Solution of Nonlinear Volterra-Fredholm Integrodifferential Equations via Hybrid of Block-Pulse Functions and Lagrange Interpolating Polynomials

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: ^𝑞_𝑗=0𝑓_𝑗(𝑡)𝑦^(𝑗)(𝑡)=𝑓(𝑡)+𝜆₁^𝑡₀𝑘₁(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+𝜆₂^𝑡𝑓₀𝑘₂(𝑡,𝑠)𝑦(𝑠)𝑑𝑠,0≤𝑡,𝑠≤𝑡_𝑓. (1.1) Maleknejad and Mahmoudi [8] developed a numerical method by using Taylor polynomials to solve the following type of nonlinear Volterra-Fredholm integrodifferential equations: ^𝑞_𝑗=0𝑓_𝑗(𝑡)𝑦^(𝑗)(𝑡)=𝑓(𝑡)+𝜆₁^𝑡₀𝑘₁(𝑡,𝑠)𝑦(𝑠)^𝑝𝑑𝑠+𝜆₂^𝑡𝑓₀𝑘₂(𝑡,𝑠)𝑦(𝑠)𝑑𝑠,0≤𝑡,𝑠≤𝑡_𝑓. (1.2) Darania and Ivaz [9] suggested an efficient analytical and numerical procedure for solving the most general form of nonlinear Volterra-Fredholm integrodifferential equations ^𝑞_𝑗=0𝑓_𝑗(𝑡)𝑦^(𝑗)(𝑡)=𝑓(𝑡)+𝜆₁^𝑡₀𝑘₁(𝑡,𝑠)𝑔₁𝑠,𝑦(𝑠)𝑑𝑠+𝜆₂^𝑡𝑓₀𝑘₂(𝑡,𝑠)𝑔₂𝑠,𝑦(𝑠)𝑑𝑠, (1.3) under the mixed conditions ^𝑞−1_𝑗=0𝛼_𝑖𝑗𝑦^(𝑗)(0)+𝛽_𝑖𝑗𝑦^(𝑗)𝑡_𝑓=𝜇_𝑖,𝑖=0,1,…,𝑞−1, (1.4) where 𝜆₁ and 𝜆₂ are constants and 0≤𝑡,𝑠≤𝑡_𝑓. Moreover, 𝑓(𝑡), 𝑘₁(𝑡,𝑠), 𝑘₂(𝑡,𝑠), 𝑔₁(𝑠,𝑦(𝑠)), 𝑔₂(𝑠,𝑦(𝑠)), and 𝑓_𝑗(𝑡), 𝑗=0,1,…,𝑞, are functions that have suitable derivatives on the interval 0≤𝑡,𝑠≤𝑡_𝑓.

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 [0,𝑡_𝑓) into 𝑁 equidistant subintervals [((𝑛−1)/𝑁)𝑡_𝑓,(𝑛/𝑁)𝑡_𝑓), for 𝑛=1,2,…,𝑁. The hybrid of block-pulse functions and Lagrange interpolating polynomials, 𝑏_𝑛𝑚(𝑡), 𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1, are defined on the interval [0,𝑡_𝑓) as, [20], 𝑏_𝑛𝑚(𝑡)=𝐿_𝑚2𝑁𝑡_𝑓𝑡−2𝑛+1,𝑡∈𝑛−1𝑁𝑡_𝑓,𝑛𝑁𝑡_𝑓,0,otherwise, (2.1) where 𝑛 and 𝑚 are the orders of block-pulse functions and Lagrange interpolating polynomials, respectively. Here, 𝐿_𝑚(𝑡) are defined in [21] as 𝐿_𝑚(𝑡)=^𝑀−1_{𝑖=0𝑖≠𝑚}𝑡−𝜏_𝑖𝜏_𝑚−𝜏_𝑖,𝑚=0,1,…,𝑀−1, (2.2) in which 𝜏_𝑖, 𝑖=0,1,…,𝑀−1 are the zeros of Legendre polynomial of order 𝑀, with the Kronecker property 𝐿_𝑚𝜏_𝑖=𝛿_𝑚𝑖=1,if𝑖=𝑚,0,if𝑖≠𝑚. (2.3) where 𝛿_𝑚𝑖 is the Kronecker delta function. No explicit formulas are known for the points 𝜏_𝑖. However, they can be computed numerically using existing subroutines [21]. Since 𝑏_𝑛𝑚(𝑡) 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 ℒ²[0,𝑡_𝑓). A more detailed error analysis concerning this type of hybrid functions was discussed in [22].

2.2. Function Approximation

A function 𝑓(𝑡)∈ℒ²[0,𝑡_𝑓) may be expanded as 𝑓(𝑡)=^∞∞_𝑛=1𝑚=0𝑓_𝑛𝑚𝑏_𝑛𝑚(𝑡). (2.4)

If the infinite series in (2.4) is truncated, then (2.4) can be written as 𝑓(𝑡)≃^𝑁_𝑛=1^𝑀−1_𝑚=0𝑓_𝑛𝑚𝑏_𝑛𝑚(𝑡)=𝐹^𝑇𝐵(𝑡), (2.5) where 𝐹=𝑓₁₀,…,𝑓_1(𝑀−1),…,𝑓_𝑁0,…,𝑓_{𝑁(𝑀−1)}^𝑇,𝐵(𝑡)=𝑏₁₀(𝑡),…,𝑏_1(𝑀−1)(𝑡),…,𝑏_𝑁0(𝑡),…,𝑏_{𝑁(𝑀−1)}(𝑡)^𝑇. (2.6) In (2.6), 𝑓_𝑛𝑚, 𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1 are the expansion coefficients of the function 𝑓(𝑡) in the 𝑛th subinterval [((𝑛−1)/𝑁)𝑡_𝑓,(𝑛/𝑁)𝑡_𝑓), and 𝑏_𝑛𝑚(𝑡), 𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1 are defined in (2.1). In order to evaluate the coefficients 𝑓_𝑛𝑚, we use Gaussian nodes which are the zeros of Legendre polynomial of order 𝑀. Let 𝜏_𝑚, 𝑚=0,1,…,𝑀−1 be the Gaussian nodes which are defined in [−1,1] and let 𝑡_𝑛𝑚, 𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1 be the corresponding Gaussian nodes in the 𝑛th subinterval [((𝑛−1)/𝑁)𝑡_𝑓,(𝑛/𝑁)𝑡_𝑓]. The relation between 𝜏_𝑚 and 𝑡_𝑛𝑚 is given by 𝑡_𝑛𝑚=𝑡_𝑓𝜏2𝑁^𝑚+2𝑛−1. (2.7) With the aid of (2.3) we have 𝑏_𝑛𝑚𝑡_𝑖𝑗=1,if𝑖=𝑛,𝑗=𝑚,0,otherwise. (2.8) Using (2.8), the coefficients 𝑓_𝑛𝑚, can be obtained as 𝑓_𝑛𝑚=𝑓𝑡_𝑛𝑚. (2.9)

Now, let 𝑔(𝑡,𝑠) be a function of two independent variables defined for 𝑡∈[0,𝑡_𝑓] and 𝑠∈[0,𝑠_𝑓]. Then 𝑔(𝑡,𝑠) can be expanded in terms of hybrid functions as follows: 𝑔(𝑡,𝑠)≃𝐵^𝑇(𝑡)𝐺𝐵(𝑠), (2.10) where 𝐺 is a matrix of order 𝑁𝑀×𝑁𝑀 whose elements can be calculated for 𝑖,𝑛=1,2,…,𝑁, and 𝑗,𝑚=0,1,…,𝑀−1 as follows: 𝐺_{(𝑛−1)𝑀+𝑚,(𝑖−1)𝑀+𝑗}=𝑔𝑡_𝑛𝑚,𝑠_𝑖𝑗, (2.11) in which 𝑠_𝑖𝑗 can be determined by the following relation: 𝑠_𝑖𝑗=𝑠_𝑓𝜏2𝑁^𝑗+2𝑖−1. (2.12) The integration of the vector 𝐵(𝑡) can also be expanded in terms of hybrid functions as ^𝑡₀𝐵(𝑠)𝑑𝑠≃𝑃𝐵(𝑡), (2.13) where 𝑃 is the 𝑁𝑀×𝑁𝑀 operational matrix of integration and given by, [20], 𝑃=,𝐸𝐻𝐻⋯𝐻0𝐸𝐻⋯𝐻00𝐸⋯𝐻⋮⋮⋮⋮000⋯𝐸 (2.14) in which 𝐸 and 𝐻 can be obtained in the following manner.

Let 𝐸=𝑒_𝑖𝑗,𝐻=ℎ_𝑖𝑗, (2.15) then, for 𝑖,𝑗=0,1,…,𝑀−1, we have 𝑒_𝑖𝑗=𝑡_𝑓2𝑁^𝜏𝑗₋₁𝐿_𝑖(𝑡)ℎ𝑑𝑡,_𝑖𝑗=𝑡_𝑓2𝑁¹₋₁𝐿_𝑖(𝑡)𝑑𝑡, (2.16) where 𝜏_𝑗, 𝑗=0,1,…,𝑀−1 are the zeros of Legendre polynomial of order 𝑀. It is noted that 𝐸 is the associated operational matrix of integration for Lagrange interpolating polynomials on interval [((𝑛−1)/𝑁)𝑡_𝑓,(𝑛/𝑁)𝑡_𝑓].

Integrating (2.5) from 0 to 𝑡 and using (2.13) imply ^𝑡₀𝑓(𝑠)𝑑𝑠≃𝐹^𝑇𝑃𝐵(𝑡). (2.17) From (2.13), by using the interpolating property of hybrid functions we get ^𝑡𝑛𝑚₀𝑓(𝑠)𝑑𝑠≃𝐹^𝑇𝑃𝐵𝑡_𝑛𝑚=^𝑛−1_𝑖=1^𝑀−1_𝑗=0𝑓_𝑖𝑗ℎ_𝑗𝑚+^𝑀−1_𝑗=0𝑓_𝑛𝑗𝑒_𝑗𝑚. (2.18)

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 𝐵(𝑡)𝐵^𝑇(𝑡) from 0 to 𝑡_𝑓, and using the orthogonality of hybrid functions, we get 𝑊=^𝑡𝑓₀𝐵(𝑡)𝐵^𝑇(𝑡)𝑑𝑡, (2.19) where 𝑊 is a block diagonal matrix of order 𝑁𝑀×𝑁𝑀 given by 𝑊=,𝐴0⋯00𝐴⋯0⋮⋮⋱⋮00⋯𝐴 (2.20) moreover 𝐴 is also a diagonal matrix of order 𝑀×𝑀 given as 𝐴=diag𝑎₀,𝑎₁,…,𝑎_𝑀−1. (2.21) According to the orthogonality of hybrid functions, the values of 𝑎_𝑚, for 𝑚=0,1,…,𝑀−1, can be computed as follows: 𝑎_𝑚=𝑡_𝑓2𝑁¹₋₁𝐿²_𝑚(𝑡)𝑑𝑡. (2.22)

2.4. The Operational Matrix of Derivative

Let 𝑑/𝑑𝑡 denote the derivative operator. By expanding (𝑑/𝑑𝑡)𝐵(𝑡) in terms of 𝐵(𝑡) we get 𝑑𝐵(𝑡)𝑑𝑡=𝐷^∗𝐵(𝑡). (2.23) In (2.23), 𝐷^∗ is called the operational matrix of derivative for hybrid functions which is a block diagonal matrix of order 𝑁𝑀×𝑁𝑀 and is given by 𝐷^∗=,𝐷0⋯00𝐷⋯0⋮⋮⋱⋮00⋯𝐷 (2.24) where 𝐷 is also a matrix of order 𝑀×𝑀 and the (𝑖,𝑗) th element of this matrix denoted by [𝐷]_𝑖𝑗 for 𝑖,𝑗=0,1,…,𝑀−1 can be calculated as follows: [𝐷]_𝑖𝑗=𝑑𝐿_𝑖𝑡𝑑𝑡^𝑖𝑗=2𝑁𝑡_𝑓𝐿^′_𝑖2𝑁𝑡_𝑓𝑡_𝑖𝑗−2𝑗+1=2𝑁𝑡_𝑓𝐿^′_𝑖𝜏_𝑗. (2.25) Let 𝑓⁽¹⁾(𝑡) denote the first derivative of 𝑓(𝑡) with respect to 𝑡. By expanding 𝑓⁽¹⁾(𝑡) in terms of hybrid functions and using (2.5) and (2.23) we obtain 𝑓⁽¹⁾(𝑡)≃^𝑁_𝑛=1^𝑀−1_𝑚=0𝑓⁽¹⁾𝑡_𝑛𝑚𝑏_𝑛𝑚(𝑡)=𝐹^𝑇𝐷^∗𝐵(𝑡). (2.26) According to (2.8) we get 𝑓⁽¹⁾𝑡_𝑛𝑚=𝐹^𝑇𝐷^∗𝐵𝑡_𝑛𝑚=^𝑀−1_𝑙=0𝑓_𝑛𝑙[𝐷]_𝑙𝑚. (2.27) Let 𝑓^(𝑗)(𝑡) denote the 𝑗th derivative of 𝑓(𝑡) with respect to 𝑡. In order to obtain the relation between the expansion coefficients of 𝑓^(𝑗)(𝑡) and 𝑓(𝑡), we use the following fact: 𝑑^𝑗𝑑𝑡^𝑗𝐵(𝑡)=𝐷^∗𝑗𝐵(𝑡), (2.28) where (𝐷^∗)^𝑗 is the 𝑗th power of the matrix 𝐷^∗ given in (2.23). Equation (2.28) can easily be obtained from (2.23) by differentiating 𝑓(𝑡), 𝑗 times with respect to 𝑡, 𝑗=2,3,…,𝑞. It should be noted that the matrix (𝐷^∗)^𝑗 has the following structure: 𝐷^∗𝑗=𝐷^𝑗0⋯00𝐷^𝑗⋯0⋮⋮⋱⋮00⋯𝐷^𝑗. (2.29) By expanding 𝑓^(𝑗)(𝑡) in terms of hybrid functions and using (2.5) and (2.28) we get 𝑓^(𝑗)(𝑡)≃^𝑁_𝑛=1^𝑀−1_𝑚=0𝑓^(𝑗)𝑡_𝑛𝑚𝑏_𝑛𝑚(𝑡)=𝐹^𝑇𝐷^∗𝑗𝐵(𝑡). (2.30) According to (2.8) it immediately follows that 𝑓^(𝑗)𝑡_𝑛𝑚=𝐹^𝑇𝐷^∗𝑗𝐵𝑡_𝑛𝑚=^𝑀−1_𝑙=0𝑓_𝑛𝑙𝐷^𝑗_𝑙𝑚. (2.31) where [𝐷^𝑗]_𝑙𝑚 denotes the (𝑙,𝑚) th element of the matrix 𝐷^𝑗.

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 ℒ²[0,𝑡_𝑓), we can expand any function in this space in terms of hybrid functions.

Let 𝑦(𝑡)=^𝑁_𝑛=1^𝑀−1_𝑚=0𝑦_𝑛𝑚𝑏_𝑛𝑚(𝑡)=𝑌^𝑇𝐵(𝑡), (3.1) where 𝑦_𝑛𝑚=𝑦𝑡_𝑛𝑚. (3.2) Using (2.30), for 𝑗=1,2,…,𝑞, we have 𝑦^(𝑗)(𝑡)=^𝑁_𝑛=1^𝑀−1_𝑚=0𝑦^(𝑗)𝑡_𝑛𝑚𝑏_𝑛𝑚(𝑡). (3.3) where 𝑦^(𝑗)𝑡_𝑛𝑚=^𝑀−1_𝑙=0𝑦_𝑛𝑙𝐷^𝑗_𝑙𝑚. (3.4) Moreover, by expanding 𝑘₂(𝑡,𝑠) in terms of hybrid functions we get 𝑘₂(𝑡,𝑠)=𝐵^𝑇(𝑡)𝐾₂𝐵(𝑠), (3.5) where 𝐾₂ is a matrix of order 𝑁𝑀×𝑁𝑀 and can be calculated similar to matrix 𝐺 in (2.10). From (2.5) and (2.9) we obtain 𝑔₂𝑡,𝑦(𝑡)=^𝑁_𝑛=1^𝑀−1_𝑚=0𝑔₂𝑡_𝑛𝑚,𝑦𝑡_𝑛𝑚𝑏_𝑛𝑚(𝑡)=^𝑁_𝑛=1^𝑀−1_𝑚=0𝑔₂𝑡_𝑛𝑚,𝑦_𝑛𝑚𝑏_𝑛𝑚(𝑡)=𝐵^𝑇(𝑡)𝐺₂, (3.6) where 𝐺₂ is a vector of order 𝑁𝑀×1. From (2.19), (3.5) and (3.6), we have ¹₀𝑘₂(𝑡,𝑠)𝑔₂𝑠,𝑦(𝑠)𝑑𝑠=¹₀𝐵^𝑇(𝑡)𝐾₂𝐵(𝑠)𝐵^𝑇(𝑠)𝐺₂𝑑𝑠=𝐵^𝑇(𝑡)𝑉, (3.7) where 𝑉=𝐾₂𝑊𝐺₂. (3.8) It should be noted that 𝑉 is a vector of order 𝑁𝑀×1. Substituting (3.1)–(3.7) in (1.3) yields ^𝑞_𝑗=0𝑓_𝑗(𝑡)𝑦^(𝑗)(𝑡)=𝑓(𝑡)+𝜆₁^𝑡₀𝑘₁(𝑡,𝑠)𝑔₁𝑠,𝑦(𝑠)𝑑𝑠+𝜆₂𝐵^𝑇(𝑡)𝑉. (3.9) We now collocate (1.3) at 𝑁𝑀 points. For a suitable collocation points, we use Gaussian nodes 𝑡_𝑛𝑚,𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1, which are introduced in (2.7). Therefore we have ^𝑞_𝑗=0𝑓_𝑗𝑡_𝑛𝑚𝑦^(𝑗)𝑡_𝑛𝑚=𝑓𝑡_𝑛𝑚+𝜆₁^𝑡𝑛𝑚₀𝑘₁𝑡_𝑛𝑚,𝑠𝑔₁𝑠,𝑦(𝑠)𝑑𝑠+𝜆₂𝐵^𝑇𝑡_𝑛𝑚𝑉. (3.10) Also, using (2.8) we get 𝐵^𝑇𝑡_𝑛𝑚𝑉=𝑉_{(𝑛−1)𝑀+𝑚}, (3.11) where 𝑉_{(𝑛−1)𝑀+𝑚} denotes the ((𝑛−1)𝑀+𝑚)th element of vector 𝑉. Consequently, with the aid of (2.18) and (3.10) we obtain ^𝑞_𝑗=0𝑓_𝑗𝑡_𝑛𝑚𝑦^(𝑗)𝑡_𝑛𝑚=𝑓𝑡_𝑛𝑚+𝜆₂𝑉_{(𝑛−1)𝑀+𝑚}+𝜆₁^𝑛−1_𝑖=1^𝑀−1_𝑗=0𝑘₁𝑡_𝑛𝑚,𝑠_𝑖𝑗𝑔₁𝑠_𝑖𝑗,𝑦_𝑖𝑗ℎ_𝑗𝑚+^𝑀−1_𝑗=0𝑘₁𝑡_𝑛𝑚,𝑠_𝑛𝑗𝑔₁𝑠_𝑛𝑗,𝑦_𝑛𝑗𝑒_𝑗𝑚, (3.12) where 𝑦^(𝑗)(𝑡_𝑛𝑚) for 𝑛=1,2,…,𝑁, 𝑚=0,1,…,𝑀−1 can be calculated from (3.4). Equation (3.12) gives 𝑁𝑀 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 ^𝑞−1_𝑗=0𝛼_𝑖𝑗^𝑀−1_𝑚=0𝑦^(𝑗)𝑡_1𝑚𝑏_1𝑚(0)+𝛽_𝑖𝑗^𝑀−1_𝑚=0𝑦^(𝑗)𝑡_𝑁𝑚𝑏_𝑁𝑚𝑡_𝑓=𝜇_𝑖,𝑖=0,1,…,𝑞−1. (3.13)

In order that the approximate solution obtained by the present method be continuous, we impose the continuity condition at the points (𝑛/𝑁)𝑡_𝑓, 𝑛=1,2,…,𝑁−1. Equations. (3.12)-(3.13) together with the continuity conditions give 𝑁(𝑀+1)+𝑞−1, nonlinear equations that can be solved by using the well-known Tau method [23] for the unknown coefficients 𝑦_𝑛𝑚, 𝑛=1,2,…,𝑁, 𝑚=0,1,…𝑀−1.

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 𝑡²𝑦^′′(𝑡)+2𝑦^′(𝑡)5=2−61𝑡+2𝑡𝑒^−𝑡2+^𝑡₀𝑡𝑠𝑒^{−𝑦2(𝑠)}𝑑𝑠+¹₀𝑡𝑦²(𝑠)𝑦(0)𝑑𝑠,=0,𝑦^′(0)=1, (4.1) whose exact solution is given in [9] as 𝑦(𝑡)=𝑡. We applied the present method to solve this problem.
Let us define 𝐸=^𝑡𝑓₀𝑦_𝑒(𝑡)−𝑦(𝑡)²𝑑𝑡^1/2, (4.2) where 𝑦(𝑡) and 𝑦_𝑒(𝑡) denote the approximate solution obtained by the present method and the exact solution, respectively. In Table 1, the computational results of the ℒ²-norm error denoted by 𝐸 between the approximate solution and the exact solution for different values of 𝑁 and 𝑀 are given. The simulation results reported in Table 1, indicate that there is an excellent agreement between the approximate and exact solution.

Table 1 

The computational results of ℒ²-norm error for Example 4.1.

Example 4.2. As the second example, consider the nonlinear Volterra integrodifferential equation 𝑦^′(𝑡)1=2𝑡−2sin𝑡⁴+^𝑡₀𝑡²𝑠cos𝑡²𝑦(𝑠)𝑦(0)𝑑𝑠,=0, (4.3)whose exact solution is given in [19] as 𝑦(𝑡) = 𝑡². We applied the present method to solve this problem. In Table 2, the computational results of the ℒ²-norm error between the approximate solution and the exact solution for different values of 𝑁 and 𝑀 are summarized. The simulation results obtained by the present method are superior to those reported in [19].

Table 2 

The computational results of ℒ²-norm error for Example 4.2.

Example 4.3. Let us consider the nonlinear Volterra integrodifferential equation 𝑦⁽⁴⁾(𝑡)=1+^𝑡₀𝑒^−𝑠𝑦²(𝑠)[],𝑦(0)𝑑𝑠,𝑡∈0,1=1,𝑦^′(0)𝑦(1)=1,=𝑒,𝑦^′(1)=𝑒, (4.4)whose exact solution is given in [24] as 𝑦(𝑡)=𝑒^𝑡. We solved this problem by using hybrid functions. In Table 3, the computational results of 𝑦(𝑡) obtained by the present method with 𝑁=2 and 𝑀=10 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 ℒ²-norm error between the approximate solution obtained by the present method and the exact solution for different values of 𝑁 and 𝑀 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.

Table 3 

The computational results of 𝑦(𝑡) for Example 4.3.

Table 4 

The computational results of ℒ²-norm error for Example 4.3.

Example 4.4. As the last example, consider the nonlinear Volterra-Fredholm integrodifferential equation 𝑦^′′′(𝑡)−𝑡𝑦^′′(𝑡)=13𝑡³−2𝑡+1−^𝑡₀𝑠𝑦(𝑠)𝑑𝑠+¹₀2𝑡𝑒^𝑠𝑦(𝑠)𝑦(0)−𝑠𝑑𝑠,=1,𝑦^′(0)=1,𝑦(1)=1+𝑒, (4.5) which has the exact solution 𝑦(𝑡)=𝑡+𝑒^𝑡. To measure the accuracy of proposed approach, the ℒ²-norm error between the approximate solution obtained by the present method for different values of 𝑁 and 𝑀 and the exact solution is given in Table 5. The computational results reveal that the method is convergent.

Table 5 

The computational results of ℒ²-norm error for Example 4.4.

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 𝑃, 𝐶, and 𝐷^∗ are sparse, hence making the method computationally attractive without sacrificing the accuracy of the solution. Moreover, It was shown that small values for 𝑁 and 𝑀 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.