key: cord-0836464-yc6oux7m
authors: Pourdarvish, Ahmad; Sayevand, Khosro; Masti, Iman; Kumar, Sunil
title: Orthonormal Bernoulli Polynomials for Solving a Class of Two Dimensional Stochastic Volterra–Fredholm Integral Equations
date: 2022-01-22
journal: Int J Appl Comput Math
DOI: 10.1007/s40819-022-01246-z
sha: fb9e76ac3a71d939100fbeef50de4d1497895426
doc_id: 836464
cord_uid: yc6oux7m

Analyzing the mathematical models involving Itô integral, in particular in science and engineering has received much attention, and the reason for this issue is the randomness and lack of access to the exact answer of this type of models. For this purpose, in this paper, the approximate solution of two dimensional (2-D) stochastic Volterra–Fredholm integral equations (SVFIEs) based on the operational matrix method and orthonormal Bernoulli polynomials (OBP) is investigated. Some results and convergence analysis are also presented. Finally, by presenting three examples and reviewing the results and numerical comparisons, we showed that the proposed method has an excellent performance.

model for the transmission of Coronavirus, which is a recent major crisis in many communities. Ganji et al. [2] presented a mathematical model of a brain tumor. This model was an extension of a simple two-dimensional mathematical model of glioma growth and diffusion. In [3] the Klein-Gordon equation (KGE) which has many scientific applications such as quantum field theory, nonlinear optics and solid state physics is investigated. Jafari et al. [4] , examined the population dynamics model, including the hunting-hunting problem where the logistics equation, is generalized using the fractional operator.

In many cases, real problems are dependent on some random factors which were ignored due to poor computational power. By increasing the computational power in recent years, these problems are modeled by various types of stochastic equations. Stochastic processes occur in many real issues such as control systems [5] , biological population growth [6] , biology and medicine [7] . In recent decades, due to the importance of stochastic differential equations (SDE) and stochastic integral equations (SIE) in modeling programs where there is considerable uncertainty, scientists have studied the stochastic process and its applications [8] [9] [10] [11] [12] .

The 2D Volterra-Fredholm integral equations are an important class of multi dimensional integral equations which arise in various physical and biological models. These equations can be rarely solved exactly and computational complexity of mathematical operations is important obstacle for solving high dimensional stochastic integral equations. Therefore obtaining their numerical solutions have attracted the attention of researchers in numerical analysis branch and some numerical approaches have been presented to solve stochastic integral equations. Such as: operational matrix method based on hat functions [13] and block-pulse [14] . Also, some studies have been performed on different types of stochastic differential integral equations [15] [16] [17] .

Until now, the number of published papers on numerical solution of multi dimensional stochastic integral equations are very few. Therefore, in this framework this study deals with the numerical solution of 2-D SVFIEs in the following form z(u, v) = g(u, v) + where (u, v) ∈ [0, 1] × [0, 1]. In this equation, g(u, v) and k i (u, v, e, w) for i = 1, 2, 3 are known functions and z(u, v) are unknown functions that we try to determine with our numerical method and B = {B(t), t ≥ 0} is a Brownian process. Also in this equation

This study applies the operational matrix method based on the OBP to obtain the approximate solution of the SVFIEs and the analysis of convergence of the proposed method is discussed.

In this section, we briefly introduce the Itô integral and its properties. The reader can refer to references [8, 16, 17] for more information. Definition 1.1 [8, 18] For t ∈ [0, T ], B(t) is called the Brownian motion, if it satisfies the following properties:

Remark 1.2 By using property (ii), we consider B(0) = 0 [16] .

Assume now that g ∈ (U , V ) and ϕ m is a sequence of elementary functions which satisfies in the following form

then, the Itô integral of g [17] is defined as follows: [17] :

v 0 g(u)dB u = g(v)B v − v 0 B(u)dg u . (1.4)

Definition 1.5 The Bernoulli polynomial of order m is defined as follows [19] :

One of the features of polynomials in the finite-dimensional, that makes them effective in numerical methods is their orthogonality. But Bernoulli polynomials are not orthogonal, despite their very useful properties [19] . Using Gram-Schmidt orthogonalization process, these polynomials can be orthogonalized as defined below:

The orthonormal Bernoulli polynomial of order m is defined as follows:

Therefore, according to the orthogonal property, we will have:

In this case, we express a set of OBP as follows:

(1.8) Definition 1.7 2-D orthonormal Bernoulli polynomial of order m, n is defined as follows: (1.9) which applies to the following property:

(1.10) 11) or

where B(u) and B(v) are one-dimensional OBP vector and ⊗ is the Kronecker product.

Then, by using Eq. (1.6) we obtain

and

Since det(A) =| A | = 0, therefore: 

where c = [c i j ] (m+1)×(m+1) and

(1.20)

By doing calculations, it can easily be shown that:

where C is defined as follows:

Similarly, an arbitrary function k(r , s, e, w) is estimated via 2-D OBP as:

where K is a matrix of order (m + 1) 2 × (m + 1) 2 .

By performing calculations from Eqs. (1.12) and (2.2), we deduce that

Theorem 2.1 Assume now that B(u) be the OBP vector. Consequently, the operational matrix of integration has the following form:

Here, A is given in Eq. (1.15) and M is defined by:

(2.5)

Proof By performing calculations, the result can be easily obtained.

(1.11) for mixed variables is as follows:

where Q = AE A −1 , A is given in Eq. (1.15 ) and E is defined as:

Now, by approximating the integral in Eq. (2.9) in sense of trapezoidal rule for 0,

(2.10) Therefore:

(2.13) Furthermore, by inserting Eq. (2.12) into Eq. (2.13) it results:

(2.14)

To obtain B(0.5) and B(0.25), in Eq. (2.12) we apply the Definition (1.1), where B(t) has normal distribution such that: 

is an arbitrary vector. The operational matrix of product C (m+1) 2 ×(m+1) 2 using 2-D OBP can be given as follows:

Proof For the first one, from Eq. (1.17) it follows where S kt is an (m + 1) 2 × (m + 1) 2 matrix, which is as follows: 

Now, we define E kt = S kt C where: 

Hereunder, we employ the operational matrix method based on 2-D OBP for solving Eq.

(1.1). For this purpose we expand z(u, v), g(v, u) and k i (u, v) for i = 1, 2, 3 with 2-D OBP so that:

with Z , G, B(u, v), K 1 , K 2 and K 3 as defined in the previous section. We will now examine the integrals in Eq. (1.1), respectively. In this way we will have: For the second integral in Eq. (1.1) we have:

Similarly: 

To calculate the unknown coefficients Z from the above equation, we use suitable collocation points. Thus Eq. (3.7) becomes a linear system with (m + 1) 2 equations and (m + 1) 2 unknowns. After calculating the unknown coefficients with the help of Equation z(u, v) = B T (u, v)Z , we can get the approximate answer of the equation. 

| z r |≤ r , (4.1) where r = ρ √ 2r + 1 r l=0 r l 2r − l r − l ,(4.

and ρ is an arbitrary constant such that | z(u) |≤ ρ.

Proof Using the OBP, z(u) can be approximated as

where z r can be determined by Thus, we will have:

which is presented in Eq. (4.2) .

. Therefore:

(4.7)

Here, O is a big-O notation.

Proof See [21] .

(4.9) Theorem 4.5 Suppose u ∈ [0, ∞). In this case, we have

where M(v) = sup 0≤r <v B(r ), P is a probability function and is cumulative standard normal distribution function.

Proof See [22] . for i = 1, 2, 3, respectively. Also assume that:

Then, we have:

(4.12)

The approximate equation of Eq. (4.12) is:

(4.13) Therefore, from Eqs. (4.12) and (4.13) we have:

(4.14)

We now estimate Eq. (4.14) by 2D-integrals mean value theorem for every (u, v) ∈ [0, 1] 2 and (u, v, e, w) ∈ [0, 1] 4 . Therefore:

On the other hand, for i = 1, 2, 3

(4.16) 1) , consider the SVFIE that was studied by [13, 14] :

(5.1)

, the exact solution of the equation is z(u, v) = 1.

In Table 1 , the values of absolute error at different points for m = 2 and m = 3 are reported which indicates the accuracy of the suggested scheme. Table 2 , provides a comparison of maximum absolute error for different values of m in points (0.1, 0.2) and (0.3, 0.7) between the presented scheme and previously published algorithms by Fallahpour et al. [14] . Also, absolute error for m = 2 and m = 3 are plotted in Figs. 1 and 2 , respectively. By examining the absolute error presented in Figs. 1 and 2 for different values of m, we will find that by increasing m, the error decreases substantially. Example 5.2 Consider the SVFIE that was studied by [14] :

where

3)

The exact solution of (5.2) is z(u, v) = u + v.

In Table 3 , the values of absolute error at different points for m = 2 and m = 3 are reported which indicates the accuracy of the suggested scheme. Table 4 , provides a comparison of [14] . Also, absolute error for m = 2 and m = 3 are plotted in Figs. 3 and 4, respectively. Three dimensional graphs of the exact and approximation solution is shown in Fig. 5 . It can easily be seen from the figures that as the number of m increases the approximate solution converges towards the exact solution. Figure 6 shows the approximate solution and the exact solution at the top and the absolute error at the bottom, for v = 0.5 and m = 3.

Consider the SVFIE that was studied by [14] : where

(5.5)

The exact solution of (5.4) is z(u, v) = uv.

In Table 5 , the values of absolute error at different points for m = 2 and m = 3 are reported which indicates the accuracy of the suggested scheme. Table 6 , provides a comparison of maximum absolute error for different values of m in points (0.6, 0.8) and (0.4, 0.9) between the presented scheme and previously published algorithms in [13, 14] . 

Obtaining approximate answers for Itô integral with their applications to modelling in science and engineering is very important offers. Also it is a comprehensive examination to the most important issues of stochastic differential equations. Due to computational complexity, in this study, we develop a new matrix approach based on 2D-OBP to solve two dimensional stochastic Volterra-Fredholm integral equations with the given boundary conditions. Based on this, 2-D operational matrix of integration and 2-D stochastic operational matrix of inte- gration is obtained, which leads to the converting the problem into a linear system. Some results and convergence analysis are also presented. Finally, by presenting three examples and reviewing the results and numerical comparisons, we showed that the proposed method has an excellent performance. In other words, numerical examples show that the suggested method is very accurate, simple and powerful tool to solve various problems. Another considerable advantage of this method in case of linear problems is that the exact solution can be obtained from mth-order approximation when the solution is a polynomial of degree equal to or less than m.

Funding This research received no external funding.

Data availability statement Ph.D. thesis dissertations are public documents, but at the moment are not internetaccessible. If one is interested in revising the investigation, information can be requested from the authors.

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The authors are thankful to the respected reviewers for their valuable comments and constructive suggestions towards the improvement of the original paper. 

By assumptions (i) and (ii) for i = 1, 2, 3 and Note (4.4) we obtain:Substituting Eq. (4.17) in Eq. (4.15) and using Note (4.4) implies that:(4.20) Note 4.7 By an argument similar to Theorem (4.6), we have:(4.21)

In this section, we show the efficiency and accuracy of the suggested method by giving three examples. All the numerical calculations are performed on an Intel Core i7 laptop by running some programming codes written in MATLAB R2016b.

Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this article.