Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions

Mirzaee, Farshid; Hadadiyan, Elham

doi:10.1016/j.cam.2016.02.015

Cited by 20 publications

(11 citation statements)

References 20 publications

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“…Matrix P is called operational matrix of integration based on MHFs, when the following relation be satisfied:

\int_{0}^{x} boldH false(y false) d y ≃ P boldH false(x false) .

The matrix P calculated in paper as

P = \frac{h}{12} (\begin{matrix} 0 & 5 & 4 & 4 & 4 & ⋯ & 4 & 4 & 4 & 4 \\ 0 & 8 & 16 & 16 & 16 & ⋯ & 16 & 16 & 16 & 16 \\ 0 & - 1 & 4 & 9 & 8 & ⋯ & 8 & 8 & 8 & 8 \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ 0 & 0 & 0 & 0 & 0 & ⋯ & - 1 & 4 & 9 & 8 \\ 0 & 0 & 0 & 0 & 0 & ⋯ & 0 & 0 & 8 & 16 \\ 0 & 0 & 0 & 0 \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

Mirzaee

Samadyar

2017

Math Methods in App Sciences

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In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion with Hurst parameter H∈false(12,1false). The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's‐Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.

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“…Matrix P is called operational matrix of integration based on MHFs, when the following relation be satisfied:

\int_{0}^{x} boldH false(y false) d y ≃ P boldH false(x false) .

The matrix P calculated in paper as

P = \frac{h}{12} (\begin{matrix} 0 & 5 & 4 & 4 & 4 & ⋯ & 4 & 4 & 4 & 4 \\ 0 & 8 & 16 & 16 & 16 & ⋯ & 16 & 16 & 16 & 16 \\ 0 & - 1 & 4 & 9 & 8 & ⋯ & 8 & 8 & 8 & 8 \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ 0 & 0 & 0 & 0 & 0 & ⋯ & - 1 & 4 & 9 & 8 \\ 0 & 0 & 0 & 0 & 0 & ⋯ & 0 & 0 & 8 & 16 \\ 0 & 0 & 0 & 0 \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

Mirzaee

Samadyar

2017

Math Methods in App Sciences

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“…, for the solution of the linear Fredholm integral equations. Solving nonlinear Stratonovich Volterra integral equations via 1D‐MHFs is presented in . Moreover, Mirzaee et al .…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, Mirzaee et al . applied two‐dimensional modification of hat functions functions for solving two‐dimensional linear Fredholm integral equations and Volterra–Fredholm integral equations, respectively. The new and basic idea in this paper is extending 1D‐MHFs to 3D‐MHFs and using them for solving with following assumption

H (x, y, z, s, t, r, u (s, t, r)) = k (x, y, z, s, t, r) G (u (s, t, r)) .

The paper is organized as follows: In Sections 2 and 3, we will introduce 1D‐MHFs and 3D‐MHFs and their properties, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…These functions are extensions of one-dimensional modification of hat functions functions (1D-MHFs), introduced by Mirzaee et al [14], for the solution of the linear Fredholm integral equations. Solving nonlinear Stratonovich Volterra integral equations via 1D-MHFs is presented in [15]. Moreover, Mirzaee et al [15,16]…”

Section: Introductionmentioning

confidence: 99%

“…Solving nonlinear Stratonovich Volterra integral equations via 1D-MHFs is presented in [15]. Moreover, Mirzaee et al [15,16]…”

Section: Introductionmentioning

confidence: 99%

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Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

Mirzaee

Hadadiyan

2016

Math Methods in App Sciences

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This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three‐dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

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A Numerical Approach for Approximating Variable-Order Fractional Integral Operator

Nemati

2021

Operator Theory, Functional Analysis and Applications

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Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions

Cited by 20 publications

References 20 publications

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

A Numerical Approach for Approximating Variable-Order Fractional Integral Operator

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