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.

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

Multidiscipline Modeling in Materials and Structures ◽

10.1108/mmms-04-2018-0075 ◽

2019 ◽

Vol 15 (3) ◽

pp. 575-598 ◽

Cited By ~ 4

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.

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

Random Operators and Stochastic Equations ◽

10.1515/rose-2020-2040 ◽

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

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691319500073 ◽

2019 ◽

Vol 17 (03) ◽

pp. 1950007 ◽

Cited By ~ 1

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.

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

Axioms ◽

10.3390/axioms10030200 ◽

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.

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

Mathematics ◽

10.3390/math8081369 ◽

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.

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Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations

Georgian Mathematical Journal ◽

10.1515/gmj-2018-0009 ◽

2020 ◽

Vol 27 (1) ◽

pp. 81-95 ◽

Cited By ~ 2

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.

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Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v4i3.4052 ◽

2015 ◽

Vol 4 (3) ◽

pp. 420 ◽

Cited By ~ 1

Author(s):

Behrooz Basirat ◽

Mohammad Amin Shahdadi

Keyword(s):

Bernstein Polynomials ◽

Operational Matrix ◽

Numerical Procedure ◽

Operational Matrices ◽

Algebraic Equations ◽

Mixed Condition ◽

Matrix Methods ◽

Numerical Examples ◽

Linear Algebraic Equations ◽

The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽

10.3390/sym12111755 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1755

Author(s):

M. S. Al-Sharif ◽

A. I. Ahmed ◽

M. S. Salim

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Science And Engineering ◽

Numerical Examples ◽

Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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