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Mathematical Problems in Engineering
Volume 2011, Article ID 341989, 11 pages
http://dx.doi.org/10.1155/2011/341989
Research Article

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials

Department of Science, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, China

Received 2 June 2011; Accepted 23 August 2011

Academic Editor: Rafael Martinez-Guerra

Copyright © 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.

Abstract

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 [1]. 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 [2], 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 [35].

In this paper, we study the numerical solution of a nonlinear fractional integrodifferential equation of the second:𝐷𝛼𝑓(𝑥)𝜆10[]𝑘(𝑥,𝑡)𝑓(𝑡)𝑚𝑑𝑡=𝑔(𝑥),𝑚>1,(1.1) with the initial condition𝑓(𝑖)(0)=𝛿𝑖,𝑖=0,1,,𝑟1,𝑟1<𝛼𝑟,𝑟𝑁(1.2) 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 [611], He’s variational iteration method [1214], homotopy perturbation method [15, 16], homotopy analysis method [3], collocation method [17], Galerkin method [18], and other methods [1921]. 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 [22] 𝐽𝛼1𝑓(𝑥)=Γ(𝛼)𝑥0(𝑥𝑡)𝛼1𝐽𝑓(𝑡)𝑑𝑡,𝛼>0,𝑥>0,0𝑓(𝑥)=𝑓(𝑥).(2.1) It has the following properties: 𝐽𝛼𝑥𝛾=Γ(𝛾+1)𝑥Γ(𝛼+𝛾+1)𝛼+𝛾,𝛾>1.(2.2)

Definition 2.2. The Caputo definition of fractal derivative operator is given by 𝐷𝛼𝑓(𝑥)=𝐽𝑚𝛼𝐷𝑚1𝑓(𝑥)=Γ(𝑚𝛼)𝑥0(𝑥𝑡)𝑚𝛼1𝑓(𝑚)(𝑡)𝑑𝑡,(2.3) where𝑚1𝛼𝑚,𝑚𝑁,𝑥>0. It has the following two basic properties: 𝐷𝛼𝐽𝛼𝐽𝑓(𝑥)=𝑓(𝑥),𝛼𝐷𝛼𝑓(𝑥)=𝑓(𝑥)𝑚1𝑘=0𝑓(𝑘)0+𝑥𝑘𝑘!,𝑥>0.(2.4)

3. Properties of Hybrid Functions

3.1. Hybrid Functions of Block-Pulse and Chebyshev Polynomials

Hybrid functions 𝑛𝑚(𝑥),𝑛=1,2,,𝑁,𝑚=0,1,2,,𝑀1,are defined on the interval [0,1) as𝑛𝑚𝑇(𝑥)=𝑚(2𝑁𝑥2𝑛+1),𝑥𝑛1𝑁,𝑛𝑁0,otherwise(3.1) and 𝜔𝑛(𝑡)=𝜔(2𝑁𝑡2𝑛+1), where 𝑛 and 𝑚are 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=0𝑐𝑛𝑚𝑛𝑚(𝑥),(3.2) where 𝑐𝑛𝑚=𝑦(𝑥),𝑛𝑚(𝑥),(3.3) in which (,)denotes the inner product.

If 𝑦(𝑥) in (3.2) is truncated, then (3.2) can be written as𝑦(𝑥)=𝑁𝑛=1𝑀1𝑚=0𝑐𝑛𝑚𝑛𝑚(𝑥)=𝐶𝑇𝐻(𝑥)=𝐻𝑇(𝑥)𝐶,(3.4) where 𝐶and 𝐻(𝑥), given by𝑐𝐶=10,𝑐11,,𝑐1𝑀1,𝑐20,,𝑐2𝑀1,𝑐𝑁0,,𝑐𝑁𝑀1𝑇,(3.5)𝐻(𝑥)=10(𝑥),11(𝑥),,1𝑀1(𝑥),20(𝑥),,2𝑀1(𝑥),𝑁0(𝑥),,𝑁𝑀1(𝑥)𝑇.(3.6) In (3.4) and (3.5), 𝑐𝑛𝑚,𝑛=1,2,,𝑁,𝑚=0,1,,𝑀1,are the coefficients expansions of the function 𝑦(𝑥) and 𝑛𝑚(𝑥),𝑛=1,2,,𝑁,𝑚=0,1,,𝑀1,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 𝑥0𝐻(𝑡)𝑑𝑡𝑃𝐻(𝑥),(3.7) see [23], where 𝑃is 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 as𝑏𝑛𝑖(𝑥)=1,𝑚𝑡𝑖+1𝑚,0,otherwise,𝑖=0,1,2,,𝑚1.(3.8) The functions 𝑏𝑖(𝑥) are disjoint and orthogonal. That is,𝑏𝑖(𝑥)𝑏𝑗𝑏(𝑥)=0,𝑖𝑗,𝑗(𝑥),𝑖=𝑗.(3.9) 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),(3.10) where 𝐵(𝑥)=[𝑏1(𝑡),𝑏2(𝑡),,𝑏𝑁𝑀(𝑡)] and Φ is an 𝑀𝑁×𝑀𝑁 product operational matrix.

In [24], Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration 𝐹𝛼as follows:𝐽𝑎𝐵(𝑥)𝐹𝛼𝐵(𝑥),(3.11) where 𝐹𝛼=1𝑙𝛼1Γ(𝛼+2)1𝜉1𝜉2𝜉3𝜉𝑙101𝜉1𝜉2𝜉𝑙2001𝜉1𝜉𝑙30000𝜉100001,(3.12) with 𝜉𝑘=(𝑘+1)𝛼+12𝑘𝛼+1+(𝑘1)𝛼+1.

Next, we derive the hybrid function operational matrix of the fractional integration. Let 𝐽𝛼𝐻(𝑥)𝑃𝛼𝐻(𝑥),(3.13) where matrix 𝑃𝛼is called the hybrid function operational matrix of fractional integration.

Using (3.10) and (3.11), we have𝐽𝛼𝐻(𝑥)𝐽𝛼Φ𝐵(𝑥)=Φ𝐽𝛼𝐵(𝑥)Φ𝐹𝛼𝐵(𝑥).(3.14) From (3.10) and (3.13), we get𝑃𝛼𝐻(𝑥)=𝑃𝛼Φ𝐵(𝑥)=Φ𝐹𝛼𝐵(𝑥).(3.15) Then, the hybrid function operational matrix of fractional integration 𝑃𝛼 is given by𝑃𝛼=Φ𝐹𝛼Φ1.(3.16) 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)𝐶𝐶𝐻(𝑥),(3.17) where 𝐶𝐶=10𝐶002𝐶000𝑁(3.18) is an 𝑀𝑁×𝑀𝑁 product operational matrix. And, 𝐶𝑖𝑖=1,2,3,𝑁 are 𝑀×𝑀 matrices given by𝐶𝑖=122𝑐𝑖02𝑐𝑖12𝑐𝑖22𝑐𝑖32𝑐𝑖,𝑀22𝑐𝑖,𝑀1𝑐𝑖12𝑐𝑖0+𝑐𝑖2𝑐𝑖1+𝑐𝑖3𝑐𝑖2+𝑐𝑖4𝑐𝑖,𝑀3+𝑐𝑖,𝑀1𝑐𝑖,𝑀2𝑐𝑖2𝑐𝑖1+𝑐𝑖32𝑐𝑖0+𝑐𝑖4𝑐𝑖1+𝑐𝑖5𝑐𝑖,𝑀4𝑐𝑖,𝑀32𝑐𝑖0+𝑐𝑖𝑢𝑐𝑖1+𝑐𝑖,𝑢+1𝑐𝑖𝑣𝑐𝑖1+𝑐𝑖𝑢2𝑐𝑖02𝑐𝑖0𝑐𝑖1𝑐𝑖,𝑀1𝑐𝑖,𝑀2𝑐𝑖,𝑀3𝑐𝑖,𝑀4𝑐𝑖12𝑐𝑖0.(3.19) We also define the matrix 𝐷 as follows:𝐷=10𝐻(𝑥)𝐻𝑇(𝑥)𝑑𝑥.(3.20) For the hybrid functions of block-pulse and Chebyshev polynomials, 𝐷 has the following form:𝐷=𝐿000𝐿000𝐿,(3.21) where 𝐿is 𝑀×𝑀 nonsingular symmetric matrix given in [23].

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).(4.1) (see [25]), Now, let𝐷𝛼𝑓(𝑥)𝐴𝑇𝐻(𝑥).(4.2) 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).(4.3) Define𝑐𝐶=0,𝑐1,𝑐2,,𝑐𝑙1𝑇=𝐴𝑇𝑃𝛼,[]𝑓(𝑡)𝑚=𝐻𝑇(𝑡)𝐶𝑚=𝐶𝑇𝐻(𝑡)𝑚=𝐶𝑇𝐻(𝑡)𝐻𝑇𝐻(𝑡)𝐶𝑇(𝑡)𝐶𝑚2.(4.4) Applying (3.17) and (4.4), []𝑓(𝑡)𝑚=𝐴𝑇𝐻𝐶𝐻(𝑡)𝑇(𝑡)𝐶𝑚2=𝐴𝑇𝐶𝐻(𝑡)𝐻𝑇𝐵(𝑡)𝐶𝑇(𝑡)𝐶𝑚3,[]𝑓(𝑡)𝑚=𝐶𝑇𝐶𝑚1𝐻(𝑡)=𝐶𝐻(𝑡).(4.5) With substituting in (1.1), we have𝐻𝑇(𝑥)𝐴𝜆10𝐻𝑇(𝑥)𝐾𝐻(𝑡)𝐻𝑇(𝑡)𝐶𝑇𝑑𝑡=𝐻𝑇𝐻(𝑥)𝐺,𝑇(𝑥)𝐴𝜆𝐻𝑇(𝑥)𝐾10𝐻(𝑡)𝐻𝑇(𝑡)𝑑𝑡𝐶𝑇=𝐻𝑇(𝑥)𝐺(4.6) Applying (3.20), we get𝐴𝜆𝐾𝐷𝐶𝑇=𝐺,(4.7) which is a nonlinear system of equations. By solving this equation, we can find the vector𝐶.

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],𝐸𝑥𝑖=𝐴𝑇𝐻𝑥𝑖𝜆10𝑘𝑥𝑖𝐶,𝑡𝑥𝐻(𝑡)𝑑𝑡𝑔𝑖0.(4.8) If max 𝐸(𝑥𝑖)=10𝑘 (𝑘is any positive integer) is prescribed, then the truncation limit 𝑁is increased until the difference 𝐸(𝑥𝑖) at each of the points 𝑥𝑖 becomes smaller than the prescribed10𝑘.

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)10[]𝑥𝑡𝑓(𝑡)2𝑥𝑑𝑡=14,0𝑥<1,0<𝛼1,(5.1) (see [26]), with the initial condition 𝑓(0)=0.
The numerical results for 𝑀=1,𝑁=2,and 𝛼=1/4,1/2,3/4,and1 are plotted in Figure 1. For𝛼=1, we can get the exact solution𝑓(𝑥)=𝑥. From Figure 1, we can see the numerical solution is in very good agreement with the exact solution when𝛼=1.

341989.fig.001
Figure 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)10[]𝑥𝑡𝑓(𝑡)4𝑑𝑡=𝑔(𝑥),0𝑥<1,(5.2) with the initial condition 𝑓(0)=0 and 𝑔(𝑥)=(1/Γ(1/2))((8/3)𝑥32𝑥)(𝑥/1260), the exact solution is𝑓(𝑥)=𝑥2𝑥. Table 1 shows the numerical results for Example 5.2.

tab1
Table 1: Absolute error for𝛼=1/2and different values of M, N for Example 5.2.

Example 5.3. 𝐷5/3𝑓(𝑥)10(𝑥+𝑡)2[]𝑓(𝑡)3𝑑𝑡=𝑔(𝑥),0𝑥<1,(5.3) (see [12]), where 6𝑔(𝑥)=Γ(1/3)3𝑥𝑥27𝑥419,(5.4) and with these supplementary conditions 𝑓(0)=𝑓(0)=0. The exact solution is𝑓(𝑥)=𝑥2. Figures 2 and 3 illustrates the numerical results of Example 5.3 with𝑁=2,𝑀=3.

341989.fig.002
Figure 2: Exact and numerical solutions of Example 5.3 for 𝑁=2, 𝑀=3.
341989.fig.003
Figure 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.

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