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

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.

Automated summary

This paper proposes a computational method to determine the approximate solution for the deflection of Euler-Bernoulli beams under stochastic dynamic loading. The method utilizes operational matrices and Legendre polynomials to approximate the functions and break down the problem into a set of algebraic equations. Through numerical examples, the practicality and effectiveness of the method are confirmed.