A. Chakrabarti

Numerical Solution of a Singular Integro-Differential Equation

Numerical solution is obtained for the singular integro-differential equation with an antisymmetric forcing function f(x) (i.e. f(- x) = - f(x)), with end conditions \phi (- 1) = \phi (1) = 0, by three different methods, the two first of which presented here, produce the solution as accurate as the one obtained by Frankel (see [7]), recently. The convergence of the first method discussed in section 2, is also analysed.

Ring currents in condensed ring systems

We have studied magnetism and aromaticity of polycyclic ring systems by analyzing ring currents for different circulations in these molecules. The technique employed for calculating ring currents uses correction vectors which implicitly includes all the eigenstates of the Hamiltonian in the space of the chosen configurations. We have employed the Pariser-Parr-Pople Hamiltonian and have carried out full configuration interaction CI calculations for small systems and approximate CI calculations for large systems. The systems studied include polyacenes, nonaromatic ring systems including the

The Wiener-Hopf solution of a class of mixed boundary value problems arising in surface water wave phenomena

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Scattering of surface water waves involving a vertical barrier with a gap

fragments pyracylene, fluoranthene, and corannulene, and heteroatomic systems with upto two six-membered rings. We find that in polyacenes, the aromaticity of the extreme phenyl rings reduces with increasing number of phenyl rings in the system, and it saturates at approx 2/3 the benzene value. In systems containing nonaromatic rings, we find paramagnetic or diamagnetic behavior for different circulations depending upon the number of atoms in the chosen ring cycle, in agreement with the 4n + 2 rule. In corannulene, the largest fragment of

Reflection of water waves by a nearly vertical porous wall

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Water wave diffraction by a surface strip

we have studied, the five-membered ring is weakly diamagnetic while the six-membered ring is more diamagnetic, although much less than in isolated benzene. The ring structures with heteroatoms studied are pyridine, pyrimidine, and its isomers, s-triazine, quinoline and its isomer, and quinazoline and its isomers. All these have similar ring currents as in their purely carbon counterparts, although ions of these molecules show interesting behavior.

The effect of surface tension in porous wave maker problems

Abstract Two mixed boundary value problems associated with two-dimensional Laplace equation, arising in the study of scattering of surface waves in deep water (or interface waves in two superposed fluids) in the linearised set up, by discontinuities in the surface (or interface) boundary conditions, are handled for solution by the aid of the Weiner-Hopf technique applied to a slightly more general differential equation to be solved under general boundary conditions and passing on to the limit in a manner so as to finally give rise to the solutions of the original problems. The first problem involves one discontinuity while the second problem involves two discontinuities. The reflection coefficient is obtained in closed form for the first problem and approximately for the second. The behaviour of the reflection coefficient for both the problems involving deep water against the incident wave number is depicted in a number of figures. It is observed that while the reflection coefficient for the first problem steadily increases with the wave number, that for the second problem exhibits oscillatory behaviour and vanishes at some discrete values of the wave number. Thus, there exist incident wave numbers for which total transmission takes place for the second problem.

Derivation of the solution of a simple hypersingular integral equation

A mixed boundary-value problem associated with scattering of surface water waves by a vertical barrier with a gap of an arbitrary length is solved completely by the aid of the solution of a special logarithmic singular integral equation in the domain (a,b), which has bounded behaviour at both the end points a(>0) and b. The reflection coefficient is obtained analytically and its numerical values are presented graphically, for different values of the ratio of the width of the gap to the position of the gap. The present method of solution replaces the existing methods, which are either more elaborate or approximate in nature.

Properties of the low-lying electronic states of phenanthrene: Exact PPP results

The problem of reflection of water waves by a nearly vertical porous wall has been investigated. A perturbational analysis has been applied for the first order correction to be employed to the corresponding vertical wall problem. The Greens function technique has been used to obtain the solution of the boundary value problem at hand, after utilising a mixed Fourier transform together with an idea involving the regularity of the transformed function along the real axis. The cases of fluid of finite as well as infinite depth have been taken into consideration. Particular shapes of the wall have been considered and numerical results are also discussed.

A unified approach to problems of scattering of surface water waves by vertical barriers

The two-dimensional problem of wave diffraction by a strip of arbitrary width is investigated here in the context of linearized theory of water waves by reducing it to a pair of Carleman-type singular integral equations. These integral equations have been solved earlier by an iterative process which is valid only for a sufficiently wide strip. A new method is described here by which solutions of these integral equations are determined by solving a set of four Fredholm integral equations of the second kind, and the process is valid for a strip of arbitrary width. Numerical solutions of these Fredholm integral equations are utilized to obtain fairly accurate numerical estimates for the reflection and transmission coefficients. Previous numerical results for a wide strip are recovered from the present analysis. Additional results for the reflection coefficient are presented graphically for moderate values of the strip width which exhibit a less oscillatory nature of the curve than the case of a wide strip.