Research Article | Open Access
Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials
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.
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 [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 block-pulse 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 , collocation method , Galerkin method , and other methods [19–21]. 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 is defined as  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 Block-Pulse and Chebyshev Polynomials
Hybrid functions are defined on the interval as and , where and are the orders of block-pulse 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.
3.3. Operational Matrix of the Fractional 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 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 NM-set of block pulse function as where and is an product operational matrix.
In , 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 Block-Pulse 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 block-pulse and Chebyshev polynomials, has the following form: where is nonsingular symmetric matrix given in .
4. Nonlinear Fredholm Integral Equations
Consider (1.1); we approximate by the way mentioned in Section 3 as (see ), 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 integro-differential equation:
(see ), 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 integro-differential equation: with the initial condition and , the exact solution is. Table 1 shows the numerical results for Example 5.2.
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 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 integro-differential equations, but some modifications are required.
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.
- J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology, vol. 15, no. 2, pp. 86–90, 1999.
- J. A. T. Machado, “Analysis and design of fractional-order digital control systems,” Systems Analysis Modelling Simulation, vol. 27, no. 2-3, pp. 107–122, 1997.
- I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674–684, 2009.
- Ü. Lepik, “Solving fractional integral equations by the Haar wavelet method,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 468–478, 2009.
- J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.
- S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006.
- S. Momani and M. A. Noor, “Numerical methods for fourth-order fractional integro-differential equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 754–760, 2006.
- V. Daftardar-Gejji and H. Jafari, “Solving a multi-order fractional differential equation using Adomian decomposition,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 541–548, 2007.
- S. S. Ray, K. S. Chaudhuri, and R. K. Bera, “Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 544–552, 2006.
- Q. Wang, “Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006.
- J.-F. Cheng and Y.-M. Chu, “Solution to the linear fractional differential equation using Adomian decomposition method,” Mathematical Problems in Engineering, vol. 2011, Article ID 587068, 14 pages, 2011.
- M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476–484, 2008.
- S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,” Physics Letters A, vol. 355, no. 4-5, pp. 271–279, 2006.
- F. Dal, “Application of variational iteration method to fractional hyperbolic partial differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 824385, 10 pages, 2009.
- S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345–350, 2007.
- N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.
- E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 1–6, 2006.
- V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
- P. Kumar and O. P. Agrawal, “An approximate method for numerical solution of fractional differential equations,” Signal Processing, vol. 86, no. 10, pp. 2602–2610, 2006.
- F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004.
- S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,” Journal of Computational Physics, vol. 216, no. 1, pp. 264–274, 2006.
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
- M. Razzaghi and H.-R. Marzban, “Direct method for variational problems via hybrid of block-pulse and Chebyshev functions,” Mathematical Problems in Engineering, vol. 6, no. 1, pp. 85–97, 2000.
- A. Kilicman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007.
- M. T. Kajani and A. H. Vencheh, “Solving second kind integral equations with hybrid Chebyshev and block-pulse functions,” Applied Mathematics and Computation, vol. 163, no. 1, pp. 71–77, 2005.
- H. Saeedi, M. M. Moghadam, N. Mollahasani, and G. N. Chuev, “A CAS wavelet method for solving nonlinear Fredholm Integro-differential equations of fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1154–1163, 2011.
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.