied to signal decompositions and reconstructions, Laplace inversions [
In this paper, we introduce a Coifletsbased wavelet Laplace method (CWLM) that can efficiently solve Riccati differential equation. This method depends on an explicit wavelet approximation scheme for the nonlinear terms of unknown function in the equation, in which series coefficients are just the function samplings at corresponding nodal points, and also by using Laplace transform, the equation with singular integral kernel was converted into equivalent nonsingular integral equation. At last, numerical simulations are performed to show the efficiency of the method proposed.
Here we give some necessary definitions and mathematical preliminaries of the fractional calculus which are used in this paper. The two most commonly used definitions are the RiemannLiouville and Caputo. The difference between the two definitions is in the order of evaluation. The RiemannLiouville fractional derivative operator of order
For
The Laplace transform of Caputo fractional derivative operator
Function
There is a sequence set
Set
where
is KroneckerDelta function.
For all
where
Function space is
setting
For any function
The form equation (
For the composite function of the function
where
For the integral operator or the derivative operator
That is, we just need to put the role of operator
As we know, wavelet series approximation is a square integrable function defined in the infinite interval, when the approximating function is defined only in a finite interval, and we need to truncate the wavelet series, which may introduce the boundary effect significantly, and the corresponding numerical method of calculation led to decreased accuracy. Traditionally, general treatment of boundary conditions is by using the zeroextension, symmetric, or periodic extension and so on. To some extent, these approaches can effectively inhibit the jitter of the border when it is a special form of approximation function, but not universal, and does not consider the wavelet expansion to meet the boundary conditions. Different from the past expansion of function, in this paper, based on Taylor series expansion of the boundary extension which is applied on the function defined on a finite interval [
First, we assume that the function
Then, (
Consider Coiflets scaling function
Thus, when specific boundary conditions are given, the differential coefficients of extension can be determined in accordance with the above process, and the corresponding improved scaling function is given by (
In this section, we will consider the modified wavelet approximation scheme to solve nonlinear Riccati differential equation (
First, applying the Laplace transform on the time variable
In this section we will give two numerical experimentsto illustrate the efficiency and apply the approach proposed in this paper.
Consider the following fractional Riccati equation given in [
Figure
Comparison between the numerical and exact solution of Example
The absolute errors of the exact solution and the results obtained by CWLM with different
Solution of (
Table
Solution of (

Methods  

HAM [ 
HWOMM [ 
CWLM  




1.0  0.6982  0.6987  0.6667  0.6837  0.6883 
2.0  0.7858  0.7857  0.7732  0.7796  0.7814 
3.0  0.8268  0.8258  0.8190  0.8224  0.8234 
4.0  0.8525  0.8499  0.8455  0.8477  0.8484 
5.0  0.8714  0.8664  0.8633  0.8648  0.8652 
6.0  0.8869  0.8785  0.8761  0.8773  0.8776 
7.0  0.9004  0.8878  0.8860  0.8869  0.8871 
8.0  0.9127  0.8953  0.8938  0.8946  0.8948 
9.0  0.9243  0.9016  0.9003  0.9009  0.9011 
10.0  0.9354  0.9068  0.9057  0.9062  0.9064 
Consider the following fractional Riccati equation [
Figure
Solution of (

Methods  

EHPM [ 
MHPM [ 
HWOMM [ 
CWLM  



1.0  1.9753  2.0874  1.8186  1.1571  1.1538 
2.0  2.2062  0.7787  2.1796  1.8206  1.8198 
3.0  2.2813  5.8102  2.2684  2.0566  2.0629 
4.0  2.3153  −0.0807  2.3062  2.1539  2.1643 
5.0  2.3340  −396.4145  2.3273  2.2038  2.2164 
6.0  2.3460 

2.3408  2.2337  2.2477 
7.0  2.3544 

2.3504  2.2537  2.2685 
8.0  2.3608 

2.3574  2.2679  2.2833 
9.0  2.3657 

2.3629  2.2787  2.2945 
10.0  2.3679 

2.3673  2.2870  2.3031 
Comparison between the numerical and exact solutions of Example
The absolute errors of the exact solution and the results obtained by CWLM with different
Solution of (
In this paper, a numerical method based on the Coiflets wavelet operational method is applied to solve the fractional differential equations. In this method, the equation with fractional differential order is transferred to an integral equation of convolution type by the Laplace transform and then the solution is approximated by the modified wavelet approximate scheme. This simple method was established by Zhou et al. [
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was supported by National Natural Science Foundation of China (11302081) and Huazhong Agricultural University Scientific and Technological Selfinnovation Foundation under Grant (529020900206074).