MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation34198910.1155/2011/341989341989Research ArticleNumerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev PolynomialsYangChangqingMartinez-GuerraRafaelDepartment of ScienceHuaihai Institute of TechnologyLianyungangJiangsu 222005Chinahhit.edu.cn201131102011201102062011230820112011Copyright © 2011 Changqing Yang.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm-type equations, which have many applications in mathematical physics, are then considered. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Chebyshev series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

1. Introduction

Over the last years, the fractional calculus has been used increasingly in different areas of applied science. This tendency could be explained by the deduction of knowledge models which describe real physical phenomena. In fact, the fractional derivative has been proved reliable to emphasize the long memory character in some physical domains especially with the diffusion principle. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives, and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow . In the fields of physics and chemistry, fractional derivatives and integrals are presently associated with the application of fractals in the modeling of electrochemical reactions, irreversibility, and electromagnetism , heat conduction in materials with memory, and radiation problems. Many mathematical formulations of mentioned phenomena contain nonlinear integrodifferential equations with fractional order. Nonlinear phenomena are also of fundamental importance in various fields of science and engineering. The nonlinear models of real-life problems are still difficult to be solved either numerically or theoretically. There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical or numerical, to nonlinear models .

In this paper, we study the numerical solution of a nonlinear fractional integrodifferential equation of the second:Dαf(x)-λ01k(x,t)[f(t)]mdt=g(x),m>1, with the initial conditionf(i)(0)=δi,i=0,1,,r-1,  r-1<αr,rN by hybrid of block-pulse functions and Chebyshev polynomials. Here, gL2([0,1)),  kL2([0,1)2)  are known functions; f(x) is unknown function. Dα is the Caputo fractional differentiation operator and mis a positive integer.

During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integrodifferential equations, and dynamic systems containing fractional derivatives, such as Adomian’s decomposition method , He’s variational iteration method , homotopy perturbation method [15, 16], homotopy analysis method , collocation method , Galerkin method , and other methods . But few papers reported application of hybrid function to solve the nonlinear fractional integro-differential equations.

The paper is organized as follows: in Section 2, we introduce the basic definitions and properties of the fractional calculus theory. In Section 3, we describe the basic formulation of hybrid block-pulse function and Chebyshev polynomials required for our subsequent. Section 4 is devoted to the solution of (1.1) by using hybrid functions. In Section 5, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples.

2. Basic Definitions

We give some basic definitions and properties of the fractional calculus theory, which are used further in this paper.

Definition 2.1.

The Riemann-Liouville fractional integral operator of order α0 is defined as  Jαf(x)=1Γ(α)0x(x-t)α-1f(t)dt,α>0,x>0,J0f(x)=f(x). It has the following properties: Jαxγ=Γ(γ+1)Γ(α+γ+1)xα+γ,γ>-1.

Definition 2.2.

The Caputo definition of fractal derivative operator is given by Dαf(x)=Jm-αDmf(x)=1Γ(m-α)0x(x-t)m-α-1f(m)(t)dt, where  m-1αm,  mN,  x>0. It has the following two basic properties: DαJαf(x)=f(x),JαDαf(x)=f(x)-k=0m-1f(k)(0+)xkk!,x>0.

3. Properties of Hybrid Functions3.1. Hybrid Functions of Block-Pulse and Chebyshev Polynomials

Hybrid functions hnm(x),  n=1,2,,N,  m=0,1,2,,M-1,  are defined on the interval [0,1) ashnm(x)={Tm(2Nx-2n+1),x[(n-1N),nN)0,otherwise and ωn(t)=ω(2Nt-2n+1), where n and m  are the orders of block-pulse functions and Chebyshev polynomials.

3.2. Function Approximation

A function y(x) defined over the interval 0 to 1 may be expanded asy(x)=n=1m=0cnmhnm(x), where cnm=(y(x),hnm(x)), in which (·,·)  denotes the inner product.

If y(x) in (3.2) is truncated, then (3.2) can be written asy(x)=n=1Nm=0M-1cnmhnm(x)=CTH(x)=HT(x)C, where C  and H(x), given byC=[c10,c11,,c1M-1,c20,,c2M-1,cN0,,cNM-1]T,H(x)=[h10(x),h11(x),,h1M-1(x),h20(x),,h2M-1(x),hN0(x),,hNM-1(x)]T. In (3.4) and (3.5), cnm,  n=1,2,,N,  m=0,1,,M-1,  are the coefficients expansions of the function y(x) and hnm(x),  n=1,2,,  N,    m=0,1,,  M-1,  are defined in (3.1).

3.3. Operational Matrix of the Fractional Integration

The integration of the vector H(x)  defined in (3.6) can be obtained as 0xH(t)dtPH(x), see , where P  is the MN×MNoperational matrix for integration.

Our purpose is to derive the hybrid functions operational matrix of the fractional integration. For this purpose, we consider an m-set of block pulse function asbn(x)={1,imti+1m,0,otherwise,i=0,1,2,,m-1. The functions bi(x) are disjoint and orthogonal. That is,bi(x)bj(x)={0,ij,bj(x),i=j. From the orthogonality of property, it is possible to expand functions into their block pulse series.

Similarly, hybrid function may be expanded into an NM-set of block pulse function asH(x)=ΦB(x), where B(x)=[b1(t),b2(t),,bNM(t)] and Φ is an MN×MN product operational matrix.

In , Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration Fα  as follows:JaB(x)FαB(x), where Fα=1lα1Γ(α+2)[1ξ1ξ2ξ3ξl-101ξ1ξ2ξl-2001ξ1ξl-30000ξ100001], with ξk=(k+1)α+1-2kα+1+(k-1)α+1.

Next, we derive the hybrid function operational matrix of the fractional integration. Let JαH(x)PαH(x), where matrix Pα  is called the hybrid function operational matrix of fractional integration.

Using (3.10) and (3.11), we haveJαH(x)JαΦB(x)=ΦJαB(x)ΦFαB(x). From (3.10) and (3.13), we getPαH(x)=PαΦB(x)=ΦFαB(x). Then, the hybrid function operational matrix of fractional integration Pα is given byPα=ΦFαΦ-1. Therefore, we have found the operational matrix of fractional integration for hybrid function.

3.4. The Product Operational of the Hybrid of Block-Pulse and Chebyshev Polynomials

The following property of the product of two hybrid function vectors will also be used.

Let H(x)HT(x)CC̃H(x), where C̃=(C̃1000C̃2000C̃N) is an MN×MN product operational matrix. And, C̃i  i=1,2,3,N are M×M matrices given byC̃i=12(2ci02ci12ci22ci32ci,M-22ci,M-1ci12ci0+ci2ci1+ci3ci2+ci4ci,M-3+ci,M-1ci,M-2ci2ci1+ci32ci0+ci4ci1+ci5ci,M-4ci,M-32ci0+ciuci1+ci,u+1civci1+ciu2ci02ci0ci1ci,M-1ci,M-2ci,M-3ci,M-4ci12ci0). We also define the matrix D as follows:D=01H(x)HT(x)dx. For the hybrid functions of block-pulse and Chebyshev polynomials, D has the following form:D=(L000L000L), where L  is M×M nonsingular symmetric matrix given in .

4. Nonlinear Fredholm Integral Equations

Consider (1.1); we approximate g(x),k(x,t) by the way mentioned in Section 3 asg(x)=HT(x)G,k(x,t)=HT(x)KH(t). (see ), Now, letDαf(x)ATH(x). For simplicity, we can assume that δi=0 (in the initial condition). Hence by using (2.4) and (3.13), we have f(x)ATPαH(x). DefineC=[c0,c1,c2,,cl-1]T=ATPα,[f(t)]m=[HT(t)C]m=[CTH(t)]m=CTH(t)HT(t)C  [HT(t)C]m-2. Applying (3.17) and (4.4), [f(t)]m=ATC̃H(t)[HT(t)C]m-2=ATC̃H(t)HT(t)C  [BT(t)C]m-3,[f(t)]m=CT[C̃]m-1H(t)=C*H(t). With substituting in (1.1), we haveHT(x)A-λ01HT(x)KH(t)HT(t)C*Tdt=HT(x)G,HT(x)A-λHT(x)K01H(t)HT(t)dtC*T=HT(x)G Applying (3.20), we getA-λKDC*T=G, which is a nonlinear system of equations. By solving this equation, we can find the vector  C.

We can easily verify the accuracy of the method. Given that the truncated hybrid function in (3.4) is an approximate solution of (1.1), it must have approximately satisfied these equations. Thus, for each xi[0,1],E(xi)=ATH(xi)-λ01k(xi,t)C*H(t)dt-g(xi)0. If max E(xi)=10-k (k  is any positive integer) is prescribed, then the truncation limit N  is increased until the difference E(xi) at each of the points xi becomes smaller than the prescribed  10-k.

5. Numerical Examples

In this section, we applied the method presented in this paper for solving integral equation of the form (1.1) and solved some examples.

Example 5.1.

Let us first consider fractional nonlinear integro-differential equation: Dαf(x)-01xt[f(t)]2dt=1-x4,0x<1,  0<α1, (see ), with the initial condition f(0)=0.

The numerical results for M=1,  N=2,  and α=1/4,1/2,3/4,  and  1 are plotted in Figure 1. For  α=1, we can get the exact solution  f(x)=x. From Figure 1, we can see the numerical solution is in very good agreement with the exact solution when  α=1.

The approximate solution of Example 5.1 for N=1, M=2.

Example 5.2.

As the second example considers the following fractional nonlinear integro-differential equation: D1/2f(x)-01xt[f(t)]4dt=g(x),0x<1, with the initial condition f(0)=0 and g(x)=(1/Γ(1/2))((8/3)x3-2x)-(x/1260), the exact solution is  f(x)=x2-x. Table 1 shows the numerical results for Example 5.2.

Absolute error for  α=1/2  and different values of M, N for Example 5.2.

xN=2, M=3N=3, M=3N=4, M=3
0.15.1985e-0031.5106e-0042.8496e-006
0.21.1372e-0032.4887e-0043.9120e-006
0.37.4698e-0043.2711e-0044.6808e-006
0.41.2729e-0037.0337e-0053.1231e-006
0.54.6736e-0034.3451e-0043.2653e-006
0.61.2160e-0032.5000e-0062.6369e-006
0.76.0767e-0046.2935e-0054.7123e-007
0.86.0442e-0043.2421e-0044.8631e-006
0.91.2039e-0036.2276e-0052.0707e-006
Example 5.3.

D5/3f(x)-01(x+t)2[f(t)]3dt=g(x),0x<1, (see ), where g(x)=6Γ(1/3)x3-x27-x4-19, and with these supplementary conditions f(0)=f(0)=0. The exact solution is  f(x)=x2. Figures 2 and 3 illustrates the numerical results of Example 5.3 with  N=2,M=3.

Exact and numerical solutions of Example 5.3 for N=2, M=3.

Absolute error of Example 5.3 for N=2, M=3.

6. Conclusion

We have solved the nonlinear Fredholm integro-differential equations of fractional order by using hybrid of block-pulse functions and Chebyshev polynomials. The properties of hybrid of block-pulse functions and Chebyshev polynomials are used to reduce the equation to the solution of nonlinear algebraic equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method. The advantages of hybrid functions are that the values of N and M are adjustable as well as being able to yield more accurate numerical solutions. Also hybrid functions have good advantage in dealing with piecewise continuous functions.

The method can be extended and applied to the system of nonlinear integral equations, linear and nonlinear integro-differential equations, but some modifications are required.

Acknowledgments

The authors are grateful to the reviewers for their comments as well as to the National Natural Science Foundation of China which provided support through Grant no. 40806011.

HeJ. H.Some applications of nonlinear fractional differential equations and their approximationsBulletin of Science, Technology19991528690MachadoJ. A. T.Analysis and design of fractional-order digital control systemsSystems Analysis Modelling Simulation1997272-31071222-s2.0-0030651602ZBL0875.93154HashimI.AbdulazizO.MomaniS.Homotopy analysis method for fractional IVPsCommunications in Nonlinear Science and Numerical Simulation2009143674684244987910.1016/j.cnsns.2007.09.014LepikÜ.Solving fractional integral equations by the Haar wavelet methodApplied Mathematics and Computation20092142468478254168310.1016/j.amc.2009.04.015ZBL1170.65106MachadoJ. T.KiryakovaV.MainardiF.Recent history of fractional calculusCommunications in Nonlinear Science and Numerical Simulation201116311401153273662210.1016/j.cnsns.2010.05.027MomaniS.ShawagfehN.Decomposition method for solving fractional Riccati differential equationsApplied Mathematics and Computation2006182210831092228255210.1016/j.amc.2006.05.008ZBL1107.65121MomaniS.NoorM. A.Numerical methods for fourth-order fractional integro-differential equationsApplied Mathematics and Computation20061821754760229208310.1016/j.amc.2006.04.041ZBL1107.65120Daftardar-GejjiV.JafariH.Solving a multi-order fractional differential equation using Adomian decompositionApplied Mathematics and Computation20071891541548233023110.1016/j.amc.2006.11.129ZBL1122.65411RayS. S.ChaudhuriK. S.BeraR. K.Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition methodApplied Mathematics and Computation20061821544552229206410.1016/j.amc.2006.04.016ZBL1108.65129WangQ.Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition methodApplied Mathematics and Computation2006182210481055228254810.1016/j.amc.2006.05.004ZBL1107.65124ChengJ.-F.ChuY.-M.Solution to the linear fractional differential equation using Adomian decomposition methodMathematical Problems in Engineering20112011145870682674394ZBL1202.65108IncM.The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration methodJournal of Mathematical Analysis and Applications20083451476484242266510.1016/j.jmaa.2008.04.007ZBL1146.35304MomaniS.OdibatZ.Analytical approach to linear fractional partial differential equations arising in fluid mechanicsPhysics Letters A20063554-52712792-s2.0-3364687810610.1016/j.physleta.2006.02.048DalF.Application of variational iteration method to fractional hyperbolic partial differential equationsMathematical Problems in Engineering20092009108243852578375ZBL1190.65185MomaniS.OdibatZ.Homotopy perturbation method for nonlinear partial differential equations of fractional orderPhysics Letters A20073655-6345350230877610.1016/j.physleta.2007.01.046ZBL1203.65212SweilamN. H.KhaderM. M.Al-BarR. F.Numerical studies for a multi-order fractional differential equationPhysics Letters A20073711-22633241927410.1016/j.physleta.2007.06.016ZBL1209.65116RawashdehE. A.Numerical solution of fractional integro-differential equations by collocation methodApplied Mathematics and Computation2006176116223332510.1016/j.amc.2005.09.059ZBL1106.65111ErvinV. J.RoopJ. P.Variational formulation for the stationary fractional advection dispersion equationNumerical Methods for Partial Differential Equations2006223558576221222610.1002/num.20112ZBL1095.65118KumarP.AgrawalO. P.An approximate method for numerical solution of fractional differential equationsSignal Processing20068610260226102-s2.0-3374571207610.1016/j.sigpro.2006.02.007ZBL1172.94436LiuF.AnhV.TurnerI.Numerical solution of the space fractional Fokker-Planck equationJournal of Computational and Applied Mathematics200416612092192-s2.0-154242510210.1016/j.cam.2003.09.028ZBL1036.82019YusteS. B.Weighted average finite difference methods for fractional diffusion equationsJournal of Computational Physics20062161264274222344410.1016/j.jcp.2005.12.006ZBL1094.65085PodlubnyI.Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Equations, to Methods of Their Solution and Some of Their Applications1999198New York, NY, USAAcademic Pressxxiv+340Mathematics in Science and Engineering1658022RazzaghiM.MarzbanH.-R.Direct method for variational problems via hybrid of block-pulse and Chebyshev functionsMathematical Problems in Engineering200061859710.1155/S1024123X000012652034822ZBL0987.65055KilicmanA.Al ZhourZ. A. A.Kronecker operational matrices for fractional calculus and some applicationsApplied Mathematics and Computation20071871250265232357710.1016/j.amc.2006.08.122ZBL1123.65063KajaniM. T.VenchehA. H.Solving second kind integral equations with hybrid Chebyshev and block-pulse functionsApplied Mathematics and Computation200516317177211557610.1016/j.amc.2003.11.044ZBL1067.65151SaeediH.MoghadamM. M.MollahasaniN.ChuevG. N.A CAS wavelet method for solving nonlinear Fredholm Integro-differential equations of fractional orderCommunications in Nonlinear Science and Numerical Simulation201116311541163273662310.1016/j.cnsns.2010.05.036