Area of Specialization:
Mathematical Modelling on Water Wave Phenomena
Integral Equations
Partial Differential Equations
Numerical Method and Simulation
Research Interests:
Very Large Floating Structures
Water wave scattering by plates, barriers, cylindrical structures and bottom topography
Hydrodynamic loading
Perturbation techniques
Fluid flow through porous medium
Fluid Flow in a Channel
Linear and Nonlinear Waves
Few Lines About my Research:
The objective in my doctoral thesis is to investigate the scattering of a train of small amplitude harmonic surface water waves by small undulation of a sea-bed for both normal and oblique incidence.
In this study of scattering, mixed boundary value problems are set up for the determination of a velocity potential where the governing partial differential equation happens to be Laplace's equation in two dimensions for normal incidence and in three dimensions for oblique incidence within the fluid with a mixed boundary condition on the free surface and a condition on the bottom boundary. As the fluid domain extends to infinity, a far-field condition or an infinity condition arises to ensure uniqueness of the problem.
Applying a perturbation analysis first, then using the mathematical tools such as Green's function technique and application of Green's integral theorem, Fourier transform technique and application of residue theorem, Finite cosine transform technique and eigenfunction expansion technique, the velocity potential, the reflection and transmission coefficients are evaluated up to the first order of epsilon in terms of integrals involving the shape function c(x) representing the bottom undulation.
Different examples for the shape functions are considered to evaluate the integrals explicitly and the numerical values for these coefficients are obtained (by using MATLAB).
In my post doctoral work, a class of mixed boundary value problems in the theory of scattering that involves floating elastic structures on the surface of water in studies on surface water wave problems, have been solved completely, by utilizing the eigenfunction expansion method leading to the mathematical problems of solving over-determined systems of linear algebraic equations. Such over-determined system of equations are best solved by the method of least-squares and numerical results for useful practical quantities such as the reflection and transmission coefficients are obtained.
Also, I have been working on problems involving integral equations and for its approximate numerical solutions, which arise in various areas of mathematical physics. Further, the scattering of water waves by floating structures in presence of small undulation on the sea-bed will be the most attractive problem to solve for its analytical and numerical solution.