Stochastic Operational Matrix Of Integration Research Articles

Numerical solution for stochastic Volterra-Fredholm integral equations with delay arguments

We present a method for computing the stochastic operational matrix of integration to advance the study of stochastic Volterra-Fredholm integral equations (SVFIEs) based on delay arguments. First, the method evaluates the combined effects of the delay and its parameters on the accuracy improvement of the convergence rate. Our results can be applied to SVFIEs, with the operational delay matrices of the block pulse function simplified to algebraic ones. Numerical calculations were performed on a PC using Python 3 programs. Results also demonstrate the accuracy of approximate solutions; arithmetic operations are carried out without the need for derivation or integration.

Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations

This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton’s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications.

A new hybrid approach for nonlinear stochastic differential equations driven by multifractional Gaussian noise

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems.

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in norm is derived and obtained as . The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.

Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations

This work is an extended version of the ICCS 2020 conference paper [1]. The paper aims to present an efficient numerical method to quantify the uncertainty in the solution of stochastic fractional integro-differential equations. The numerical method presented here is a wavelet collocation method based on Legendre polynomials, and their deterministic and stochastic operational matrix of integration. The operational matrices are used to convert the stochastic fractional integro-differential equation to a linear system of algebraic equations. The accuracy and efficiency of the proposed method are validated through numerical experiments. Moreover, the results are compared with the numerical methods based on the Gaussian radial basis function (GA RBF) and thin plate splines radial basis function (TBS RBF) to show the superiority of the proposed method. Finally, concerning the real-world application, a stock market model has been simulated and the results are demonstrated.

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.

Fibonacci wavelet based numerical method for the solution of nonlinear Stratonovich Volterra integral equations

This article provides an effective technique to solve nonlinear Stratonovich Volterra integral equations (NSVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown Fibonacci coefficients, by using Fibonacci wavelets, their operational matrix of integration and stochastic operational matrix of integration and these equations can be solved by numerical methods such as Newton's method. Error estimate of the proposed method is given. Moreover, the results obtained by the method proposed are compared to block pulse functions and Legendre wavelets method with two numerical examples to show that the method described is precise and accurate.

Solving Ito integral equations with time delay via basis functions

In this paper, a direct method for solving Volterra-Fredholm integral equations with time delay by using orthogonal functions and their stochastic operational matrix of integration is proposed. Stochastic integral equations can be reduced to a sparse system which can be directly solved. Numerical examples show that the proposed scheme has a suitable degree of accuracy.

A COMPUTATIONAL METHOD FOR SOLVING STOCHASTIC INTEGRAL EQUATIONS USING HAAR WAVELETS

In this article, we have developed a new technique for solving stochastic integral equations. A new Haar wavelets stochastic operational matrix of integration (HWSOMI) is developed in order to obtain efficient and accurate solution for stochastic integral equations. In the beginning we study the properties of stochastic integrals and Haar wavelets. Convergence and error analysis of Haar wavelet method is presented. Accuracy of the method investigated is justified through some examples.

Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itô-Volterra integral equations

PurposeThe purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.Design/methodology/approachFor this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.FindingsBy applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.Originality/valueTwo numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h2). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.

Application of operational matrices for solving system of linear Stratonovich Volterra integral equation

In this paper, we give a brief review on Block pulse functions (BPFs) and obtain operational matrix and stochastic operational matrix of integration based on BPFs. Then these operational matrices are used to solve system of linear Stratonovich Volterra integral equations. By applying proposed method, the system of linear Stratonovich Volterra integral equations reduce to system of linear algebraic equations which can be solved by a convenient numerical method. Also, the error analysis is proved under several mild conditions. We show the rate of convergence is O(h). Finally, by applying this method on two examples, we demonstrate accuracy and efficiency of the proposed method. All of the numerical calculation are performed on computer using a program written in MATLAB.

Numerical method for solving linear stochasticIto--Volterra integral equations driven by fractional Brownian motion using hat functions

In this paper, we present a numerical method to approximate the solution of linear stochastic Ito-Volterra integral equations driven by fractional Brownian motion with Hurst parameter $ H \in (0,1)$ based on a stochastic operational matrix of integration for generalized hat basis functions. We obtain a linear system of algebraic equations with a lower triangular coefficients matrix from the linear stochastic integral equation, and by solving it we get an approximation solution with accuracy of order $ \emph{O}(h^2)$. This numerical method shows that results are more accurate than the block pulse functions method where the rate of convergence is $ \emph{O}(h)$. Finally, we investigate error analysis and with some examples indicate the efficiency of the method.

Numerical Solution Based on Hat Functions for Solving Nonlinear Stochastic Itô Volterra Integral Equations Driven by Fractional Brownian Motion

This paper presents a numerical method for solving nonlinear stochastic Ito Volterra integral equations driven by fractional Brownian motion with Hurst parameter \( H \in (0,1)\) via of hat functions. Using properties of the generalized hat basis functions and fractional Brownian motion, new stochastic operational matrix of integration is achieved and the nonlinear stochastic equation is transformed into nonlinear system of algebraic equations which by solving it, an approximation solution with high accuracy is obtained. In addition, error analysis of the method is investigated, and by some examples, efficiency and accuracy of the suggested method are shown.

Solving system of linear Stratonovich Volterra integral equations via modification of hat functions

This paper proposes an efficient method for solving system of linear Stratonovich Volterra integral equations. Stochastic operational matrix of modification of hat functions (MHFs) is determined. By using MHFs and their stochastic operational matrix of integration, a system of linear Stratonovich Volterra integral equations can be reduced to a linear system of algebraic equations. Thus we can solve the problem by direct methods. Also, we prove that the rate of convergence is O(h3). Efficiency of this method and good degree of accuracy are confirmed by numerical examples.

A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.

A computational method for solving nonlinear stochastic Volterra integral equations

In this paper, an efficient method for solving nonlinear stochastic Itô–Volterra integral equations (NSIVIEs) is proposed. By using new adjustment of hat basis functions and their stochastic operational matrix of integration, the NSIVIE is reduced to a nonlinear system of algebraic equations. Also, the error analysis of the proposed method is investigated and proved that the order of convergence is O(h4). Finally, numerical examples affirm the efficiency of the proposed method.

Numerical approach for solving stochastic Volterra–Fredholm integral equations by stochastic operational matrix

In this paper, we obtain stochastic operational matrix of block pulse functions on interval [0,1) to solve stochastic Volterra–Fredholm integral equations. By using block pulse functions and their stochastic operational matrix of integration, the stochastic Volterra–Fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is O(h). Furthermore, the results show that the approximate solutions have a good degree of accuracy.

A numerical method for solving [formula omitted]-dimensional stochastic Itô–Volterra integral equations by stochastic operational matrix

The multidimensional Itô–Volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise sources. In this paper, we obtain a stochastic operational matrix of block pulse functions on interval [ 0 , 1 ) to solve m-dimensional stochastic Itô–Volterra integral equations. By using block pulse functions and their stochastic operational matrix of integration, m -dimensional stochastic Itô–Volterra integral equations can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is O ( h ) . Furthermore, a 95% confidence interval of the errors’ mean is made, the results shows that the approximate solutions have a credible degree of accuracy.

Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions

This article proposes an efficient method for solving stochastic Volterra integral equations. By using block pulse functions and their stochastic operational matrix of integration, a stochastic Volterra integral equation can be reduced to a linear lower triangular system, which can be directly solved by forward substitution. The results show that the approximate solutions have a good degree of accuracy.