Abstract: 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.
Bio: Dr. Athanasios A. Pantelous is an Associate Professor at the Department of Econometrics and Business Statistics at Monash University, Australia. Dr. Pantelous' primary research interests focus on the general area of quantitative research and mathematical modeling under risk and uncertainty. Dr. Pantelous has published more than 140 technical papers in peer reviewed international journals including the Proceedings of the National Academy of Sciences of the United States of America, European Journal of Operational Research, Journal of Banking and Finance, IEEE Transactions of Fuzzy Systems, ASCE Journal of Engineering Mechanics, Journal of Sound and Vibration, Expert Systems with Applications, Nonlinear Dynamics, The Journal of the Franklin Institute – Engineering and Applied Mathematics, Mechanical Systems and Signal Processing, Systems & Control Letters, Insurance: Mathematics & Economics, Computers & Industrial Engineering, Quantitative Finance among others and conference proceedings (sponsored by IEEE, IFAC, ASCE, ASME etc.). He has served in the scientific and/or organizing committees of several international technical conferences and has founded and chaired the Quantitative Finance and Risk Analysis (QFRA) symposium series. He is also an invited Visiting Professor in the School of Management at Shanghai University (China) and Associate Director of Quantitative Finance and Risk Analysis in the Center of Technology and Systems Management at University of Maryland (US). He is currently an Associate Editor of the ASCE-ASME J. of Risk and Uncertainty in Engineering Systems and has served as a Guest Editor for several special issues in international journals (including, Quantitative Finance, Annals of Operations Research, International Journal of Finance & Economics). He has already secured funding of AUS$12.6 million (from EPSRC, ESRC, ERC, CSIRO Data61, London Mathematical Society, Bank of England, ACANTO RE, MEMSI among others).
