Block-by-Block Method for Solving Nonlinear Volterra-Fredholm Integral Equation

Abstract

We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results.

1. Introduction

Different methods are used to solve integral equations which are investigated from many physical applications such as the mixed problems in the theory of elasticity. Popov [1] applied the orthogonal polynomials method to solve the mixed problem in the mechanics of continuous media. Badr [2] applied Toeplitz matrix method to solve a NVFIE. Abdou et al. [3] discussed the solution of Harmmerstein-Volterra integral equation of the second kind. In [4], Haci obtained, numerically, the solution of a system of Harmmerstein integral equations in the space 𝐿₂([𝑎,𝑏]). The equivalence between Volterra integral equation with degenerate kernel and a linear system of differential equations is mentioned by Cochran [5]. Although there are some works on Hermite-type collocation method for the second-kind VIEs with smooth kernels, not too many studies have dealt with weakly singular kernel. For example, Papatheodorou and Jesanis [6] used the collocation method and obtained the solution of Volterra integrodifferential equation with weakly singular kernels. More information about different analytical and numerical solutions of Volterra equations can be found in Davis [7], Linz [8], Volterra [9], and Wolkenfelt [10].

In this paper, we consider the following NVFIE: 𝜙(𝑥,𝑡)=𝑓(𝑥,𝑡)+𝜆^𝑡_𝜏=0^𝑏_𝑥=𝑎𝐹(𝑡,𝜏)𝐾𝑥,𝑦𝛾𝜏,𝑦,𝜙𝑦,𝜏𝑑𝑦𝑑𝜏,0≤𝑡≤𝑇<1. (1.1) The existence of a unique solution for the above equation, under certain conditions, is granted using fixed point theorem, where 𝐾(𝑥,𝑦) is the Fredholm kernel and 𝐹(𝑡,𝜏) is the Volterra kernel. 𝑓(𝑥,𝑡) is called the free term, and the unknown function, 𝜙(𝑥,𝑡), is called the potential function in the applied mathematics, and it will be determined. Both two functions 𝜙(𝑥,𝑡) and 𝑓(𝑥,𝑡) are assumed in the same space. The parameter 𝜆 has many physical meanings. A numerical method is applied to this equation, and it is reduced it to a system of Volterra integral equations of the second kind. Finally, the block-by-block method is used to obtain the numerical solution of this system. Some examples are stated to illustrate the results.

2. Existence and Uniqueness of Solution

To guarantee the existence and uniqueness of solution to (1.1), we write (1.1) in the integral operator form 𝑊𝜙(𝑥,𝑡)=𝑓(𝑥,𝑡)+𝑊𝜙(𝑥,𝑡), (2.1) where 𝑊𝜙(𝑥,𝑡)=𝜆^𝑡₀^𝑏_𝑎𝐹(𝑡,𝜏)𝑘𝑥,𝑦𝛾𝜏,𝑦,𝜙𝑦,𝜏𝑑𝑦𝑑𝜏. (2.2) Also, we assume the following conditions: (i)𝑘(𝑥,𝑦)∈𝐿₂([𝑎,𝑏]×[𝑎,𝑏]) and satisfies, in general, the condition {∫^𝑏_𝑎∫^𝑏_𝑎|𝑘(𝑥,𝑦)∣²𝑑𝑥𝑑𝑦}^1/2=𝑐^∗ (𝑐^∗ is a constant). (ii)𝐹(𝑡,𝜏)∈𝐶([0,𝑇]×[0,𝑇]),0≤𝜏≤𝑡≤𝑇 and satisfies 0<𝐹(𝑡,𝜏)≤𝑀, where 𝑀 is a constant. (iii)The given function 𝑓(𝑥,𝑡) with its partial derivatives with respect to position and time is continuous in the space 𝐿₂([𝑎,𝑏]×[0,𝑇]), and its norm is defined as ‖𝑓‖=max_𝑡,𝑥|∫^𝑡₀{∫^𝑏_𝑎|𝑓(𝑥,𝜏)|²𝑑𝑥}^1/2𝑑𝜏|=𝐻 (𝐻 is a constant). (iv)The known function 𝛾(𝜏,𝑦,𝜙(𝑦,𝜏)) satisfies the following conditions:(a)‖𝛾(𝜏,𝑦,𝜙(𝑦,𝜏))‖_{𝐿2([𝑎,𝑏]×[0,𝑇])}≤𝐵, (b)|𝛾(𝜏,𝑦,𝜙(𝑦,𝜏))−𝛾(𝜏,𝑦,𝜓(𝑦,𝜏))|≤𝑁|𝜙(𝑦,𝜏)−𝜓(𝑦,𝜏)|.

Under these conditions, using the Cauchy-Schwarz and Minkowski inequalities in conjunction with Schauders fixed point theorem, we easily can prove the following theorem.

Theorem 2.1. If the conditions (i)–(iv) are satisfied, then (1.1) has a unique solution in the space 𝐿₂([𝑎,𝑏]×𝐶[0,𝑇]).

3. The Numerical Solution

In this section, we present a numerical method to solve (1.1). This method consists of two phases.

In phase one, we rewrite this equation as a system of linear Volterra integral equations. To do so, we choose a sufficiently small step size ℎ_𝑥, and we assume 𝑎=𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑁𝑥=𝑏 is a partition of the interval [𝑎,𝑏] with ℎ_𝑥=𝑏−𝑎/𝑁_𝑥. Then, at each point 𝑥_𝑛, (1.1) becomes𝜙_𝑛(𝑡)=𝑓_𝑛(𝑡)+𝜆^𝑡₀^𝑏_𝑎𝐹(𝑡,𝜏)𝐾𝑥_𝑛,𝑦𝛾𝜏,𝑦,𝜙𝑦,𝜏𝑑𝑦𝑑𝜏,0≤𝑡≤𝑇, (3.1) where 𝜙_𝑛(𝑡)=𝜙𝑥_𝑛,𝑡,𝑓_𝑛(𝑡)=𝑓𝑥_𝑛,𝑡,𝑛=0,1,2,…,𝑁_𝑥. (3.2)

Replacing the integral ∫^𝑏_𝑎𝐾(𝑥_𝑛,𝑦)𝛾(𝜏,𝑦,𝜙(𝑦,𝜏))𝑑𝑦 by a numerical integration rule of the form ∑^𝑛_𝑗=0𝑤_𝑗𝑘_𝑛𝑗𝛾_𝑗(𝜏,𝑦_𝑗,𝜙_𝑗(𝜏)), where 𝑘_𝑛𝑗=𝑘(𝑥_𝑛,𝑥_𝑗), then an approximate to (1.1) can be found as a system of Volterra integral equations:𝜙_𝑛(𝑡)=𝑓_𝑛(𝑡)+𝜆^𝑛_𝑗=0𝑤_𝑗𝑘_𝑛,𝑗^𝑡₀𝐹(𝑡,𝜏)𝛾_𝑗𝜏,𝑦_𝑗,𝜙_𝑗(𝜏)𝑑𝜏,𝑛=0,1,2,…,𝑁_𝑥. (3.3) In phase two, we use the block-by-block method to solve such system of integral equations of Volterra type. The following subsection explains how this method works.

3.1. Block-by-Block Method, See [8]

Assume that we need to solve an equation of the form𝑢(𝑡)=𝑣(𝑡)+𝜆^𝑡₀𝐻(𝑡,𝑠,𝑢(𝑠))𝑑𝑠. (3.4) The idea behind the block-by-block method is to divide the interval [0,𝑡] into a mesh 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑛<⋯<𝑡_𝑁𝑡=𝑡<𝑇, and then we try to evaluate the value of the unknown function 𝑢(𝑡) at these points except at 𝑡=0, where we have that 𝑢(0)=𝑣(0). Using any known rule, say Simpson's rule, we have𝑢𝑡₂=𝑣𝑡₂ℎ+𝜆^𝑡3𝐻𝑡₂,𝑡₀,𝑢𝑡₀+4𝐻𝑡₂,𝑡₁,𝑢𝑡₁+𝐻𝑡₂,𝑡₂,𝑢𝑡₂. (3.5) To obtain a value for 𝑢(𝑡₁), we introduce the point 𝑡_1/2=ℎ_𝑡/2,  ℎ_𝑡=𝑡/𝑁_𝑡, and then we use Simpson's rule again to obtain 𝑢𝑡₁=𝑣𝑡₁ℎ+𝜆^𝑡3𝐻𝑡₁,𝑡₀,𝑢𝑡₀+4𝐻𝑡₁,𝑡_1/2,𝑢𝑡_1/2+𝐻𝑡₁,𝑡₁,𝑢𝑡₁. (3.6)

Replacing the value 𝑢(𝑡_1/2) by a quadratic interpolation using the values 𝑢₀, 𝑢₁, and 𝑢₂, we have 𝑢𝑡_1/2=38𝑢𝑡₀+34𝑢𝑡₁−18𝑢𝑡₂ (3.7) so that we can compute 𝑢(𝑡₁) by 𝑢𝑡₁=𝑣𝑡₁ℎ+𝜆^𝑡3𝐻𝑡₁,𝑡₀,𝑢𝑡₀+4𝐻𝑡₁,𝑡_1/2,38𝑢𝑡₀+34𝑢𝑡₁−18𝑢𝑡₂+𝐻𝑡₁,𝑡₁,𝑢𝑡₁. (3.8) Equations (3.5) and (3.8) are a pair of simultaneous equations for 𝑢(𝑡₁) and 𝑢(𝑡₂). For sufficiently small ℎ_𝑡, 𝑢(𝑡₁) and 𝑢(𝑡₂) can be found uniquely using any procedure such as Netwon's method.

In general, for 𝑚=0,1,2,…,𝑁_𝑡−1, the approximate solution of (3.4) is evaluated using the following two equations: 𝑢𝑡_2𝑚+1=𝑣𝑡_2𝑚+1+𝜆ℎ_𝑡^2𝑚_𝑠=0𝜅_𝑠𝐻𝑡_2𝑚+1,𝑡_𝑠,𝑢𝑡_𝑠+ℎ_𝑡6𝐻𝑡_2𝑚+1,𝑡_2𝑚,𝑢𝑡_2𝑚+4𝐻𝑡_2𝑚+1,𝑡_2𝑚+1/2,38𝑢𝑡_2𝑚+34𝑢𝑡_2𝑚+1−18𝑢𝑡_2𝑚+2+𝐻𝑡_2𝑚+1,𝑡_2𝑚+1,𝑢𝑡_2𝑚+1,𝑢𝑡_2𝑚+2=𝑣𝑡_2𝑚+2+𝜆ℎ_𝑡^2𝑛_𝑠=0𝜅_𝑠𝐻𝑡_2𝑚+2,𝑡_𝑠,𝑢𝑡_𝑠+ℎ_𝑡3𝐻𝑡_2𝑚+2,𝑡_2𝑚,𝑢𝑡_2𝑚+4𝐻𝑡_2𝑚+2,𝑡_2𝑚+1,𝑢𝑡_2𝑚+2+𝐻𝑡_2𝑚+2,𝑡_2𝑚+2,𝑢𝑡_2𝑚+2, (3.9) where𝜅_𝑠=13{}𝑡1,4,2,2,…,2,4,1,𝑠=0,1,2,…,2𝑚,_2𝑚+1/2=𝑡_2𝑚+ℎ_𝑡2. (3.10)(3.11)

At each subinterval [𝑡_2𝑚+1,𝑡_2𝑚+2], we solve these two equations simultaneously for the unknowns 𝑢(𝑡_2𝑚+1) and 𝑢(𝑡_2𝑚+2), so that we obtain a block of unknowns at a time.

3.2. Application of the Block-by-Block Method on the NVFIE

Combining the previous results, we obtain a new scheme defined by the following two equations: 𝜙_𝑛𝑡_2𝑚+1=𝑓_𝑛𝑡_2𝑚+1+𝜆^𝑛_𝑗=0𝑤_𝑗⋅ℎ_𝑡^2𝑚_𝑠=0𝜅_𝑠𝐹𝑡_2𝑚+1,𝑡_𝑠𝑘𝑥_𝑛,𝑦_𝑗𝛾_𝑗𝑡_𝑠,𝜙_𝑗𝑡_𝑠+ℎ_𝑡6𝐹𝑡_2𝑚+1,𝑡_2𝑚𝛾_𝑛𝑡_2𝑚,𝜙_𝑛𝑡_2𝑚+4𝐹𝑡_2𝑚+1,𝑡_2𝑚+1/2×38𝛾_𝑛𝑡_2𝑚,𝜙_𝑛𝑡_2𝑚+34𝛾_𝑗𝑡_2𝑚+1,𝜙_𝑛𝑡_2𝑚+1−18𝛾_𝑛𝑡_2𝑚+2,𝜙_𝑛𝑡_2𝑚+2+𝐹𝑡_2𝑚+1,𝑡_2𝑚+1𝛾_𝑛𝑡_2𝑚+1,𝜙_𝑛𝑡_2𝑚+1,𝜙_𝑛𝑡_2𝑚+2=𝑓_𝑛𝑡_2𝑚+2+𝜆^𝑛_𝑗=0𝑤_𝑗ℎ_𝑡^2𝑚_𝑠=0𝜅_𝑠𝐹𝑡_2𝑛+2,𝑡_𝑠𝑘𝑥_𝑛,𝑦_𝑗𝛾_𝑗𝑡_𝑠,𝜙_𝑗𝑡_𝑠+ℎ_𝑡3𝐹𝑡_2𝑚+2,𝑡_2𝑚𝛾_𝑛𝑡_2𝑚,𝜙_𝑛𝑡_2𝑚+4𝐹𝑡_2𝑚+2,𝑡_2𝑚+1𝛾_𝑛𝑡_2𝑚+2,𝜙_𝑛𝑡_2𝑚+2+𝐹𝑡_2𝑚+2,𝑡_2𝑚+2𝛾_𝑛𝑡_2𝑚+2,𝜙_𝑛𝑡_2𝑚+2, (3.12) where 𝑔_𝛼(𝑡_𝛽)=𝑔(𝑥_𝛼,𝑡_𝛽).

These two equations are two nonlinear equations in the two unknown 𝜙(⋅,𝑡_2𝑚+1) and 𝜙(⋅,𝑡_2𝑚+2), and so they can be determined uniquely using any procedure such as Netwons method. By repetitions, we get the solution 𝜙(⋅,𝑡_𝑚) for each 𝑚=1,2,…,2𝑁_𝑡.

4. Examples

We use the block-by-block method to solve numerically the following examples when (i)case 1: 𝛾(𝑡,𝑥,𝜙(𝑥,𝑡))=𝜙(𝑥,𝑡),(ii)case 2: 𝛾(𝑡,𝑥,𝜙(𝑥,𝑡))=𝜙²(𝑥,𝑡).

Example 4.1. Solve the Volterra-Fredholm integral equation 𝜙(𝑥,𝑡)=𝑓(𝑥,𝑡)+𝜆^𝑡₀¹₀𝑒^𝑥+𝑦⋅𝜏²𝛾𝜏,𝜙𝑦,𝜏𝑑𝑦𝑑𝜏,𝜆=0.1,0≤𝑡≤𝑇. (4.1) The exact solution of this integral equation is 𝜙(𝑥,𝑡)=𝑡²𝑒^𝑥. Tables 1 and 2 list the error at different values of 𝑥 and 𝑡 for different value of 𝑁_𝑥 (number of partitions of the interval [𝑎,𝑏] and 2𝑁_𝑡 (number of partitions of the interval [0,𝑇];𝑇=0.5).

Table 1 

Table 2 

Example 4.2. Solve the Volterra-Fredholm integral equation 𝜙(𝑥,𝑡)=𝑓(𝑥,𝑡)+𝜆^𝑡₀¹₀sin𝑥+𝑦⋅𝜏²𝜙𝑦,𝜏𝑑𝑦𝑑𝜏,𝜆=0.2,0≤𝑡≤1. (4.2) The exact solution of this integral equation is 𝜙(𝑥,𝑡)=𝑡²⋅sin(𝑥). Tables 3 and 4 list the error at different values of 𝑥 and 𝑡 for different value of 𝑁𝑥 and 2𝑁𝑡, 𝑇=0.5.

Table 3 

Table 4 

5. Conclusion

Two rules of numerical integration methods are used to solve (1.1). One of these rules is of order 𝑂(ℎ_𝑥), the rectangle method, which is applied on the integral with respect to the variable 𝑥. The other method is of higher-order, 𝑂(ℎ⁴_𝑡), the block-by-block method, and is applied on the integral with respect to the variable t where we used the fact that 𝜙(𝑥,0)=𝑓(𝑥,0). More information about the solution could allow us to use higher-order method with the block-by-block method.