A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

Abstract

In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in L2$$ {L}^2 $$ norm is derived and obtained as ‖‖eNfalse(t,xfalse)2≤6Kfalse(16N4false)2false(1−6Sfalse)$$ {\left\Vert {e}_N\left(t,x\right)\right\Vert}^2\le \frac{6K}{{\left(16{N}^4\right)}^2\left(1-6S\right)} $$. The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.

The Construction of Online Interactive Teaching Mode of College English in the Background of Internet

“…The disseminator, or the disseminated in the information dissemination, has been in the process of information interaction and completes the knowledge transfer in the interaction. Therefore, information dissemination in online learning interactive networks can be better portrayed with the SI model compared to other models [25].…”

Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

“…So, interested researchers with numerical methods of stochastic problems (see [6,24,33]). Let (Ω, G, µ) be a probability space where Ω is a sample space, G is a σ-algebra of subsets of Ω and µ is the probability measure (see [7,32,36]).…”

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

“…Since stochastic partial differential equations with variable coefficients can better explain physical phenomena in nature, many researchers have considered incorporating stochastic perturbation terms into these equations. Uma et al [28] proposed an approximate solution of the stochastic Fisher-KPP equation with variable coefficients using two-dimensional shifted Legendre polynomials. Chen and Xie [29], by white noise analysis, Hermite transform, and extended F-expansion method, obtained periodically modulated and periodic wave solutions for the Wick-type stochastic nonlinear Schrödinger equation with variable coefficients, respectively, and also provided soliton-like wave solutions under certain conditions.…”