Theoretical Error Analysis of Solution for Two-Dimensional Stochastic Volterra Integral Equations by Haar Wavelet

The finding an efficient way to the approximate solutions of the stochastic integral equations is an essential requirement. In this paper we discuss the convergence analysis of the two-dimensional Haar wavelet functions (2D-HWFs) method for solve 2D linear stochastic Volterra integral equation. The illustrative examples are included to demonstrate the validity and applicability of this numerical method.


Introduction
In very of the engineering problems, we see the important role of 2D integral equations [1,2], where are produced from a differential equation. Also if we import the statistical noise into a general hyperbolic differential equation we can obtain 2D linear stochastic Volterra integral equation of the second kind, i.e.

s, t)g(s, t)d B(s)d B(t)
where B(t) is Brownian motion process. In the last years, the numerous numerical methods have been introduced to estimate the solution of 2D ordinary integral equations [3][4][5][6][7][8] and many other papers, whereas it is worked few for 2D stochastic integral equations. Recently, some authors have proposed the methods to solve 2D stochastic integral equations [9][10][11][12][13][14]. Especially, Fallahpour et al. [9], included HWFs method to solve such equations without the error analysis. In this paper, we use HWFs method with the error and convergence analysis to derive approximate solution of (1). This paper is organized as follows: In next section we present HWFs. In section "Haar Wavelets Numerical Method", the method is applied to solve Eq. (1). The error analysis of this method is discussed in section "Error Analysis". Section "Numerical Example", uses some numerical examples to show the convergence of the proposed method and compares it with the block-pulse functions (BPFs) method as proposed in [10,11]. Finally, conclusion is given, in section "Conclusion".

Haar Wavelets
As we know, for HWFs we can write The integer and n, are the wavelet level and the translation parameter, respectively. Also Haar wavelets h i (z) are pairwise orthonormal in the interval [0, 1) as where δ i j is Kronecker delta. We display the maximum value of by J where M = 2 J . Therefore for any square integrable function f (z) we have We show the ordinary Volterra ihtegral of Haar wavelet as where by using HWFs we get Also for the stochastic Volterra integral we can introduce where similarly by Haar wavelet definition we have elsewhere. (4)

Haar Wavelets Numerical Method
In this section, we solve Eq. (1) by using 2D-HWFs. For Haar wavelet approximation of a function f (x, y), we use the collocation points

2D-HWFs System
A real-valued function E(x, y) can be approximated using 2D-HWFs as where the unknown coefficients b d,e 's, have calculated in [9]. Also we consider a function E(x, y, s, t) where is approximated using 2D-HWFs as so by substituting the collocation points and we get where the solution of System (7) is calculated from Corollary (1) in [9].

Error Analysis
In this section, we investigate the convergence and error analysis of HWFs method for solution (1). Here norm 2 is defined as To obtaining the typical convergence rate of HWFs method, we have: where D and M are a convex open set and a real number, respectively, we have Then for (s, t) ∈ D 2 , we have e m ≤ M 3m 2 .
where a = 2 j 1 + k, b = 2 j 2 + k, c = 2 j 3 + k, d = 2 j 4 + k, m = 2 J , J > 0 and Based on Haar wavelet definition, mean value theorem, Theorems 1 and 2 there are Finally we conclude By using Eq. (2) and substituting Eq. (11) in (10) we get In other words (9) and U i 's be defined in Theorem 3, then by using hypothesises

Theorem 4 If g(s, t) be the exact solution of Eq. (1) andĝ m (s, t) be HWFs approximate solution of (1) that is obtained by
.

Proof
We can obtain

s, t)ĝ m (s, t) d B(s)d B(t).
The mean value theorem conclude By using two first hypothesis and Theorem 3 we obtain Similarly for the stochastic case we get By Theorem 2 and substituting Eqs. (12) and (13) in (14), we have By taking sup we have finally Theorem 3 of [11] complete the proof.

Numerical Example
To illustrate the effectiveness of the proposed method, are carried out three numerical examples in this section as the solution mean, the error mean, %95 confidence interval and the length of the confidence interval are shown byḡ(x, y),ē(x, y), C I and LC I , respectively in Tables 1, 2 and 3. According to these tables by applying HWFs method, as J increases, e(x, y) and LC I decrease, nearly. Also the results of this method are compared with the BPFs method [10,11] and are displayed in Table 4. According to Table 4 we can say that HWFs mehod is better than the BPFs method at some points and also there are some other points that the BPFs method is an able method in those. In addition, 3D graphs of the these examples are shown in Figs. 1, 2 and 3. Consider the following 2D-linear stochastic Volterra integral equations of the second kind:    with the exact solution g(x, y) = x + y.

Conclusion
In this paper, we developed HWFs numerical method for approximate a solution of Eq. (1). The error analysis and the numerical examples show accuracy of this method. Furthermore, the current method can be run with increasing J until the results settle down to an appropriate accuracy that it leads to solving 2 2J +2 linear systems, that have its difficulties. Finally, this method can be extended and applied to 2D linear or non-linear multi-noise stochastic Volterra-Fredholm integral equations of the first or second kind.
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