Research Works
http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1122
2020-10-24T15:15:49ZNumerical convergence of a one step approximation of an integro-differential equation
http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1187
Numerical convergence of a one step approximation of an integro-differential equation
Bhowmik, Samir Kumar
We consider a linear partial integro-differential equation that arises in modeling various physical and biological processes. We study the problem in a spatial periodic domain. We analyze numerical stability and numerical convergence of a one step approximation of the problem with smooth and non-smooth initial functions.
2012-08-17T00:00:00ZFinite to Infinite Steady State Solutions, Bifurcations of an Integro-Differential Equation
http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1186
Finite to Infinite Steady State Solutions, Bifurcations of an Integro-Differential Equation
Bhowmik, Samir K.; Duncan, Dugald B.; Grinfeld, Michael; Lord, Gabriel J.
We consider a bistable integral equation which governs the sta- tionary solutions of a convolution model of solid–solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi– discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.
2011-07-01T00:00:00ZNumerical approximation of a convolution model of ˙ θ-neuron networks
http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1185
Numerical approximation of a convolution model of ˙ θ-neuron networks
Bhowmik, Samir Kumar
In this article, we consider a nonlinear integro-differential equation that arises in a ˙ θ-neural networks modeling. We analyze boundedness and invertibility of the model operator, construct approximate solutions using piecewise polynomials in space, and estimate the theoretical convergence rate of such spatial approximations. We present some numerical experimental results to demonstrate the scheme.
2010-12-22T00:00:00ZPreconditioners based on windowed Fourier frames applied to elliptic partial differential equations
http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1184
Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations
Bhowmik, Samir K.; Stolk, Christiaan C.
We investigate the application of windowed Fourier frames to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given by the symbol of the PDO evaluated at the wave number and central position of the windowed plane wave. This can be exploited in a preconditioning method for use in iterative inversion. For domains with periodic boundary conditions we find that the condition number with the preconditioning becomes bounded and the iteration converges well. For problems with a Dirichlet boundary condition, some large and small singular values remain. However the iterative inversion still appears to converge well.
2011-02-17T00:00:00Z