Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion

Heydari, Mohammad Hossein; Mahmoudi, Mohammad Reza; Shakiba, Ali; Avazzadeh, Z.

doi:10.1016/j.cnsns.2018.04.018

Cited by 72 publications

(26 citation statements)

References 30 publications

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“…The stochastic model driven with fBm for stochastic volatilities, stock prices and electricity prices can be expressed similar to Heydari et al (2018) by: Where f(x)=exp⁡(exp⁡(−κx)[∫0xtrue(κμ−σ2Hy2H−1true)exp⁡true(κytrue)dy+σ∫0xexp⁡true(κytrue)dBHtrue(ytrue) +ln⁡(f0`)true]true) is the exact solution. Stochastic operational matrix for Bernstein polynomials presented in Section 3 has been implemented in solving the above stochastic integral equation.…”

Section: Numerical Examplesmentioning

confidence: 99%

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

Ray

Singh

2020

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Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

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Section: Numerical Examplesmentioning

confidence: 99%

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

Ray

Singh

2020

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“…For the fractional diffusion equation, Zhang et al 32 used the time‐discrete scheme based on the finite difference method, Hajipour et al 33 applied the Chebyshev wavelet approach for solving multiterm VO time‐fractional diffusion‐wave equation, Cao et al 34 provided the compact finite difference operator for solving VO‐fractional reaction–diffusion equation. For fractional subdiffusion equations, Yu et al 35 applied the compact finite difference scheme for VO‐fractional subdiffusion equations, Yaseen et al 24 proposed the generalized Laguerre spectral Petrov–Galerkin method, Ghaffari and Ghoreishi 36 proposed a cubic trigonometric B‐spline collocation approach, and so forth (see References 22,37‐39).…”

Section: Introductionmentioning

confidence: 99%

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

Dehestani

Ordokhani

Razzaghi

2020

Numerical Linear Algebra App

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In this article, we study the numerical technique for variable‐order fractional reaction‐diffusion and subdiffusion equations that the fractional derivative is described in Caputo's sense. The discrete scheme is developed based on Lucas multiwavelet functions and also modified and pseudo‐operational matrices. Under suitable properties of these matrices, we present the computational algorithm with high accuracy for solving the proposed problems. Modified and pseudo‐operational matrices are employed to achieve the nonlinear algebraic equation corresponding to the proposed problems. In addition, the convergence of the approximate solution to the exact solution is proven by providing an upper bound of error estimate. Numerical experiments for both classes of problems are presented to confirm our theoretical analysis.

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“…This motivates our interest to propose an efficient and accurate computational method for solving stochastic integral equations. In [24] M. H. Heydari & al. used Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Новый Численный Метод Решения Нелинейных Стохастических Интегральных Уравнений

Zeghdane¹

2020

Владикавказский Математический Журнал

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The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.

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Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion

Cited by 72 publications

References 30 publications

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

Новый Численный Метод Решения Нелинейных Стохастических Интегральных Уравнений

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