Research Article  Open Access
Solution of Nonlinear VolterraFredholm Integrodifferential Equations via Hybrid of BlockPulse Functions and Lagrange Interpolating Polynomials
Abstract
An efficient hybrid method is developed to approximate the solution of the highorder nonlinear VolterraFredholm integrodifferential equations. The properties of hybrid functions consisting of blockpulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear VolterraFredholm integrodifferential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results.
1. Introduction
Integral and integrodifferential equations have many applications in various fields of science and engineering such as biological models, industrial mathematics, control theory of financial mathematics, economics, electrostatics, fluid dynamics, heat and mass transfer, oscillation theory, queuing theory, and so forth [1].
It is well known that it is extremely difficult to analytically solve nonlinear integrodifferential equations. Indeed, few of these equations can be solved explicitly. So it is required to devise an efficient approximation scheme for solving these equations. So far, several numerical methods are developed. The solution of the first order integrodifferential equations has been obtained by the numerical integration methods such as EulerChebyshev [2] and RungeKutta methods [3].
Moreover, a differential transform method for solving integrodifferential equations was introduced in [4]. Shidfar et al. [5] applied the homotopy analysis method for solving the nonlinear Volterra and Fredholm integrodifferential equations. As a concrete example, we can express the mathematical model of celltocell spread of HIV1 in tissue cultures considered by Mittler et al. [6]. Yalcinbas and Sezer [7] proposed an approximation scheme based on Taylor polynomials for solving the highorder linear VolterraFredholm integrodifferential equations of the following form: Maleknejad and Mahmoudi [8] developed a numerical method by using Taylor polynomials to solve the following type of nonlinear VolterraFredholm integrodifferential equations: Darania and Ivaz [9] suggested an efficient analytical and numerical procedure for solving the most general form of nonlinear VolterraFredholm integrodifferential equations under the mixed conditions where and are constants and . Moreover, , , , , , and , , are functions that have suitable derivatives on the interval .
These kinds of equations can be found in numerous applications such as electrodynamics, electromagnetic, biomechanics, and elasticity [10–12]. Akyüz and Sezer [13] presented a Taylor polynomial approach for solving highorder linear Fredholm integrodifferential equations in the most general form. Streltsov [14] developed an effective numerical method based on the use of Chebyshev and Legendre polynomials for solving Fredholm integral equations. Babolian et al. [15] suggested an effective direct method to determine the numerical solution of the specific nonlinear VolterraFredholm integrodifferential equations. Their approach was based on triangular functions. In [16, 17], the variational iteration method (VIM) was employed for solving integral and integrodifferential equations. In addition, iterative and noniterative methods for the solution of nonlinear Volterra integrodifferential equations were presented and their local convergence was proved. The iterative methods provide a sequence solution and make use of fixedpoint theory whereas the noniterative ones result in series solutions and also make use of the fixedpoint principles [18]. In recent years, the meshless methods have gained more attention not only by mathematicians but also in the engineering community. In [19], the meshless moving least square method was employed for solving nonlinear Fredholm integrodifferential equations.
In the present paper, we introduce a new numerical method for solving (1.3) with the mixed conditions (1.4). The method is based on a hybrid of blockpulse functions and Lagrange interpolating polynomials. The associated operational matrices of integration and product together with the Kronecker property of hybrid functions are then used to reduce the solution of (1.3) and (1.4) to the solution of nonlinear algebraic equations whose solution is much more easier than the original one.
The paper is organized as follows. In Section 2, we describe the basic properties of hybrid functions. Section 3 is devoted to the solution of the nonlinear VolterraFredholm integrodifferential equations. Finally, in Section 4, various types of integrodifferential equations are given to demonstrate the efficiency and the accuracy of the proposed method.
2. Hybrid Functions
2.1. Properties of Hybrid Functions
In order to define hybrid functions, we first divide the interval into equidistant subintervals , for . The hybrid of blockpulse functions and Lagrange interpolating polynomials, , , , are defined on the interval as, [20], where and are the orders of blockpulse functions and Lagrange interpolating polynomials, respectively. Here, are defined in [21] as in which , are the zeros of Legendre polynomial of order , with the Kronecker property where is the Kronecker delta function. No explicit formulas are known for the points . However, they can be computed numerically using existing subroutines [21]. Since consists of blockpulse functions and Lagrange interpolating polynomials, which are both complete and orthogonal, the set of hybrid of blockpulse functions and Lagrange interpolating polynomials is a complete orthogonal set in the Hilbert space . A more detailed error analysis concerning this type of hybrid functions was discussed in [22].
2.2. Function Approximation
A function may be expanded as
If the infinite series in (2.4) is truncated, then (2.4) can be written as where In (2.6), , , are the expansion coefficients of the function in the th subinterval , and , , are defined in (2.1). In order to evaluate the coefficients , we use Gaussian nodes which are the zeros of Legendre polynomial of order . Let , be the Gaussian nodes which are defined in and let , , be the corresponding Gaussian nodes in the th subinterval . The relation between and is given by With the aid of (2.3) we have Using (2.8), the coefficients , can be obtained as
Now, let be a function of two independent variables defined for and . Then can be expanded in terms of hybrid functions as follows: where is a matrix of order whose elements can be calculated for , and as follows: in which can be determined by the following relation: The integration of the vector can also be expanded in terms of hybrid functions as where is the operational matrix of integration and given by, [20], in which and can be obtained in the following manner.
Let then, for , we have where , are the zeros of Legendre polynomial of order . It is noted that is the associated operational matrix of integration for Lagrange interpolating polynomials on interval .
Integrating (2.5) from to and using (2.13) imply From (2.13), by using the interpolating property of hybrid functions we get
2.3. The Integration of the CrossProduct of Two Hybrid Function Vectors
The following property of the crossproduct of two hybrid function vectors will also be used. Integrating the square matrix from to , and using the orthogonality of hybrid functions, we get where is a block diagonal matrix of order given by moreover is also a diagonal matrix of order given as According to the orthogonality of hybrid functions, the values of , for , can be computed as follows:
2.4. The Operational Matrix of Derivative
Let denote the derivative operator. By expanding in terms of we get In (2.23), is called the operational matrix of derivative for hybrid functions which is a block diagonal matrix of order and is given by where is also a matrix of order and the th element of this matrix denoted by for can be calculated as follows: Let denote the first derivative of with respect to . By expanding in terms of hybrid functions and using (2.5) and (2.23) we obtain According to (2.8) we get Let denote the th derivative of with respect to . In order to obtain the relation between the expansion coefficients of and , we use the following fact: where is the th power of the matrix given in (2.23). Equation (2.28) can easily be obtained from (2.23) by differentiating , times with respect to , . It should be noted that the matrix has the following structure: By expanding in terms of hybrid functions and using (2.5) and (2.28) we get According to (2.8) it immediately follows that where denotes the th element of the matrix .
3. Solution of the Nonlinear VolterraFredholm Integrodifferential Equations
Consider the nonlinear VolterraFredholm integrodifferential equation given in (1.3). Since the set of hybrid functions is a complete orthogonal set in the Hilbert space , we can expand any function in this space in terms of hybrid functions.
Let where Using (2.30), for , we have where Moreover, by expanding in terms of hybrid functions we get where is a matrix of order and can be calculated similar to matrix in (2.10). From (2.5) and (2.9) we obtain where is a vector of order . From (2.19), (3.5) and (3.6), we have where It should be noted that is a vector of order . Substituting (3.1)–(3.7) in (1.3) yields We now collocate (1.3) at points. For a suitable collocation points, we use Gaussian nodes , , which are introduced in (2.7). Therefore we have Also, using (2.8) we get where denotes the th element of vector . Consequently, with the aid of (2.18) and (3.10) we obtain where for , can be calculated from (3.4). Equation (3.12) gives nonlinear equations. Finally, we approximate the mixed conditions given in (1.4). From (1.4), (3.1), (3.3), and the definition of hybrid functions we obtain
In order that the approximate solution obtained by the present method be continuous, we impose the continuity condition at the points , . Equations. (3.12)(3.13) together with the continuity conditions give , nonlinear equations that can be solved by using the wellknown Tau method [23] for the unknown coefficients , , .
4. Illustrative Examples
In this section, four examples are given to demonstrate the efficiency, the accuracy, and the applicability of the proposed method.
Example 4.1. Consider the nonlinear VolterraFredholm integrodifferential equation
whose exact solution is given in [9] as . We applied the present method to solve this problem.
Let us define
where and denote the approximate solution obtained by the present method and the exact solution, respectively. In Table 1, the computational results of the norm error denoted by between the approximate solution and the exact solution for different values of and are given. The simulation results reported in Table 1, indicate that there is an excellent agreement between the approximate and exact solution.

Example 4.2. As the second example, consider the nonlinear Volterra integrodifferential equation whose exact solution is given in [19] as = . We applied the present method to solve this problem. In Table 2, the computational results of the norm error between the approximate solution and the exact solution for different values of and are summarized. The simulation results obtained by the present method are superior to those reported in [19].

Example 4.3. Let us consider the nonlinear Volterra integrodifferential equation whose exact solution is given in [24] as . We solved this problem by using hybrid functions. In Table 3, the computational results of obtained by the present method with and together with the exact solution and the solution determined by the Tau method [24] are summarized. The computational results obtained by the present method are superior to that reported in [24]. Moreover in Table 4, the norm error between the approximate solution obtained by the present method and the exact solution for different values of and is given. It should be pointed out that only a very small numbers of hybrid basis functions are needed to obtain a quite satisfactory approximation to the exact solution.


Example 4.4. As the last example, consider the nonlinear VolterraFredholm integrodifferential equation which has the exact solution . To measure the accuracy of proposed approach, the norm error between the approximate solution obtained by the present method for different values of and and the exact solution is given in Table 5. The computational results reveal that the method is convergent.

5. Conclusion
In the present work, a hybrid approximation method was developed for solving the highorder nonlinear VolterraFredholm integrodifferential equations. The nice properties of hybrid functions together with the associated operational matrices of integration, derivative, and crossproduct are used to reduce the solution of problem to the solution of nonlinear algebraic equations whose solution is much more easier than the original one. The matrices , , and are sparse, hence making the method computationally attractive without sacrificing the accuracy of the solution. Moreover, It was shown that small values for and are needed to achieve high accuracy and a satisfactory convergence. The numerical results support this claim. The method is general, easy to implement and yields the desired accuracy in a few terms of hybrid basis functions. The simulation results indicate the convergence and the effectiveness of the proposed approach.
Acknowledgments
The authors are very grateful to the referees for their comments and valuable suggestions which improved the paper.
References
 A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2008. View at: Publisher Site  Zentralblatt MATH
 P. J. Van der Houwen and B. P. Sommeijer, “EulerChebyshev methods for integrodifferential equations,” Applied Numerical Mathematics, vol. 24, no. 23, pp. 203–218, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 W. H. Enright and M. Hu, “Continuous RungeKutta methods for neutral Volterra integrodifferential equations with delay,” Applied Numerical Mathematics, vol. 24, no. 23, pp. 175–190, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 P. Darania and A. Ebadian, “A method for the numerical solution of the integrodifferential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 657–668, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Shidfar, A. Molabahrami, A. Babaei, and A. Yazdanian, “A series solution of the nonlinear Volterra and Fredholm integrodifferential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 205–215, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, “Influence of delayed viral production on viral dynamics in HIV1 infected patients,” Mathematical Biosciences, vol. 152, no. 2, pp. 143–163, 1998. View at: Publisher Site  Google Scholar
 S. Yalcinbas and M. Sezer, “The approximate solution of highorder linear VolterraFredholm integrodifferential equations in terms of Taylor polynomials,” Applied Mathematics and Computation, vol. 112, no. 23, pp. 291–308, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 K. Maleknejad and Y. Mahmoudi, “Taylor polynomial solution of highorder nonlinear VolterraFredholm integrodifferential equations,” Applied Mathematics and Computation, vol. 145, no. 23, pp. 641–653, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 P. Darania and K. Ivaz, “Numerical solution of nonlinear VolterraFredholm integrodifferential equations,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2197–2209, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 F. Bloom, “Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 524–542, 1980. View at: Publisher Site  Google Scholar
 K. Holmåker, “Global asymptotic stability for a stationary solution of a system of integrodifferential equations describing the formation of liver zones,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 116–128, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. K. Forbes, S. Crozier, and D. M. Doddrell, “Calculating current densities and fields produced by shielded magnetic resonance imaging probes,” SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 401–425, 1997. View at: Publisher Site  Google Scholar
 A. Akyüz and M. Sezer, “A Taylor polynomial approach for solving highorder linear Fredholm integrodifferential equations in the most general form,” International Journal of Computer Mathematics, vol. 84, no. 4, pp. 527–539, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 I. P. Streltsov, “Application of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations,” Computer Physics Communications, vol. 126, no. 12, pp. 178–181, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. Babolian, Z. Masouri, and S. HatamzadehVarmazyar, “Numerical solution of nonlinear VolterraFredholm integrodifferential equations via direct method using triangular functions,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 239–247, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 N. Bildik, A. Konuralp, and S. Yalçınbaş, “Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integrodifferential equations,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 1909–1917, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S.Q. Wang and J.H. He, “Variational iteration method for solving integrodifferential equations,” Physics Letters A, vol. 367, no. 3, pp. 188–191, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. I. Ramos, “Iterative and noniterative methods for nonlinear Volterra integrodifferential equations,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 287–296, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Dehghan and R. Salehi, “The numerical solution of the nonlinear integrodifferential equations based on the meshless method,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2367–2377, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. R. Marzban, S. M. Hoseini, and M. Razzaghi, “Solution of Volterra's population model via blockpulse functions and Lagrangeinterpolating polynomials,” Mathematical Methods in the Applied Sciences, vol. 32, no. 2, pp. 127–134, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1987.
 H. R. Marzban, H. R. Tabrizidooz, and M. Razzaghi, “A composite collocation method for the nonlinear mixed VolterraFredholmHammerstein integral equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1186–1194, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. L. Ortiz, “The tau method,” SIAM Journal on Numerical Analysis, vol. 6, pp. 480–492, 1969. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 G. Ebadi, M. Y. RahimiArdabili, and S. Shahmorad, “Numerical solution of the nonlinear Volterra integrodifferential equations by the tau method,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1580–1586, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
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Copyright © 2012 Hamid Reza Marzban and Sayyed Mohammad Hoseini. 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.