A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

Abstract

The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

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

“…The existence and uniqueness of solutions to stochastic differential equations have been studied by many authors see [14,16,18,25]. In [24], the author discussed a computational method to get an approximate solution of a stochastic beam equation. A simulation analysis of this problem is carried out with matlab, author constructed the stochastic partial 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]).…”