Solving fractional integral equations by the Haar wavelet method

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Abstract

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.

Introduction

Although the conception of the fractional derivatives was introduced already in the middle of the 19th century by Riemann and Liouville, the first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [1] published in 1974. After that the number of publications about the fractional calculus has rapidly increased. The reason for this is that some physical processes as anomalous diffusion, complex viscoelasticity, behaviour of mechatronic and biological systems, rheology etc. cannot be described adequately by the classical models.

At the present time we possess several excellent monographs about fractional calculus for example the book [2] by Kilbas et al., to which is also included a rather large and up-to-date Bibliography (928 items). Because of the enormous number of the papers about this topic we shall cite here only some papers which are more close to subject of this paper.

In a number of papers fractional differential equations are discussed; mostly these equations are transformed to fractional Volterra integral equations. For solution different techniques, as Fourier and Laplace transforms, power spectral density, Adomian decomposition method, path integration etc., are applied.

One-dimensional fractional harmonic oscillator is analysed in [3], [4], [5], [6]. In [3], [4] the solution is obtained in terms of Mittag–Leffler functions using Laplace transforms; several cases of the forcing function equation are considered. In [5] the fractional equation of motion is solved by the path integral method. In [6] the case, where the fractional derivatives only slightly differ from the ordinary derivatives, is analysed. Fractional Hamilton’s equations are discussed in [7]. In [8] multiorder fractional differential equations are solved by using the Adomian decomposition. In several papers fractional chaotic systems are discussed. In [8] a three-dimensional fractional chaotic oscillator model is proposed. Chaotic dynamics of the fractionally damped Duffing equation is investigated in [9]. Two chaotic models for third-order chaotic nonlinear systems are analysed in [10].

It is somewhat surprising that among different solution techniques the wavelet method has not attained much attention. We found only one paper [11] in which the wavelet method is applied for solving fractional differential equations; for this purpose the Daubechies wavelet functions are used.

Among the different wavelet families mathematically most simple are the Haar wavelets. Due to the simplicity the Haar wavelets are very effective for solving differential and integral equations (see e.g. [12], [13], [14], [15]). Therefore the idea, to apply Haar wavelet technique also for solving problems of fractional calculus, arises. This is the main aim of the present paper.

The paper is organized as follows. Sections 2 About the fractional calculus, 3 Haar wavelets are preparative: in Section 2, basic equations of the fractional calculus are briefly reviewed, in Section 3, the Haar wavelet method is described. In Section 4, two error estimates for the results, obtained by the Haar wavelet method, are introduced. In Section 5, the wavelet solution for fractional Volterra integral equations is presented. The one-dimensional fractional harmonic vibrations are discussed in Section 6. The solution for the fractional Fredholm integral equation is presented in Section 7.

Section snippets

About the fractional calculus

Let us briefly consider some basic formulae about the fractional calculus.

The Riemann–Liouville fractional integrals of order α are defined byIA+αf(x)=1Γ(α)Axf(t)(x-t)α-1dt(x>A,[α]>0)andIB-αf(x)=1Γ(α)xBf(t)(t-x)α-1dt(x<B,[α]>0).Here Γ(α) is the gamma function and [α] the integer part of α. The integrals (1), (2) are called left-sided and right-sided fractional integrals.

As to the fractional derivatives, then in this paper we shall use the Caputo derivatives defined asDαu(x)=1Γ(n-α)Axu(n)(t)(x

Haar wavelets

Usually the Haar wavelets are defined for the interval x[0,1]. In this paper the more general case x[A,B], is considered. Let us define the quantity M=2J, where J is the maximal level of resolution. We shall divide the interval [A,B] into 2M subintervals of equal length; each subinterval has the length Δx=(B-A)/(2M). Next two parameters are introduced: the dilatation parameter j for which j=0,1,,J and the translation parameter k=0,1,,m-1 (here the notation m=2j is introduced). The wavelet

Error estimates

It is essential to estimate the exactness of the obtained solutions, for this purpose in the following two error estimates are defined. Here we have to distinguish the two following situations.

  • (i)

    If the exact solution of the problem u=uex(x) is known we shall calculate the differences Δex(l)=u(xl)-uex(xl),l=1,2,,2M and define the error estimates as δex=maxl|Δex(l)| (local estimate) or σex=u-uex/2M (global estimate).

  • (ii)

    Mostly the exact solution is unknown. For this case the following procedure is

Fractional Volterra integral equation

The fractional Volterra integral equation has the form [2]u(x)-1Γ(α)0xK(x,t)(x-t)α-1u(t)dt=f(x),0x1..The kernel K(x,t) and the right-side function f(x) are given, α>0 is a real number. The value α=1 corresponds to the ordinary (nonfractional) Volterra equation.

According to the Haar wavelet method, the solution of (15) is sought in the form (8). Replacing (8), (15) and satisfying this equation in the collocation points we obtaini=12Mai[hi(xl)-gi(xl)]=f(xl),l=1,2,,2M.Here the symbol gi(xl)

Fractional harmonic vibrations

Consider the equationDαu(x)+λDβu(x)+νu(x)=f(x),x[0,B],where 1<α<2,0<β<1,λ,ν are prescribed constants, f(x) is the forcing term. To (23) belong initial conditions u(0)=u0,u(0)=v0. If α=2 and β=1 we get the usual differential equation of the harmonic oscillator.

The symbols Dα,Dβ denote left-sided Caputo derivatives, which are defined by (3). Since in the present case nα=[α]+1=2,nβ=[β]+1=1 the equation (23) gets the form1Γ(2-α)0x(x-t)1-αu(t)dt+λΓ(1-β)0x(x-t)-βu(t)dt+νu(x)=f(x).This is a

Fractional Fredholm integral equation

The usual Fredholm integral equation has the formu(x)-ABK(x,t)u(t)dt=f(x),x[A,B].If we want to get its fractional analogue we must take into account the fact that in the case of non-integer α the expression (x-t)α has sense only for xt. Therefore it seems reasonable to define the fractional Fredholm integral equation in the formu(x)-1Γ(α)Ax(x-t)α-1K(x,t)u(t)dt+xB(t-x)α-1K(x,t)u(t)dt=f(x).In (36) stand the left-sided and right-sided Riemann–Liouville integrals, respectively. As in Section 4

Conclusion

In the present paper three nonfractional equations are solved by the Haar wavelet method, these problems must be considered as examples for the recommended method of solution, since this approach can be easily carried over to solving some other problems, as e.g.

  • (i)

    first kind linear integral equations [12]:

  • (ii)

    integro–differential equations [12], [14];

  • (iii)

    weakly singular integral equations [12].

In the present paper only linear equations are considered, but the method is applicable also for nonlinear

Acknowledgement

Financial support from the Estonian Science Foundation under Grant ETF-6697 is gratefully acknowledged.

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