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.

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.

scholarly journals Multi-Wavelets Galerkin Method for Solving the System of Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1369
Author(s):  
Hoang Viet Long ◽  
Haifa Bin Jebreen ◽  
Stefania Tomasiello
Keyword(s):  
Wavelet Transform ◽  
Galerkin Method ◽  
Integral Equations ◽  
Operational Matrix ◽  
Computational Effort ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Linear Type ◽  
Numerical Tests ◽  
Operational Matrix Of Integration

In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order O(2−Jr), where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method.

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.

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.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 200
Author(s):  
Ji-Huan He ◽  
Mahmoud H. Taha ◽  
Mohamed A. Ramadan ◽  
Galal M. Moatimid
Keyword(s):  
Linear System ◽  
Computer Program ◽  
Integral Equations ◽  
Operational Matrix ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

scholarly journals Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Mashayekhi ◽  
M. Razzaghi ◽  
O. Tripak
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Fredholm Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Hybrid Functions ◽  
Block Pulse Functions

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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