Numerical solution of nonlinear mixed Volterra-Fredholm integro-differential equations by two-dimensional block-pulse functions

Abstract: This paper proposed an effective numerical method to obtain the solution of nonlinear two-dimensional mixed Volterra-Fredholm integro-differential equations. For this purpose, the two-dimensional block-pulse functions (2D-BPFs) operational matrix of integration and differentiation has been presented. The 2D-BPFs method converts nonlinear two-dimensional mixed Volterra-Fredholm integro-differential equations to an algebraic system of equations which is computable as well. Error analysis and some numerical examples are presented to illustrate the effectiveness and accuracy of the method.


Introduction
The modeling of the most phenomena in real life from engineering and physics to mechanics and etc leads to nonlinear equations. Meanwhile, the analytical solution of them in most of the time are not exist, so we need a powerful method to approximate the exact solutions.
The mixed Volterra-Fredholm integro-differential equation is one of the most important equation release in theory of parabolic boundary value problems, population dynamics, the mathematical ABOUT THE AUTHOR The first author of this article is Mostafa Safavi. His research interests include Numerical Analysis, The theory, and Applications of Applied and Computational Mathematics, Fractional, Partial and Integral Differential Equations. The second author of this article is Amir Ahmad Khajehnasiri. He is currently Ph.D. appliicant and the main research interests of him include numerical solution of Fredholm and Volterra integral equations as well as Fractional differential equations.

PUBLIC INTEREST STATEMENT
The solutions of Partial Differential and Integral Equations (PDEs and IEs) with numerical methods play the most important role among researchers and scientific communities due to the fact that the analytical solution of them in most of the time does not exist. The concept of approximation almost along with error estimation, accuracy, efficiency, the reliability of the methods and etc. In this approach, we use the method which known as Black-Pulse Functions (BPFs) to find the solution of an equation which release in theory of parabolic boundary value problems, population dynamics, the mathematical modeling of the spatio-temporal development and in many physical and biological models.
In 1977, Harmuth (Harmuth, 1969) presented the Block-Pulse Functions (BPFs) as a mathematical tool for approximate the problems. The BPFs which are defined in the time interval ½0; T 1 are a set of orthogonal functions with piecewise constant values such as: (1:1) where i ¼ 0; . . . ; m À 1 with m as a positive integer.
The solution of Fredholm and Volterra integral equations of the second kind have been approximated by using BPFs in (Kung & Chen, 1978). Maleknejad and Mahmoudi in (Maleknejad & Mahmoudi, 2004) have applied a combination of Taylor and Block-Pulse Functions to solve linear Fredholm integral equation.
This paper is organized as follows. In Section 2, definition and some properties of the 2D-BPFs has presented. The 2D-BPFs are applied to solve Equation (1.2) in Section 3. The error analysis of the proposed method has been investigated in Section 4. Some numerical results has been presented in Section 5 to show accuracy and efficiency of the proposed method. Finally, some concluding remarks are given in section 6.

Disjointness
The two-dimensional block-pulse functions are disjoined with each other, i.e.
The two-dimensional block-pulse functions are orthogonal with each other, i.e.
The block-pulse coefficients g i 1 ;i 2 are obtained as ( 2:13) such that the error between gðt; xÞ, and its block-pulse expansion (2.11) in the region of t 2 ½0; T 1 Þ, ( 2:14) is minimal. Since each two-dimensional block-pulse function takes only one value in its subregion, the 2D-BPFs can be expressed by the two 1D-BPFs: ( 2:15) where, Ψ i 1 ðtÞ and Ψ i 2 ðxÞ are 1D-BPFs related to the variables t and s, respectively.

Operational matrix of integration
So we get Similarly, for the partial derivative of uðt; xÞ with respect to t, it can be shown that (2:26) Moreover, for the second-order partial derivatives of uðt; xÞ, the following equations can be written: (2:27) by using (2.23) and (2.27), we have (2:28) In the similar way, to approximate the second-order partial derivatives of uðt; xÞ with respect to t, the following equation has been obtained: (2:29) Finally, the following procedure can be applied to approximate u tx ðt; xÞ, u t ðt; xÞ À u t ðt; 0Þ ¼ so we can obtain, then we have (2:31)

Applying the method
In this section, we use the 2DBPFs to solve the nonlinear two-dimensional mixed Volterra-Fredholm integro-differential equations. According to the previous section we have: (3:1) where the m 1 m 2 -vectors U; F; Λ; U x ; U t ; U xx ; U tt ; U tx and matrix K are the BPFs coefficients of uðt; xÞ; f ðt; xÞ; ½uðs; yÞ p ; u x ðt; xÞ; u t ðt; xÞ; u xx ðt; xÞ; u tt ðt; xÞ; u tx ðt; xÞ and kðx; y; s; tÞ respectively, and Θ is a column vector whose elements are pth power of the elements of the vector U. Now, we consider the following equation, (3:2) Using Equations (2.15) and (2.7), denoting L j for the jth row of the conventional integration operational matrix E and considering Finally, we can determine the block-pulse coefficients of ½uðt; xÞ p by solving: where g i;j ¼ ðu i;j Þ p and Q ¼ ðQ lz Þ is a lower triangular block matrix with 1 l m 2 and 1 z m 1 and (3:6) for m 1 ðl À 1Þ þ 1 i m 1 l and ðz À 1Þm 2 þ 1 j zm 2 ; 0 is a zero matrix. If we have uðt; xÞ ! 0 for every ðt; xÞ 2 D ¼ ½0; 1Þ Â ½0; 1Þ; then an approximate solution uðt; xÞ ¼ Ψ T ðt; xÞU can be computed for Equation (1.2) by setting u i;j ¼ ðg i;j Þ 1 p ; where u i;j and g i;j are the elements of vectors U and G respectively. hence, we have (3:7) Now, by using the Equations (2.25), (2.26), (2.28), (2.29), (2.31) and (3.7), we can obtain: where A and G are the combination of block-pulse coefficient matrix, After which without using any projection method, we can evaluate the approximation of the solution of u ' U T Ψðx; yÞ for Equation (1.2). The Equation (3.8) can be solved by using Newton iterative method.

Numerical results
In this section, we will use the 2D-BPFs to nonlinear mixed volterra-fredholm integro-differential equations with variable coefficients. To demonstrate the superiority and the practicability of this approach, two test examples are carried out in this section. The exact solution of this problem is uðt; xÞ ¼ x þ sinðtÞ. In Table 1, the numerical results are presented.

Conclusion
In this work, some orthogonal functions known as Block-Pulse Functions have successfully used to approximate the solution of a general form of nonlinear mixed Volterra-Fredholm integro-differential equations. The error of the method estimated and according to this estimation and the numerical results, we found that the proposed method is accurate and effective to solve the nonlinear equations, especially for mixed Volterra-Fredholm integro-differential equations.