Time fractional diffusion equation for shipping water events simulation

Abstract

Shipping water (SW) events occur when waves, usually during extreme weather conditions, overtop the deck of vessels or structures. Since SW events take place independently but close enough in time to interact with one another, their simulation requires strong nonlinear equations to achieve an appropriate precision. To avoid using equations with high complexity and computational cost, we explore the use of the time fractional diffusion equation (TFDE) for modeling SW events. The idea behind this proposal is to keep the simplicity of the widely used standard diffusion equation, but employing a generalized derivative of order α∈[1,2]. This order can be used to describe an intermediate behavior between diffusion and wave propagation. For the time derivative, we adopted the Caputo fractional derivative. To demonstrate that the TFDE is a suitable model to describe SW events, we present its results compared against data sets of water free surface elevation recorded during SW experiments carried out in a wave flume. The best agreement was found for a fractional order derivative between 1.69 and 1.75.