Abstract
A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.
Highlights
Many models in physics, chemistry, and engineering reveal stochastic effects and are introduced as stochastic partial differential equations (SPDEs) [1,2]
We assess the applicability of our proposed approach to solve some stochastic heat equations of fractional order
From the definition of Brownian motion B(t) on (Ω B, FB, PB ), we know that B (0) = 0 with the probability 1
Summary
Chemistry, and engineering reveal stochastic effects and are introduced as stochastic partial differential equations (SPDEs) [1,2]. Scientists proposed models for numerous phenomena in engineering, fluid mechanics, physics [7,8,9,10,11], finance [12,13], geomagnetic [14] and hydrology [15] based on fractional differential and integral equations. Stochastic functional equations have arisen in many situations and numerous problems in different fields of science are modeled as fractional. Ralchenko and Shevchenko [21] surveyed the existence and uniqueness of mild solution for a special type of stochastic heat equations of fractional order. Proved the existence and uniqueness of solution for some delay stochastic differential equations of fractional order.
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