Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra–Fredholm integral equations

Mirzaee, Farshid; Solhi, Erfan; Samadyar, Nasrin

doi:10.1016/j.apnum.2020.11.013

Cited by 28 publications

(3 citation statements)

References 31 publications

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“…Our goal is to obtain the numerical solutions of Equation (1.1) using the method based on clique polynomials. The clique polynomial related to the maximal clique problem was first introduced by Hoede and Li which is defined as

normalΩ ((),, Q, τ) = \sum_{ι = 0}^{ȷ} m_{e} ((), Q) τ^{e}

where

{frakturm}_{0} (scriptQ) = 1

and

{frakturm}_{frakture} (scriptQ)

is the number of

e

‐cliques of

ℰ

33,34 . The maximal clique problem, which is a hybrid optimization problem, is used in various fields such as economics, information retrieval and biomedical engineering, graph coloring, classification theory, set packing, and optimal winner determination 35 .…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Eidinejad,

Saadati,

Vahidi

et al. 2023

Int J Numerical Modelling

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We study the two‐dimensional stochastic time‐fractional Sine–Gordon equation (2D ST‐FS‐G) and provide a solution for it. We use the new clique polynomial method to obtain a numerical solution to the 2D TFS‐G equation. In this technique, the clique polynomial is considered as a basic function for operational matrices. By converting the 2D ST‐FS‐G equation to algebraic equations, a solution is obtained for the desired equation, which clearly shows that this approach is suitable and accurate for dealing with the equation. We further present the error boundary for the obtained approximation of the desired three‐variable function based on the clique polynomial. Finally, we compare some numerical results obtained with the exact ones by a few practical examples.

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normalΩ ((),, Q, τ) = \sum_{ι = 0}^{ȷ} m_{e} ((), Q) τ^{e}

where

{frakturm}_{0} (scriptQ) = 1

and

{frakturm}_{frakture} (scriptQ)

is the number of

e

‐cliques of

ℰ

Section: Introductionmentioning

confidence: 99%

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Eidinejad,

Saadati,

Vahidi

et al. 2023

Int J Numerical Modelling

View full text Add to dashboard Cite

show abstract

“…In recent years, some important progresses for the numerical algorithm of the SVIE (1.1) with 𝜎 = 0 has been proposed, for example, Galerkin methods, 25 block pulse approximation, [26][27][28] spectral collocation method, [29][30][31] operational matrix method, [32][33][34] improved rectangular method, [35][36][37] B-spline collocation method, [38][39][40] Milstein method, 41,42 meshless discrete collocation method, 43 and Euler-Maruyama (EM) method. [44][45][46][47] In this paper, we are interested in numerically solving the nonlinear SVIE (1.1) for which 𝜎 > 0 by using EM method and devoting to two main goals.…”

Section: Introductionmentioning

confidence: 99%

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

Wang

Chen

Deng

et al. 2022

Math Methods in App Sciences

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In this paper, the Euler-Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs), which can be derived from stochastic fractional substantial integro-differential equations (SFSIDEs). The existence and uniqueness of the solution are proved for the nonlinear SVIEs ] , respectively, is established for the nonlinear SVIEs, which can be revised as supplemented theory results of our recent work (J Comput Appl Math 356 (2019) 377-390). In particular, it also closes a gap that was left on the convergence analysis in (J Comput Appl Math 317 (2017) 447-457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

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“…Recently, many innovative fractional derivative operators beyond the singular kernel have been explored, such as the Mittag–Leffler function 23 and exponential function 24 . In particular, researchers who would like to develop and address a real‐life problem have recommended fractional operators 25–39 . A problem involving these operators can be solved quickly and reliably because they include a nonsingular kernel.…”

Section: Introductionmentioning

confidence: 99%

Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

Rashid

Kubra

Abualnaja

2021

Math Methods in App Sciences

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In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional-order (FO) heat-like equations. To show this, we first establish the Elzaki transform in the context of the Atangana-Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat-like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer-order solutions.EHPTM is considered to be an appropriate and convenient approach for solving FO time-dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis.

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Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra–Fredholm integral equations

Cited by 28 publications

References 31 publications

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

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