Theoretical error analysis and validation in numerical solution of two-dimensional linear stochastic Volterra-Fredholm integral equation by applying the block-pulse functions

In this paper, we introduce an efficient method based on two-dimensional block-pulse functions (2D-BPFs) to approximate the solution of the 2D-linear stochastic Volterra–Fredholm integral equation. Also, we present convergence analysis of the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.


PUBLIC INTEREST STATEMENT
Two-dimensional stochastic integral equations result from the incorporation of either internally or externally originating random fluctuations in the dynamical description of a system. The numerical solution of such equations because of the randomness is very difficult or sometimes impossible as we can say that solving these equations has been worked very little. Here, we have successfully developed the two-dimensional BPFs numerical method to approximate a solution for two-dimensional linear stochastic Volterra-Fredholm integral equation in which error analysis and the numerical example show accuracy of this method. Engineers, physicists and others with a more technical background in mathematical methods who are interested in applying stochastic integral equations and in implementing efficient numerical schemes or developing new schemes for specific classes of applications, could use this paper.
operations because of the randomness is the most important obstacle for solving stochastic integral equations in higher dimensions. In this paper, we consider 2D-linear stochastic Volterra-Fredholm integral equation of the second kind where (x, y) ∈ E = [0, 1] × [0, 1], s ⩽ x < t ⩽ y. The function f(x, y) defined on E and the kernels v i (x, y, s, t); i = 1, 2, 3 defined on E × E, in (1) are known functions, whereas g(x, y) defined on E, is unknown function and is called the solution of (1). Also, B(t) is a Brownian motion process and ∫ y 0 ∫ x 0 v 3 (x, y, s, t)g(s, t)dB(s)dB(t) is the double Wiener-Itô integral. The condition s ⩽ x < t ⩽ y is necessary for adaptability to the filtration {F t ;0 ≤ t ≤ 1}, where F t = {B(s); 0 ≤ s ≤ t}. Recently, authors Fallahpour, Khodabin, and Maleknejad (2016) have used the BPFs method for solving Equation (1) without investigating the error analysis. Also, authors of Fallahpour, Khodabin, and Maleknejad (2015) have proposed Haar wavelet method to solve 2D-linear stochastic Fredholm integral equation without investigating the error analysis. Here, the BPFs method with the error and convergence analysis is introduced to derive approximate solution of (1). First, for validation the stochastic double Wiener-Itô integral, we need the following lemma and definition.
Lemma 1 (Kuo, 2006) Put (t, s) = v(x, y, s, t) g (s, t). Let be a function in L 2 ([0, 1] 2 ). Then, there exists a sequence n of off-diagonal step functions such that Definition 1 (Kuo, 2006) Let ∈ L 2 ([0, 1] 2 ). Then, the double Wiener-Itô integral of is defined as follows: where Ω is the sample space of random variables. This paper is organized as follows: In Section 2, we review the proposed numerical method in Fallahpour et al. (2016) for solving Equation (1) based on BPFs. The error analysis of this method is discussed in Section 3. We show the accuracy of the proposed method by some examples in Section 4. Also, in this section, we construct an 95% confidence interval for each solution. Finally, Section 5, gives brief conclusions.
Moreover, for every (n 1 n 2 ) × (n 1 n 2 ) matrix A , we have where the elements of Â , n 1 n 2 -vector, are the diagonal entries of matrix A.
where ⊗ denotes the Kronecker product defined by where x ij is ij-element of X matrix and O is the operational matrix of the 1D-BPFs defined over [0, T) with k = T n and T = T 1 = T 2 as follows: Therefore, using Equation (4) in (11) follows that where G 3 is an (n 1 n 2 )-vector with the components equal to the diagonal entries of matrix Γ 2G1 P.
Using Equation (4) in (13) follows that where G 4 is an (n 1 n 2 )-vector with the components equal to the diagonal entries of matrix Γ 3G1 P s .

Error analysis
In this section, we present error analysis of the proposed method in Section 2. For convenience, we put n 1 = n 2 = n, and hence k 1 = k 2 = 1 n . Also in this section, we consider for simplicity the 2-norms defined in this paper based on following definition.
Also, we need the following theorems and definition.

Theorem 4 Suppose that g(s, t), is the exact solution of Equation (1) and ĝ n (s, t) is the block-pulse series approximate solution of (1) that their elements are obtained by
then for every (x, y) ∈ [0, 1) 2 , we have Proof From Equation (1), we get then by mean value theorem for 2D-integrals, for every (x, y) ∈ [0, 1) 2 and (x, y, s, t) ∈ [0, 1) 4 , we have By using Hypothesises 1 and 2 and Theorem 3 we obtain and Similarly for the stochastic case, we get By substituting (21), (22), and (23) in (20) and using Theorem 2, we can write ‖g −ĝ n ‖ 2 ≤ ‖f −f n ‖ 2 + ‖v 1 g −v 1,nĝn ‖ 2 By taking sup and the Hypothesis 3, we have so and by Theorem 4, we get ✷

Numerical example
In this section, we consider some numerical examples to illustrate the efficiency and reliability of the BPFs method for solving the 2D-linear stochastic Volterra-Fredholm integral equations.
Example 1 Consider the following linear 2D-stochastic Volterra-Fredholm integral equation of the second kind:

The exact solution of this equation is
For convenience, we put n 1 = n 2 = n so k 1 = k 2 = 1 n . The solution mean (ḡ(x, y)), error mean (ē(x, y)) and %95 confidence interval (L, U) at arbitrary points ( where The exact solution of this equation is Also for this example, ḡ(x, y), ē(x, y) and (L, U) at arbitrary points (0.1, 0.2) and (0, 0.6) for some values of n are shown in Tables 3 and 4. 3D-graph of the exact and approximation solutions of this example for some values of n are shown in Figure 2.
Example 3 Consider the following linear 2D-stochastic Volterra-Fredholm integral equation of the second kind: g(x, y) = x + y.