Solving a nonlinear fractional differential equation using Chebyshev wavelets

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Abstract

Chebyshev wavelet operational matrix of the fractional integration is derived and used to solve a nonlinear fractional differential equations. Some examples are included to demonstrate the validity and applicability of the technique.

Introduction

The use of fractional differential and integral operators in mathematical models has become increasingly widespread in recent years. Several forms of fractional differential equations have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solution. These methods include Laplace transforms [1], Fourier transforms [2], eigenvector expansion [3], Adomian decomposition method (ADM) [4], [5], Variational Iteration Method (VIM) [6], [7], Fractional Differential Transform Method (FDTM) [8], [9], Fractional Difference Method (FDM) [10] and Power Series Method [11]. But, few papers reported application of wavelet to solve the fractional order differential equations [12], [13].

In view of successful application of wavelet operational matrix in system analysis [14], [15], system identification [16], [17], optimal control [18], [19], [20] and numerical solution of integral and differential equations [21], [22], [23], [24], [25], [26], together with the characteristic of wavelet functions, we hold that they should be applicable to solve the fractional order systems.

So my purpose is to introduce the method to solve multi-order arbitrary differential equations, which include the linear and nonlinear differential equations.

Similar to the integer-order case, firstly, the underlying fractional differential equation is converted into a fractional integral equation via fractional integration; subsequently, the various signals involved in the fractional integral equation are approximated by representing them as linear combinations of the wavelet functions and truncating them at optimal levels; finally, the integral equation is converted to an algebraic equation by introducing the wavelet operational matrix of the fractional integration. Therefore, there are some questions to be answered:

  • (1)

    How to derive Chebyshev wavelet operational matrix of the fractional integration.

  • (2)

    How to analyze the fractional differential equations via Chebyshev wavelet operational matrices of the fractional integration.

The paper is organized as follows: I begin by introducing some necessary definitions and mathematical preliminaries of the fractional calculus theory which are required for establishing our results. In Section 3, after describing the basic formulation of wavelets and Chebyshev wavelets, I derive Chebyshev wavelet operational matrix of the fractional integration. In Section 4, I present three examples to show the efficiency and simplicity of the method.

Section snippets

Preliminaries and notations

I give some necessary definitions and mathematical preliminaries of the fractional calculus theory which are used further in this paper. The Riemann–Liouville fractional integration of order α>0 is defined as [1](Iαf)(t)=1Γ(α)0t(t-τ)α-1f(τ)dτ,(I0f)(t)=f(t),and its fractional derivative of order α>0 is normally used:(Dαf)(t)=ddtn(In-αf)(t)(n-1<αn),where n is an integer. For Riemann–Liouvilles definition,one hasIαtv=Γ(v+1)Γ(α+v+1)tα+v.The Riemann–Liouville derivative have certain disadvantages

Chebyshev wavelet

Wavelets are a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets as [23].ψa,b(t)=a-1/2ψt-ba,a,bR,a0.If we restrict the parameters a and b to discrete values as a=a0-k,b=nb0a0-k,a0>1,b0>0, where n and k are positive integers, the family of discrete wavelets are defined asψk,n(t)=a0k/2ψ(a0kt-nb0),where ψk,n

Applications and results

In this section, we will use the Chebyshev wavelet operational matrices of the fractional integration to solve nonlinear fractional (arbitrary) order differential equation. These examples are considered because closed form solutions are available for them, or they have also been solved using other numerical schemes. This allows one to compare the results obtained using this scheme with the analytical solution or the solutions obtained using other schemes.

Example 1

Following Odibat and Momani [29], we

Conclusion

We derive Chebyshev wavelet operational matrix of the fractional integration, and use its to solve nonlinear fractional (arbitrary) order differential equation. Several examples are given to demonstrate the powerfulness of the proposed method. Using wavelet operational matrix of the fractional integration to solve the fractional differential equations has several advantages: (1) The method is computer oriented, thus solving higher order differential equation becomes a matter of dimension

Acknowledgements

The author would like to thank the reviewers for their suggestions to improve the quality of the paper.The work was supported by the Foundation of NUIST under Grant (20080305), Foundation of NUIST under Grant (20080153), Foundation of NUIST under Grant (20080256) and in part by Jiangsu Ordinary University Science Research Project under Grant 09KJB510007.

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