scholarly journals A numerical method for solving Fredholm–Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

2011 ◽  
Vol 235 (14) ◽  
pp. 3965-3971 ◽  
Author(s):  
E. Babolian ◽  
K. Maleknejad ◽  
M. Mordad ◽  
B. Rahimi
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Block Pulse Functions

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.

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 Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mengting Deng ◽  
Guo Jiang ◽  
Ting Ke
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Hurst Parameter ◽  
Stochastic Integral Equation ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

scholarly journals Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1326
Author(s):  
Sanda Micula
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Cubature Formula ◽  
Volterra Integral Equations ◽  
Picard Iteration ◽  
Successive Approximations ◽  
Future Research ◽  
Two Dimensional ◽  
Uniqueness Of The Solution

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed.

Sign in / Sign up

Export Citation Format

Share Document