Tchebychev Polynomial Approximations for mthm^{th} Order Boundary Value Problems

Higher order boundary value problems (BVPs) play an important role modeling various scientific and engineering problems. In this article we develop an efficient numerical scheme for linear $m^{th}$ order BVPs. First we convert the higher order BVP to a first order BVP. Then we use Tchebychev orthogonal polynomials to approximate the solution of the BVP as a weighted sum of polynomials. We collocate at Tchebychev clustered grid points to generate a system of equations to approximate the weights for the polynomials. The excellency of the numerical scheme is illustrated through some examples.Comment: 21 pages, 10 figure

Spatiotemporal Orthogonal Polynomial Approximation for Partial Differential Equations

Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and Techbychef orthogonal polynomials for spatiotemporal approximation of PDEs as a weighted sum of polynomials. We use collocation at some clustered grid points to generate a system of equations to approximate the weights for the polynomials. We finish the study by demonstrating approximate solutions of some PDEs in one space dimension.Comment: 9 pages, 9 figure

Radiatively and thermally driven self-consistent bipolar outflows from accretion discs around compact objects

We investigate the role of radiative driving of shock ejected bipolar outflows from advective accretion discs in a self consistent manner. Radiations from the inner disc affects the subsonic part of the jet while those from the pre-shock disc affects the supersonic part, and there by constitutes a multi stage acceleration process. We show that the radiation from the inner disc not only accelerate but also increase the mass outflow rate, while the radiation from the pre-shock disc only increases the kinetic energy of the flow. With proper proportions of these two radiations, very high terminal speed is possible. We also estimated the post-shock luminosity from the pre-shock radiations, and showed that with the increase of viscosity parameter the disc becomes more luminous, and the resulting jet simultaneously becomes faster. This mimics the production of steady mildly relativistic but stronger jets as micro-quasars moves from low hard to intermediate hard spectral states.Comment: 18 pages, 11 figures. Accepted for publication in MNRAS on 31 October 201

Fast and Efficient Numerical Methods for an Extended Black-Scholes Model

An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and accuracy of the numerical scheme considered. In this article we consider a PIDE that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. In the first part of the article, we study several iterative techniques (preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis have been used to design various preconditioners to improve the convergence criteria of iterative solvers. We implement a multigrid (MG) iterative method. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.Comment: 29 pages; 10 figure

Toward an Alternative Intrinsic Probe for Spectroscopic Characterization of a Protein

The intrinsic fluorescent amino acid tryptophan is the unanimous choice for the spectroscopic investigation of proteins. However, several complicacies in the interpretation of tryptophan fluorescence in a protein are inevitable and an alternative intrinsic protein probe is a longstanding demand. In this contribution, we report an electron-transfer reaction in a human transporter protein (HSA) cavity which causes the tryptophan residue (Trp214) to undergo chemical modification to form one of its metabolites kynurenine (Kyn214). Structural integrity upon modification of the native protein is confirmed by dynamic light scattering (DLS) as well as near and far circular dichroism (CD) spectroscopy. Femtosecond-resolved fluorescence transients of the modified protein describe the dynamics of solvent molecules in the protein cavity in both the native and denatured states. In order to establish general use of the probe, we have studied the dipolar interaction of Kyn214 with a surface-bound ligand (crystal violet, CV) of the protein. By using the sensitivity of FRET, we have determined the distance between Kyn214 (donor) and CV (acceptor). Our study is an attempt to explore an alternative intrinsic fluorescence probe for the spectroscopic investigation of a protein. In order to establish the efficacy of the modification technique we have converted the tryptophan residues of other proteins (bovine serum albumin, chymotrypsin and subtilisin Carlsberg) to kynurenine and confirmed their structural integrity. We have also shown that catalytic activity of the enzymes remains intact upon the modification