Research Article  Open Access
Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of BlockPulse Functions and Chebyshev Polynomials
Abstract
A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholmtype equations, which have many applications in mathematical physics, are then considered. The method is based upon hybrid function approximate. The properties of hybrid of blockpulse 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 fluiddynamic 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 reallife 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 [3–5].
In this paper, we study the numerical solution of a nonlinear fractional integrodifferential equation of the second: with the initial condition by hybrid of blockpulse functions and Chebyshev polynomials. Here, are known functions; is unknown function. is the Caputo fractional differentiation operator and is 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 [6–11], He’s variational iteration method [12–14], homotopy perturbation method [15, 16], homotopy analysis method [3], collocation method [17], Galerkin method [18], and other methods [19–21]. But few papers reported application of hybrid function to solve the nonlinear fractional integrodifferential 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 blockpulse 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 RiemannLiouville fractional integral operator of order is defined as [22] It has the following properties:
Definition 2.2. The Caputo definition of fractal derivative operator is given by where. It has the following two basic properties:
3. Properties of Hybrid Functions
3.1. Hybrid Functions of BlockPulse and Chebyshev Polynomials
Hybrid functions are defined on the interval as and , where and are the orders of blockpulse functions and Chebyshev polynomials.
3.2. Function Approximation
A function defined over the interval 0 to 1 may be expanded as where in which denotes the inner product.
If in (3.2) is truncated, then (3.2) can be written as where and , given by In (3.4) and (3.5), are the coefficients expansions of the function and are defined in (3.1).
3.3. Operational Matrix of the Fractional Integration
The integration of the vector defined in (3.6) can be obtained as see [23], where is the operational matrix for integration.
Our purpose is to derive the hybrid functions operational matrix of the fractional integration. For this purpose, we consider an mset of block pulse function as The functions are disjoint and orthogonal. That is, From the orthogonality of property, it is possible to expand functions into their block pulse series.
Similarly, hybrid function may be expanded into an NMset of block pulse function as where and is an product operational matrix.
In [24], Kilicman and Al Zhour have given the block pulse operational matrix of the fractional integration as follows: where with .
Next, we derive the hybrid function operational matrix of the fractional integration. Let where matrix is called the hybrid function operational matrix of fractional integration.
Using (3.10) and (3.11), we have From (3.10) and (3.13), we get Then, the hybrid function operational matrix of fractional integration is given by Therefore, we have found the operational matrix of fractional integration for hybrid function.
3.4. The Product Operational of the Hybrid of BlockPulse and Chebyshev Polynomials
The following property of the product of two hybrid function vectors will also be used.
Let where is an product operational matrix. And, are matrices given by We also define the matrix as follows: For the hybrid functions of blockpulse and Chebyshev polynomials, has the following form: where is nonsingular symmetric matrix given in [23].
4. Nonlinear Fredholm Integral Equations
Consider (1.1); we approximate by the way mentioned in Section 3 as (see [25]), Now, let For simplicity, we can assume that (in the initial condition). Hence by using (2.4) and (3.13), we have Define Applying (3.17) and (4.4), With substituting in (1.1), we have Applying (3.20), we get which is a nonlinear system of equations. By solving this equation, we can find the vector.
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 , If max (is any positive integer) is prescribed, then the truncation limit is increased until the difference at each of the points becomes smaller than the prescribed.
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 integrodifferential equation:
(see [26]), with the initial condition .
The numerical results for and and are plotted in Figure 1. For, we can get the exact solution. From Figure 1, we can see the numerical solution is in very good agreement with the exact solution when.
Example 5.2. As the second example considers the following fractional nonlinear integrodifferential equation: with the initial condition and , the exact solution is. Table 1 shows the numerical results for Example 5.2.

Example 5.3. (see [12]), where and with these supplementary conditions . The exact solution is. Figures 2 and 3 illustrates the numerical results of Example 5.3 with.
6. Conclusion
We have solved the nonlinear Fredholm integrodifferential equations of fractional order by using hybrid of blockpulse functions and Chebyshev polynomials. The properties of hybrid of blockpulse 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 and 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 integrodifferential 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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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.