Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.


Introduction
Wavelets are mathematical functions that isolate the data and analyze each variable with the corresponding resolution in various frequency components. As a mathematical tool, wavelet can be used to extract information from the different forms of data, including, seismic waves, earthquakes, music, image processing, signal processing, acoustics, nuclear engineering, and astronomy.
We lack enough knowledge in certain issues related to behavioral analysis for some processes to decide how it conforms, or its operation is so complex that it is irrelevant or difficult to explain precisely. In such a scenario a probabilistic model is always helpful. Due to their significance in modeling, science and technological phenomenon, nonlinear stochastic and deterministic functions have been widely investigated and studied.
The Itô integral typically used in applied mathematics is named after Kiyoshi Itô. In analytical forms, stochastic integral equations cannot be resolved normally and therefore a necessary prerequisite for their solution in applied mathematics become necessary. Throughout recent years, the computational methods such as [20][21][22][23][24] have been used to solve stochastic integral equations. However, very few articles exist on stochastic integral equations. The operational matrix method using Bernoulli wavelets for solving linear Itô-Volterra integral equations was recently employed by Mirzaee and Samadyar [25]. This article however attempts to frame a stochastic operational matrix of integration of Bernoulli wavelets (SOMIBW) and is employed to obtain the solution to the particular case of the NSIVIE as follows are functions of x and t, )) ( ( x y µ and )) ( ( x y σ are known functions, ) (x y is the unknown that is to be determined and ) (x W is the Brownian motion process defined on probability space ) , , ( P F Ω that consists of the sample space , Ω a σ -algebra F of subsets of Ω which we call events, and a real-valued set function P defined on F that is called probability.

Equation
(1) appears in various fields including engineering, mathematics, biology, health and social science. It is very hard or even difficult to solve this equation, so we develop an efficient method for solving it. In this article, equation (1) is numerically solved by the use of operational matrix of integration (OMI) and stochastic OMI based on Bernoulli wavelets. This equation is reduced to a nonlinear system of algebraic equations by the use of collocation points and these matrices can be solved by an effective numerical method such as Newton's method.

397
The rest of the work is structured accordingly. Section 2 provides some basic definitions and characteristics of stochastic calculus, wavelets, and Bernoulli wavelets. Also, in this section, OMI and stochastic OMI based on Bernoulli wavelets are obtained. The proposed method of solution is given to estimate the solution of non linear Itô-Volterra integral equation in Section 3. In Section 4, Numerical examples are presented to show the efficiency and reliability of the proposed method. Finally, the conclusion of the article is given in Section 5.

Brownian motion
has continuous sample paths.
• The process has independent, stationary increments.

Definition 2.2.
An n-dimensional process, ( is a standard Brownian motion and the i x W ) ( 's are independent of each other.

Wavelets
A family of functions is generated by mother wavelets by dilating and translating itself, which we call wavelets. The wavelet family is as given below [26]: where a and b are denoted respectively for the dilation parameter and translation parameter. Nevertheless, b varies continously.
the family of discrete wavelets may be given as, where the wavelet bases in )

Bernoulli polynomials
Bernoulli polynomials are defined in general as [25]: where , are Bernoulli numbers. For instance, starting five Bernoulli numbers are: . 30 And the first four Bernoulli polynomials are:

Bernoulli wavelets
Bernoulli wavelets are defined as follows [25]: denotes the order of the Bernoulli polynomials and

Approximation of function
∈ is expanded in terms of the Bernoulli wavelets as Truncating the above infinite series, we get ,

OMI and stochastic OMI of Bernoulli wavelets (SOMIBW)
OMI of Bernoulli wavelets is given in detail in [25]. Now, we derive the stochastic OMI of Bernoulli wavelets as follows: The stochastic integral of ) (x ψ can be obtained as follows: where s P is a m m× matrix and is called the SOMIBW. In particular, for 2 = M and 10 10 0 Using equations (9) to (12), we get where, ) (x ψ is the Bernoulli wavelet coefficient matrix and F is a m m× matrix given by , where .
Also, for a m m× matrix C, is a m -vector and .

Remark 2. If µ is an analytic function on R and
) (x C T ψ be the the expansion of ) (x f in terms of Bernoulli wavelets, where C is given in equation (6), then where [ ].
Remark 3. If µ is an analytic function on R and ) (x C T ψ be the the expansion of ) (x f in terms of Bernoulli wavelets, where C is given in equation (6), then is the Bernoulli wavelets coefficient matrix given in (7) and is given in Remark 2.

Bernoulli Wavelets Method of Solution (BWM)
In this section, we use the newly derived SOMIBW for the numerical solution of NSIVIE. Here we consider the equation in (1) as: with respect to Bernoulli wavelets as follows: , where C is given in equation (6) and is the unknown vector to be determined.
where C and F are Bernoulli wavelet coefficient vectors and 1 K and 2 K are Bernoulli wavelet matrices. Substituting (19), (20), (21) and (22) in (18), we get Now, by using Remark 3, equation (23) can be rewritten as, Using equation (24) and Remark 1, we get where 1 X and 2 X are m m× matrices described in Remark 1. Applying the OMI of Bernoulli wavelets P explained in [25] and the stochastic OMI of Bernoulli wavelets described in Section 2, equation (25) reduces to: Let us assume that P X K 1 1 1= δ and .
Again using Remark 1, where 1 δ and 2 δ are m -vectors containing a nonlinear combination of elements of C.
Replacing − by = , equation (27) reduces to a nonlinear system of equations .
Solving this nonlinear system, we get the unknown vector C. Substituting this obtained vector in equation (19), we obtain the solution of NSIVIE (1).

Numerical Experiments
Test problem 1. Consider the SLSVIE [27], is the unknown stochastic process defined on the probability space is the Brownian motion process. Table 1 where the exact solution of this problem is found to be , ) 20 where ) (x y is the unknown stochastic process defined on the probability space is the Brownian motion process. Table 3 shows the numerical results obtained by the method described in Section 3 (BWM), exact solution and absolute errors (AE) for Table 4 shows the comparison of absolute errors of test problem 2 for different values of k and M and Figure  2 shows the graph of exact and approximate values of test problem 2 for

Conclusion
In this article, an effective strategy is provided to solve NSIVIE. This technique reduces these equations to a system of nonlinear algebraic equations with unknown Bernoulli coefficients, by using Bernoulli wavelets, their operational matrix of integration and stochastic operational matrix of integration which are solved by Newton's method. Error analysis of the proposed method is given. Moreover, the results obtained are compared with the exact solution with two Volterra integral equations and two NSIVIE in order to show the difference between the nonlinear Volterra integral equations and NSIVIE, and these examples show that the method described is precise and accurate and the results are in good agreement with the exact solution. By this we can conclude that the predicted algorithm is well organized and efficient.