Numerical Study for a Class of Variable Order Fractional Integral-differential Equation in Terms of Bernstein Polynomials

Jinsheng Wang,Liqing Liu,Yiming Chen,Lechun Liuand Dayan Liu

Numerical Study for a Class of Variable Order Fractional Integral-differential Equation in Terms of Bernstein Polynomials

Jinsheng Wang1,Liqing Liu2,Yiming Chen2,Lechun Liu2and Dayan Liu3

The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sense.Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable.By solving the algebraic equations,the numerical solutions are acquired.The method in general is easy to implement and yields good results.Numerical examples are provided to demonstrate the validity and applicability of the method.

Bernstein polynomials,variable order fractional integral-differential equation,operational matrix,numerical solution,convergence analysis,the absolute error.

1 Introduction

In recent years,fractional calculus has attracted many researchers successfully in different disciplines of science and engineering.One of the main advantages of the fractional calculus is that the fractional derivatives provide a superior approach for the description of memory and hereditary properties of various materials and processes[Galue,Kalla and Al-Saqabi(2007)].Many numerical methods using different kinds of fractional derivative operators for solving different types of fractional differential equations have been proposed.The most commonly used ones are Adomian decomposition method(ADM)[EI-Sayed(1998)],Variational iteration method(VIM)[Odibat(2010)],Generalized differential transform method(GDTM)[Momani and Odibat(2007)],Generalized block pulse operational matrix method[Li and Sun.(2011)]and wavelet method[Yi and Chen(2012)]and so on.

Recently,more and more researchers are finding that numerous important dynamical problems exhibit fractional order behavior which may vary with space and time.This fact illustrates that variable order calculus provides an effective mathematical framework for the description of complex dynamical problems.The concept of a variable order operator is a much more recent development,which is a new orientation in science.Different authors have proposed different definitions of variable order differential operators,each of these with a specific meaning to suit desired goals.The variable order operator definitions recently proposed in the science include the Riemann-Liouvile definition,Caputo definition,Marchaud definition,Coimbra definition and Grünwald definition[Lorenzo and Hartley(2007);Coimbra(2003);Samko and Ross(1993);Samko.(1995);Soon,Coimbra and Kobayashi(2003)].

Since the kernel of the variable order operators is too complex for having a variable exponent,the numerical solutions of variable order fractional differential equations are quite difficult to obtain,and have not attracted much attention.To the best of the authors’knowledge,there are few references appeared on the discussion the numerical of variable order fractional differential equation.Coimbra[Coimbra(2003)]employed a consistent approximation with first-order accurate for the solution of variable order differential equations.Soon et al.[Soon,Coimbra and Kobayashi(2003)]proposed a second-order Runge–Kutta method which is consisting of an explicit Euler predictor step followed by an implicit Euler corrector step to numerically integrate the variable order differential equation.Lin et al.[Lin,Liu,and Anh(2009)]studied the stability and the convergence of an explicit finite difference approximation for the variable-order fractional diffusion equation with a nonlinear source term.Zhuang et al.[Zhuang,Liu and Anh(2009)]obtained explicit and implicit Euler approximations for the fractional advection–diffusion nonlinear equation of variable-order.Aiming a variable-order anomalous subdiffusion equation,Chen et al.[Chen,Liu and Anh(2010)]employed two numerical schemes one fourth order spatial accuracy and with first order temporal accuracy,the other with fourth order spatial accuracy and second order temporal accuracy.However,as far as we know,no one had attempted to seek the numerical solution of the variable order fractional differential equations.

So in this paper,we introduce the Bernstein polynomials to seek the numerical solution of the variable order fractional equation.With the simple structure and perfect properties[Yousefi,Behroozifar and Dehghan(2011)],the Bernstein polynomials play an important role in various areas of mathematics and engineering.Those polynomials have been widely used in the solution of integral equations and differential equations[Chen,Yi and Chen(2011)].

In this paper,our study focuses on a class of variable order fractional integral differential equation as follows:

Subject to the initial conditions

whereDα(x)(u(x,t)g(x,t)),0＜α(x)≤1 is fractional derivative in Caputo sense,wheng(x,t)=u(x,t),the initial problem is changed to nonlinear equation.Amongf(x,t),g(x,t),u(x,t),k(x,t)are assumed to be casual functions of time and space on the section(x,t)∈ [0,1]×[0,1],wheref(x,t),g(x,t),k(x,t)are known whileu(x,t)is the unknown.

The reminder of the paper is organized as follows:Sections 2 and 3 are preparative.In Section 4,four kinds of operational matrixes with Bernstein polynomials are derived and we applied the operational matrixes to solve the equation as given at beginning.In Section 5,we present some numerical examples to illustrative the method and to demonstrate efficiency of the method.We end the paper with a few concluding remarks in Section 6.

2 Basic definitions and properties of the variable order fractional derivatives

In this section,we firstly provide some basic definitions and properties of the variable order fractional order derivatives[Lorenzo and Hartley(2007);Coimbra(2003)].

Definition 2.1:Captuo’s fractional derivate with order α(t),(0 ＜ α(t)≤ 1)

With the definition above,we can get the following formula(0＜α(t)≤1):

3 Bernstein polynomials and their properties

3.1 The definition of Bernstein polynomials basis

The Bernstein polynomials of degreenare defined by

By using the binomial expansion of(1−x)n−i,Eq.(6)can be expressed as:

Now,we define:

where we can have

where

Clearly

3.2 Function approximation

A functionf(x)∈L2(0,1)can be expressed in terms of the Bernstein Polynomials basis.In practice,only the first(n+1)terms of Bernstein Polynomials are considered.Hence

Then we have

whereis the Hilbert matrix:

We can also approximate the functionu(x,t)∈L2([0,1]×[0,1])as following:

where

3.3 Convergence analysis

Suppose that the functionf:[0,1]→R ism+1 times continuously differentiable,Ifis the best approximation offout of Y,then the mean error bound is presented as follows:

Proof:We consider the Taylor polynomials

which we know

4 The operational matrix in terms of Bernstein polynomials

4.1 The operational matrix of the section asin terms of Bernstein polynomials

The differentiation of vector Φ (t)in Eq.(9)can be expressed as:

Define the(n+1)×(n)matrixand vectoras:

Eq.(22)may then be restated as

where

Then we have

Therefore we get the operational matrix of the section as as follows:

4.2 The operational matrix of the section as Dα(t)(u(x,t)g(x,t))in terms of Bernstein polynomials

If we approximate the functionu(x,t),g(x,t)with Bernstein polynomials,it can be written aswhereis unknown andis known.So we have:

Now we define

So we have

4.3 The operational matrix of the section asu(x,T)dT in terms of Bernstein polynomials

The integration of the vector Φ (t)in Eq.(9)can be expressed as:

Now we approximate the elements of vectorin terms of(t).By Eq.(12),then we have:

We define

So we have

4.4 The operational matrix of the section asu(x,T)k(x,T)dT in terms of Bernstein polynomials

Firstly,we approximate the functionk(x,t)with Bernstein polynomials,it can be written asandis known.So we have

We define

So the initial equation is transformed into the products of several dependent matrixes as follows:

Dispersing Eq.(43)with(xi,tj),(i=1,2,···,n;j=1,2,···,n)by using a symbolic software such as"Mathematica",we can obtain U.So the numerical solution of the original problem is obtained ultimately.

5 Numerical examples

Example1

where

The exact solution of the above equation is

Takingn=2,dispersingwe can obtain the matrixU U Uas follows:

The absolute error between the exact solution and the numerical solution is displayed asFigure 1.

Takingn=3,dispersingwe can obtain the matrixU U Uas follows:

Figure 1:The absolute error whenn=2.

Figure 2:The absolute error whenn=3.

Example 2

Where

The exact solution of the above problem is

Takingn=3,dispersingthe matrixis displayed as follows:

The absolute error between the exact solution and the numerical solution is displayed inTable 1.

Table 1:The absolute error when n=3.

Takingn=4,dispersingthe matrixis displayed as follows:

The absolute error between the exact solution and the numerical solution is displayed asTable 2.

Table 2:The absolute error when n=4.

Wheng(x,t)=u(x,t),the initial equation becomes nonlinear equation.Example 3describes the situation.

Example 3:

where

The exact solution of the above equation isu(x,t)=x2+t2

This is a nonlinear variable order fractional differential equation,the numerical solution can also be gained with the method proposed in Section 4 whenn≤2.

Takingn=2,dispersingwe can obtain the matrixas follows:

The

The absolute error between the exact solution and the numerical solution is displayed asFigure 3.

Figure 3:The absolute error for Example 3 of n=2.

Whenn≥3,the computation is very large,getting the numerical solution is a very difficult thing.

FromFigures 1-3,Tables 1-2,we can see that the absolute error is very tiny and only a small number of Bernstein polynomials are needed whenn≥3.The calculating results also show that combining with Bernstein polynomials,the method in this paper can be effectively used in the numerical solution of the fractional equation.At the same time the feasibility of the method can be also proved.From the above results,the numerical solutions are in good agreement with the exact solution.

6 Conclusions

This article uses Bernstein polynomials method to solve a class of the variable order fractional integral-differential equation by combining Bernstein polynomials with the properties of fractional differentiation.Actually we derive four kinds of operational matrixes using Bernstein polynomials.The matrixes are used to solve the numerical solutions of a class of fractional integral-differential equations effectively.We translate the initial equation into the product of some relevant matrixes which can also be regarded as the system of linear equations after dispersing the variable.And it is easy to solve by the least square method.Numerical examples illustrate the powerful of the proposed method.The solutions obtained using the suggested method show that numerical solutions are in very good coincidence with the exactsolution.The method can be applied by developing for the other fractional problem.

However,there are many issues to be resolved,such as the section Ω1=[0,1]×[0,1]is transformed to Ω2=[0,X]× [0,T],or the equations are nonlinear and so on.This requires the efforts of all of us.

Acknowledgement:This work is supported by the Natural Foundation of Hebei Province(A2012203407).The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers,which have improved the presentation.

Chen,C.M.;Liu,F.W.;Anh,V.;Turner,I.W.(2010):Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation.SIAM J.Sci.Comput:The DBLP Computer Science Bibliography,vol.32,no.4,pp.17401760.

Chen,Y.M.;Yi,M.X.;Chen,C.;Yu,C.X.(2011):Bernstein polynomials method for fractional convection-diffusion equation with variable coefficients.Computer Modeling in Engineering&Sciences,vol.83,no.6,pp.639-653.

Coimbra,C.F.M.(2003):Mechanics with variable-order differential operators.Ann.Phys,General&Introductory Physics,vol.12,no.1112,pp.692703.

EI-Sayed,A.M.A.(1998):Nonlinear functional differential equations of arbitrary orders.Nonliear Analysis:Theory,Methods&Applications,vol.33,no.2,pp.181-186.

Galue,L.;Kalla,S.L.;Al-Saqabi,B.N.(2007):Fractional extensions of the temperature field problems in oil strata.Applied Mathematics and Computation,vol.186,no.1,pp.3544.

Li,Y.L.;Sun,N.(2011):Numerical solution of fractional differential equations using the generalized block pulse operational matrix.Computers and Mathematics with Application,vol.62,no.3,pp.1046-1054.

Lin,R.;Liu,F.;Anh,V.;Turner,I.(2009):Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation.Applied Mathematics and Computation,vol.212,no.2,pp.435445.

Lorenzo,C.F.;Hartley,T.T.(2007):Initialization,conceptualization,and application in the generalized fractional calculus.Critical Reviews in Biomedical Engineering,vol.35,no.6,pp.447-553.

Momani,S.;Odibat,Z.(2007):Generalized differential transform method for solving a space and time-fractional diffusion-wave equation.Physics Letters A,vol.370,no.5-6,pp.379-387.

Odibat,Z.(2010):A study on the convergence of variational iteration method.Mathematical and Computer Modelling,vol.51,no.9-10,pp.1181-1192.

Samko,S.G.;Ross,B.(1993):Intergation and differentiation to a variable fractional order.Integral Transforms and Special Functions,vol.1,no.4,pp.277300.

Samko,S.G.(1995):Fractional integration and differentiation of variable order.Analysis Mathematica,vol.21 no.3,pp.213236.

Soon,C.M.;Coimbra,F.M.;Kobayashi,M.H.(2003):The variable viscoelasticity oscillator.Ann.Phys.,vol.12,no.11-12,pp.692-703.

Yi,M.X.;Chen,Y.M.(2012):Haar wavelet operational matrix method for solving fractional partial differential equations.Computer Modeling in Engineering&Sciences,vol.88,no.3,pp.229-244.

Yousef i,S.A.;Behroozifar,M.;Dehghan,M.(2011):The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass.Journal of Computational and Applied Mathematics,vol.235,no.17,pp.52725283.

Zhuang,P.;Liu,F.;Anh,V.;Turner,I.(2009):Numerical methods for the variable-order fractional advection-diffusion equation with anonlinear source term.SIAM J.Numer.Anal:Society for Industrial and Applied Mathematics Philadelphia,vol.47,no.3,pp.17601781.

1School of Continuing Education,Yanshan University,Qinhuangdao,066004,Hebei,China.

2College of Science,Yanshan University,Qinhuangdao,066004,Hebei,China.

3INSA Centre Val de Loire,Université d’Orléans,PRISME EA 4229,Bourges 18022 France.