Shallow Water Wave Equations with Uncertain Parameters

Abstract

Shallow water equations are a set of hyperbolic partial differential equations, also called Saint-Venant equations, named after Adhemar Jean Claude Barre de Saint-Venant. Sometimes, these describe the flow below a pressure surface in a fluid. The water waves whose horizontal scale of the flow is much greater than the depth of the fluid are treated as shallow water waves. These waves exist near the shore when the waves approach the coastal area. Tsunamis are examples of such waves and so shallow water equations are widely used in tsunami simulations and various other physical problems such as shock waves, tidal flows, Coastal waves, etc. The shallow water equations are also appropriate to describe phenomena on a smaller scale, such as broken-wave (bore) propagation and run-up on beaches, which play a major role in sediment transport and beach evolution. The shallow water equations have applications to a wide range of phenomena other than water waves, like avalanches and atmospheric flow etc.
One may find different methods for solving one as well as two dimensional shallow water wave and KdV equations but sometimes those are not efficient and problem dependent. As such firstly recent computationally efficient methods are used in crisp environment for few of the above said problems. Next, new methods have also been developed to handle some of the above problems. Further this thesis targets to investigate shallow water wave equations with uncertain parameters. In case of water waves, the factors like basin depth, initial wave motion, boundary conditions and environmental conditions etc., affects the wave motion. Small change in these factors may change the wave motion, which leads to uncertain environment. This uncertainty may be modeled well by considering parameters involved in governing equations as interval or fuzzy numbers. Basin depth and initial conditions have been considered as uncertain in term of interval/fuzzy to handle the above said involved uncertainty. The uncertain parameters lead to interval/fuzzy differential equations which are to be solved with utmost care as interval/fuzzy computations have to be used. Further, Korteweg-de Vries (KdV) like equations are also used by various authors for modeling shallow water waves. It may be noted that the works related to KdV equations may be found in crisp environment only. To the best of our knowledge, no work has been reported till date on uncertain (interval/fuzzy) KdV equations. In this regard, fractional KdV equations are also interesting to study, where some studies are done for solving the same.
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But those methods are problem dependent and sometimes may not be efficient in particular to handle higher order KdV equations. Accordingly, efficient methods for solutions of nonlinear KdV and fractional KdV equations of different orders in crisp as well as uncertain environment are developed in this investigation.