Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations

2020 ◽  
Vol 27 (1) ◽  
pp. 81-95 ◽  
Author(s):  
Mohammad Hossein Heydari ◽  
Mohammad Reza Hooshmandasl ◽  
Carlo Cattani
Keyword(s):  
Nonlinear Systems ◽  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
Computational Method ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
Stochastic Operational Matrix ◽  
A New Technique

AbstractIn this paper, a new computational method based on the Chebyshev wavelets (CWs) is proposed for solving nonlinear stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix (SOM) for the CWs is obtained. By using these basis functions and their SOM, such problems can be transformed into nonlinear systems of algebraic equations which can be simply solved. Moreover, a new technique for computation of nonlinear terms in such problems is presented. Further error analysis of the proposed method is also investigated and the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient.

Stochastic operational matrix of Chebyshev wavelets for solving multi-dimensional stochastic Itô–Volterra integral equations

2019 ◽  
Vol 17 (03) ◽  
pp. 1950007 ◽  
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
General Procedure ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
System Of Integral Equations ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

In this paper, the numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations have been obtained by second kind Chebyshev wavelets. The second kind Chebyshev wavelets are orthonormal and have compact support on [Formula: see text]. The block pulse functions and their relations to second kind Chebyshev wavelets are employed to derive a general procedure for forming stochastic operational matrix of second kind Chebyshev wavelets. The system of integral equations has been reduced to a system of nonlinear algebraic equations and solved for obtaining the numerical solutions. Convergence and error analysis of the proposed method are also discussed. Furthermore, some examples have been discussed to establish the accuracy and efficiency of the proposed scheme.

Numerical study of stochastic Volterra–Fredholm integral equations by using second kind Chebyshev wavelets

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Fakhrodin Mohammadi ◽  
Parastoo Adhami
Keyword(s):  
Integral Equations ◽  
Numerical Study ◽  
Operational Matrix ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Efficiency And Reliability

AbstractIn this paper, we present a computational method for solving stochastic Volterra–Fredholm integral equations which is based on the second kind Chebyshev wavelets and their stochastic operational matrix. Convergence and error analysis of the proposed method are investigated. Numerical results are compared with the block pulse functions method for some non-trivial examples. The obtained results reveal efficiency and reliability of the proposed wavelet method.

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

2019 ◽  
Vol 15 (3) ◽  
pp. 575-598 ◽  
Author(s):  
Farshid Mirzaee ◽  
Nasrin Samadyar
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Stochastic Operational Matrix ◽  
Bernoulli Wavelet ◽  
Operational Matrix Of Integration

Purpose The 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/approach For 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. Findings By 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/value Two 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.

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

2020 ◽  
pp. 395-410
Author(s):  
S. C. Shiralashetti ◽  
Lata Lamani
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Effective Strategy ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Stochastic Operational Matrix ◽  
Nonlinear Algebraic Equations ◽  
Operational Matrix Of Integration

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.

Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions

2018 ◽  
Vol 19 (3-4) ◽  
pp. 289-297 ◽  
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Integral Equations ◽  
Compact Support ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Results ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

AbstractIn this article, the numerical solutions of stochastic Volterra–Fredholm integral equations have been obtained by hybrid Legendre block-pulse functions (BPFs) and stochastic operational matrix. The hybrid Legendre BPFs are orthonormal and have compact support on$[0, 1)$. The numerical results obtained by the above functions have been compared with those obtained by second kind Chebyshev wavelets. Furthermore, the results of the proposed computational method establish its accuracy and efficiency.

Numerical Solution of Stochastic Ito-Volterra Integral Equations using Haar Wavelets

2016 ◽  
Vol 9 (3) ◽  
pp. 416-431 ◽  
Author(s):  
Fakhrodin Mohammadi
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
General Procedure ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Computational Method ◽  
Haar Wavelets ◽  
Stochastic Operational Matrix

AbstractThis paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

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