Baofeng Li

1 Introduction

Fractional calculus has been known for more than 300 years.These fractional phenomena allow us to describe a real object more accurately than the classical integer order methods.As we all know,the nature of real objects is fractional.However,for many of them the fractionality is very low.The fractional system describes many typical examples,such us the voltage current relation of a semi-infinite lossy transmission line[Wang(1987)],the diffusion of heat through a semi-infinite solid,where heat flow is equal to the half derivative of the temperature[Westerlund(2002)].In recent years,there are a lot of methods for approximation of fractional derivatives and integrals can be used in wide filed of applications.Fractional order calculus plays an important roles in electrical engineering[Nakagava and Sorimachi(1992)],physics[Valdes-Parada;Ochoa-Tapia;Alvarez-Ramirez(2007)],signal processing[Vinagre and Chen(2003);Tseng(2007)],robotics[Maria da Graca Marcos,Duarte,Tenreiro Machado(2008)],chemistry[Oldham and Spanier(1974)],chaos[Tavazoei and Haeri(2008)],and so on.In general,it is difficult to derive the analytical solutions to most of the fractional differential equations.Therefore,it is important to develop some reliable and efficient techniques to solve fractional differential equations[Chen,Yi,Chen and Yu(2012);Yi and Chen(2012);Chen,Sun,Li and Fu(2013)].The numerical solutions of fractional differential equations have attracted considerable attention from many researchers.The most commonly used methods are Variational Iteration Method[Zaid M.Odibat(2010)],Adomian Decomposition Method[EI-Kalla(2008)and Hosseini(2006)],and Generalized Differential Transform Method[Shaher and Zaid(2007);Zaid and Shaher(2008)].Wavelet basis approach has also been successfully employed to solve the factional differential equations.

The motivation of this paper is to extend the application of generalized hat functions to provide approximate solution of linear and nonlinear integro-differential equations of fractional order.The linear and nonlinear integro-differential equations of fractional order can be solved by many numerical methods.Saeedi and Moghadam[Saeedi and Moghadam(2011)]applied CAS wavelets method to solve the numerical solution of nonlinear Volterra integro-differential equations of fractional order and nonlinear Fredholm integro-differential equations of fractional order.In Refs.[Zhu and Fan(2013),Zhu and Fan(2012)],the authors solved the same integrodifferential equations by using the second kind Chebyshev wavelets[Babolian and Mordad(2011)].

The structure of this paper is as follows:In Section 2,the generalized hat functions are introduced.The generalized hat functions operational matrix of fractional integration is also introduced and the error analysis of generalized hat functions is given in Section 3.In Section 4,we summarize the application of generalized hat functions operational matrix method to the solution of the fractional integro-differential equation.Four numerical examples are provided to clarify the approach in Section 5.The conclusion is given in Section 6.

2 Generalized hat functions and their properties

Using the definition of generalized hat functions,we can obtain

An arbitrary functionu∈L2[0,T]is approximated in vector form as

whereUn+1=[u0,u1,...,un]Tand Ψn+1(x)=[ψ0(x),ψ1(x),...,ψn(x)]T.

Substituting Eq.(1)-(3)into the Eq.(6),we get the coefficients in Eq.(6)as following

3 Operational matrix of the integration for generalized hat functions

3.1　Fractional calculus

Before we introduce the generalized hat functions operational matrix of the fractional integration,we first review some basic definitions of fractional calculus,which have been given in[Li and Sun(2011)].

Definition 1.The Riemann-Liouville fractional integral of orderαis given by

Definition 2.The Caputo definition of fractional differential operator is given by

The Caputo fractional derivatives of orderαis also defined asDαu(x)=Jr?αDru(x),whereDris the usual integer differential operator of orderr.The relation between the Riemann-Liouville operator and Caputo operator is given by the following expressions:

3.2　Fractional order generalized hat functions operational matrix of integration.

IfJαis fractional integration operator of generalized hat functions,we can get:

Apart from the generalized hat functions,we consider another basis set of block pulse functions.The set of these functions,over the interval[0,T),is defined as

LetBn(x)=[b0(x),b1(x),...,bn?1(x)]T.SupposeJα(Bn(x))≈FαnBn(x),thenFαnis called the block pulse operational matrix of fractional integration[21],here

There is a relation between the block pulse functions and generalized hat functions,namely

3.3　Error analysis

In this section,from Eq.(6),we suppose

whereJαnu(x)denotes the approximation ofαorder Riemann-Liouville fractional integral ofu(x).Letεn(x)=|Jαu(x)?Jαnu(x)|,then we have the following theorem.

Theorem 3.1Ifu(x),x∈[0,T]is approximated by the Eq.(6),then

(ii)Ifjh

Using the Taylor’s series ofu(x),in the powers of(x?jh),we have

whereu(k)denotes thekth order derivative ofu(x).From Eq.(24)and Eq.(25),we get

2.1 兩組患者臨床療效比較 觀(guān)察組患者治愈率高于對(duì)照組，組間比較差異有統(tǒng)計(jì)學(xué)意義(χ2=8.362，P<0.05)，見(jiàn)表1。

(iii)According to the definition of the absolute errorεn(x),we obtain

Forjh

Substituting Eq.(27)into Eq.(29),we have

IfMax|u00(kh)|≤M,k=0,1,2,...,j,then we obtain

This completes the proof.

Whenα=0.5,m=32,the comparison results for the fractional integration is shown in Figure 1

4 The algorithm for finding numerical solution of fractional integro-differential equations

4.1　Linear fractional integro-differential equations

Consider the linear fractional integro-differential equations subject to initial conditions

Figure 1:0.5-order integration of the function u(t)=t.

whereu(s)(x)stands for thesth-order derivative ofu(x),Dα(·)denotes the Caputo fractional order derivative of orderα,f(x)is input term andu(x)is the output response.k1(x,t),k2(x,t)are given functions.λ1,λ2are real constants.

Now we approximateDαu(x),k1(x,t),k2(x,t)andf(x)in terms of generalized hat functions as follows

Now using Eq.(35)and Eq.(12),we obtain

Substituting Eq.(20)into Eq.(37),we have

Substituting the above equations into Eq.(33),we have

which is a linear system of algebraic equations.By solving this system we can obtain the approximation of Eq.(37).

4.2　Nonlinear fractional integro-differential equations

In this section we deal with nonlinear fractional integro-differential equation of the form

subject to initial conditions

wherep,q∈N,and the other parameters and variables are the same as the section 4.1.While dealing with such a situation,the same procedure(as in linear case)of expansion of fractional order derivatives via generalized hat functions is adopted with exception at the term containing[u(t)]p,[u(t)]q.

From Eq.(38),we haveu(x)≈EBn(x)and hence

Following the procedure of section 4.1 and using the Eq.(45)and Eq.(46),the Eq.(44)is transformed into a nonlinear system of algebraic equations

Solving the system of equations given by Eq.(47),the approximate numerical solutionu(x)is obtained.The Eq.(47)can be solved by iterative numerical technique such as Newton’s method.Also the Matlab function “fsolve”is available to deal with such a nonlinear system of algebraic equations.

5 Numerical examples

In order to illustrate the effectiveness of the proposed method,we consider numerical examples of linear and nonlinear nature.

Example 5.1Consider this equation:

Figure 2:Comparison of Num.sol.and Exa.Sol.of n=8.

Example 5.2Consider the following nonlinear equation:

Figure 3:Comparison of Num.sol.and Exa.Sol.of n=16.

Figure 4:Comparison of Num.sol.and Exa.Sol.of n=32.

Figure 5:Comparison of Num.sol.and Exa.Sol.of n=64.

Table 1:The absolute errors for different values of n.

We can see that the numerical solutions are more and more close to the exact solution with the value ofnbecomes large by taking a closer look at Figures 6-8.

Example 5.3Consider this equation:

Figure 6:Comparison of Num.sol.and Exa.Sol.of n=16 for Example 3.

Figure 7:Comparison of Num.sol.and Exa.Sol.of n=32for Example 3.

The comparison of numerical results forα=0.7,α=0.8,α=0.9,α=1 and the exact solution forα=1 are shown in Figure.9.

Figure 8:Comparison of Num.sol.and Exa.Sol.of n=64for Example 3.

Figure 9:Numerical solution and exact solution of α=1.

From Figure 9,we can see clearly that the numerical solutions are in very good agreement with the exact solution whenα=1.It is evident from the Figure 9 that,asαclose to 1,the numerical solutions by the generalized hat functions converge to the exact solution.

6 Conclusion

In this work,we introduce the generalized hat functions and operational matrix of the fractional integration.Using the operational matrix to solve the fractional linear and nonlinear integro-differential equations numerically.By solving the linear and nonlinear system,numerical solutions are obtained.The error analysis of generalized hat functions is proposed.The numerical results show that the approximations are in very good coincidence with the exact solution

Acknowledgement:This work is supported by the Natural Science Foundation of Tangshan Normal University(2014D09).

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