Euler polynomial solutions of nonlinear stochastic Itô–Volterra integral equations

Mirzaee, Farshid; Samadyar, Nasrin; Hoseini, Seyede Fatemeh

doi:10.1016/j.cam.2017.09.005

Cited by 55 publications

(18 citation statements)

References 14 publications

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“…Euler studied Bernoulli polynomials, and these polynomials are employed in the integral representation of differentiable periodic functions and play an important role in the approximation of such functions using polynomials. Many early Euler and Bernoulli polynomial implementations can be contained in [18][19][20]. Euler polynomials are strictly related to Bernoulli and are used in Taylor's expansion in the trigonometric and hyperbolic secant function districts.…”

Section: Matrix Relations For Euler Polynomialsmentioning

confidence: 99%

An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

Basirat

Elahi

2020

International Journal of Differential Equations

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This paper deals with an approach to obtaining the numerical solution of the Lotka–Volterra predator-prey models with discrete delay using Euler polynomials connected with Bernoulli ones. By using the Euler polynomials connected with Bernoulli ones and collocation points, this method transforms the predator-prey model into a matrix equation. The main characteristic of this approach is that it reduces the predator-prey model to a system of algebraic equations, which greatly simplifies the problem. For these models, the explicit formula determining the stability and the direction is given. Numerical examples illustrate the reliability and efficiency of the proposed scheme.

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Section: Matrix Relations For Euler Polynomialsmentioning

confidence: 99%

An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

Basirat

Elahi

2020

International Journal of Differential Equations

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“…Recently, many research has been carried out on solving the stochastic Itô‐Volterra integral equation. In these researches, the numerical methods based on the least squares, stochastic operational matrix, radial basis functions (RBFs), Euler polynomial, orthonormal Bernoulli polynomials, Haar wavelets, and cubic B‐spline approximation are introduced. In Saffarzadeh et al, an iterative numerical algorithm to approximate the solution of stochastic Itô‐Volterra integral equations with

m

‐dimensional Brownian motion process is provided.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Saffarzadeh

Heydari

Loghmani

2020

Math Methods in App Sciences

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In this paper, an efficient and accurate numerical iterative algorithm based on the linear spline interpolation for solving the system of nonlinear stochastic Itô-Volterra integral equations is presented. The most important merit of this method is that it does not need to solve any system of nonlinear algebraic equations. An upper bound for the linear spline approximation of the stochastic function is provided. Using this upper bound and under the Lipschitz and linear growth conditions, the convergence analysis of the suggested method is studied.Finally, to verify the efficiency of the proposed scheme, some problems in the finance, physics, and biology are investigated, and the obtained results are compared with the stochastic -method. KEYWORDSlinear spline interpolation, multiple stock models, pendulum problem, predator-prey models, successive approximations method, system of stochastic Volterra integral equations MSC CLASSIFICATION 60H10; 60H20; 65C30Math Meth Appl Sci. 2020;43:5212-5233. wileyonlinelibrary.com/journal/mma

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“…Both mathematicians and physicists have devoted considerable effort to find robust and stable analytical and numerical methods for solving stochastic differential equations, Adomian method [2], implicit Taylor methods [3,4] and recently the operational matrices ofintegration for orthogonal polynomials Legendre wavelets, Chebyshev polynomials, etc. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Several analytical and numerical methods have been proposed for solving various types of stochastic problems with the classical Brownian motion [10,12,14,[21][22][23].…”

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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Euler polynomial solutions of nonlinear stochastic Itô–Volterra integral equations

Cited by 55 publications

References 14 publications

An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Новый Численный Метод Решения Нелинейных Стохастических Интегральных Уравнений

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