Numerical solution of Volterra–Fredholm integral equations via modification of hat functions

Mirzaee, Farshid; Hadadiyan, Elham

doi:10.1016/j.amc.2016.01.038

Cited by 36 publications

(23 citation statements)

References 19 publications

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“…Lemma 3.1 Suppose that H m (v) and Ψ m (v) are respectively given in (10) and 4, H m (v) can be written in accordance with BPFs as follows:…”

Section: Haar Wavelets and Bpfsmentioning

confidence: 99%

“…As we all know, many stochastic Volterra integral equations do not have exact solutions, so it makes sense to find more precise approximate solutions to stochastic Volterra integral equations. There are different numerical methods to stochastic Volterra integral equations, for example, orthogonal basis methods [1][2][3][4][5][6][7][8][9][10], wash series methods [11,12], and polynomials methods [13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

Jiang

Sang

2019

Adv Differ Equ

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In this paper, an efficient numerical method is presented for solving nonlinear stochastic Itô-Volterra integral equations based on Haar wavelets. By the properties of Haar wavelets and stochastic integration operational matrixes, the approximate solution of nonlinear stochastic Itô-Volterra integral equations can be found. At the same time, the error analysis is established. Finally, two numerical examples are offered to testify the validity and precision of the presented method.

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“…Lemma 3.1 Suppose that H m (v) and Ψ m (v) are respectively given in (10) and 4, H m (v) can be written in accordance with BPFs as follows:…”

Section: Haar Wavelets and Bpfsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

Jiang

Sang

2019

Adv Differ Equ

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“…Example Consider the following nonlinear FPQIDE with given initial conditions

\{\begin{matrix} left \\ \frac{\partial u (x, y)}{\partial x} + \frac{\partial u (x, y)}{\partial y} + \frac{\partial^{2} u (x, y)}{\partial x \partial y} + u (x, y) = \frac{x}{12} \sin (\frac{xy}{4}) + H 1 . H 2, \\ H 1 = \frac{1}{Γ (\frac{9}{2}) Γ (\frac{11}{2})} \int_{0}^{x} \int_{0}^{y} {((), x - s)}^{\frac{7}{2}} {((), y - t)}^{\frac{9}{2}} (x + \cos (t)) u^{2} (s, t) dtds \\ H 2 = \frac{1}{Γ (\frac{7}{2}) Γ (\frac{13}{2})} \int_{0}^{x} \int_{0}^{y} {((), x - s)}^{\frac{5}{2}} {((), y - t)}^{\frac{11}{2}} (\frac{y \sin ((), s + t)}{8}) u^{2} (s, t) dtds, \\ u (x, 0) = 0, u (0, y) = 0, \end{matrix}

whose exact solution u ( x , y ) is unknown. Since the exact solution of this example is not known, the maximum absolute error is calculated using the following double mesh principle

E_{n} = \sup R_{n} (x, y), (x, y)

…”

Section: Numerical Examplesmentioning

confidence: 99%

“…whose exact solution u(x, y) is unknown. Since the exact solution of this example is not known, the maximum absolute error is calculated using the following double mesh principle [23] E n = sup R n (x, y), (x, y) ∈ ,…”

Section: Proof Letmentioning

confidence: 99%

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

Mirzaee

Alipour

2018

Numerical Methods Partial

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In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.

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“…MHFs have the following properties:

h_{i} false(j h false) = ({, \begin{matrix} array 1 & array i = j, \\ array 0 & array i \neq j . \end{matrix})

Let

boldH false(x false) = {([], h_{0} false(x false), h_{1} false(x false), ⋯, h_{n} false(x false))}^{T} .

According to Equation and expanding entries of H ( x ) H T ( x ) by using MHFs, we have

boldH false(x false) H^{T} false(x false) = diag ((), boldH false(x false)) .

If F be an arbitrary vector of order ( n +1), by using Equation , we have

boldH false(x false) H^{T} false(x false) F = \overset{F}{true} boldH false(x false),

where

\overset{F}{true}

is an ( n +1)×( n +1) diagonal matrix with diagonal elements equal to the entries of vector F . Also, if A be a ( n +1)×( n +1) matrix, we have

H^{T} false(x false) A boldH false(x false) = H^{T} false(x false) Â = Â^{T} boldH false(x false),

where

Â

is an ( n +1) vector with elements equal to the diagonal entries of matrix A .…”

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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Numerical solution of Volterra–Fredholm integral equations via modification of hat functions

Cited by 36 publications

References 19 publications

Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

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

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