TY - JOUR

T1 - Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms

AU - Liaskos, Konstantinos B.

AU - Pantelous, Athanasios A.

AU - Kougioumtzoglou, Ioannis A.

AU - Meimaris, Antonios T.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - Analytic solutions in implicit-form are derived for a linear stochastic partial differential equation (SPDE) with fractional derivative terms, which can model the dynamics of a stochastically excited Euler–Bernoulli beam resting on a viscoelastic foundation. Specifically, the original initial–boundary value problem of the SPDE is reduced to an initial value problem of a second-order stochastic differential equation in an appropriate Hilbert space. Next, addressing the abstract Cauchy problem, employing cosine and sine families of operators, and representing the fractional derivative term in a suitable form, a variation of parameters treatment yields the solution in implicit-form. The limiting purely viscous and purely elastic modeling cases are also studied within the same framework. The herein proposed technique and derived implicit-form solutions can be construed as an extension of available results in the literature to account for fractional derivative terms. This generalization is of significant importance given the vast utilization of fractional calculus modeling in engineering mechanics, and in viscoelastic material behavior in particular. In this regard, the herein proposed analytical treatment also supplements existing more numerically oriented solution schemes available in the engineering mechanics literature.

AB - Analytic solutions in implicit-form are derived for a linear stochastic partial differential equation (SPDE) with fractional derivative terms, which can model the dynamics of a stochastically excited Euler–Bernoulli beam resting on a viscoelastic foundation. Specifically, the original initial–boundary value problem of the SPDE is reduced to an initial value problem of a second-order stochastic differential equation in an appropriate Hilbert space. Next, addressing the abstract Cauchy problem, employing cosine and sine families of operators, and representing the fractional derivative term in a suitable form, a variation of parameters treatment yields the solution in implicit-form. The limiting purely viscous and purely elastic modeling cases are also studied within the same framework. The herein proposed technique and derived implicit-form solutions can be construed as an extension of available results in the literature to account for fractional derivative terms. This generalization is of significant importance given the vast utilization of fractional calculus modeling in engineering mechanics, and in viscoelastic material behavior in particular. In this regard, the herein proposed analytical treatment also supplements existing more numerically oriented solution schemes available in the engineering mechanics literature.

KW - Euler–Bernoulli beam

KW - Fractional derivative

KW - Hilbert space

KW - Implicit analytic solution

KW - Stochastic partial differential equation

UR - http://www.scopus.com/inward/record.url?scp=85054816138&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2018.09.001

DO - 10.1016/j.sysconle.2018.09.001

M3 - Article

AN - SCOPUS:85054816138

VL - 121

SP - 38

EP - 49

JO - Systems and Control Letters

JF - Systems and Control Letters

SN - 0167-6911

ER -