#### Abstract

An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential 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 Euler-Chebyshev [2] and Runge-Kutta 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 cell-to-cell spread of HIV-1 in tissue cultures considered by Mittler et al. [6]. Yalcinbas and Sezer [7] proposed an approximation scheme based on Taylor polynomials for solving the high-order linear Volterra-Fredholm 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 Volterra-Fredholm integrodifferential equations: Darania and Ivaz [9] suggested an efficient analytical and numerical procedure for solving the most general form of nonlinear Volterra-Fredholm 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 high-order 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 Volterra-Fredholm 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 fixed-point theory whereas the noniterative ones result in series solutions and also make use of the fixed-point 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 block-pulse 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 Volterra-Fredholm 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 block-pulse functions and Lagrange interpolating polynomials, , , , are defined on the interval as, [20], where and are the orders of block-pulse 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 block-pulse functions and Lagrange interpolating polynomials, which are both complete and orthogonal, the set of hybrid of block-pulse 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 Cross-Product of Two Hybrid Function Vectors

The following property of the cross-product 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 Volterra-Fredholm Integrodifferential Equations

Consider the nonlinear Volterra-Fredholm 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 well-known 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 Volterra-Fredholm 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 Volterra-Fredholm 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 high-order nonlinear Volterra-Fredholm integrodifferential equations. The nice properties of hybrid functions together with the associated operational matrices of integration, derivative, and cross-product 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.