Author

# Aloknath Chakrabarti

Other affiliations: University of Surrey

Bio: Aloknath Chakrabarti is an academic researcher from Indian Institute of Science. The author has contributed to research in topics: Boundary value problem & Integral equation. The author has an hindex of 16, co-authored 82 publications receiving 872 citations. Previous affiliations of Aloknath Chakrabarti include University of Surrey.

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##### Papers

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21 Jan 2000

TL;DR: In this article, the basic equations of the wave-scattering problem are discussed. But they do not specify a solution to the boundary value problem, which is a special case of boundary value problems.

Abstract: Contents: Introduction The Basic Equations Some Important Mathematical Concepts and Results Explicit Solutions to Some Barrier Problems Vertical Wall with a Narrow Gap Approximate Solution Oblique Wave Scattering by Barriers Nearly Vertical Barriers and Special Boundary Value Problems Thin Vertical Barriers in Finite Depth Water Thick Rectangular Barriers in Finite Depth Water Interface Wave Scattering by Barrier Incoming Water Waves Against a Vertical Cliff Second-Order Wave Scattering Appendices Bibliography Index.

56 citations

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TL;DR: In this paper, the mixed boundary value problem arising in the study of scattering of two-dimensional time-harmonic surface-water waves by a discontinuity on the surface boundary conditions, separating the clean surface and an ice-covered surface, is solved completely in the case of an infinite depth of water.

Abstract: The mixed boundary–value problem arising in the study of scattering of two–dimensional time–harmonic surface–water waves by a discontinuity on the surface boundary conditions, separating the clean surface and an ice–covered surface, is solved completely in the case of an infinite depth of water. The main problem is reduced to that of solving a singular integral equation, of the Carleman type, over a semi–finite range and the explicit solution of the original problem is determined. Neat and computable expressions are derived for the two most important quantities, known as the reflection and transmission coefficients, occurring in such scattering problems and tables of numerical values of these quantities are presented for specific choices of a parameter modelling the ice cover. The absolute values of the reflection and transmission coefficients are presented graphically. The present method of solution of the boundary–value problem produces simple expressions for the principal unknowns of the problem at hand and thus provides an easily understandable alternative to the rather complicated Wiener–Hopf method used previously.

55 citations

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TL;DR: The methodology of the present work is expected to be useful for solving singular integral equations of the first kind, involving partly singular and partly regular kernels, as well as equation of the second kind involving similar kernels, with appropriate adjustments regarding the endpoint behaviours of the unknown function.

Abstract: An approximate method is developed for solving singular integral equations of the first kind, over a finite interval. The singularity is assumed to be of the Cauchy type, and the four basically different cases of singular integral equations of practical occurrence are dealt with simultaneously. The presently obtained results are found to be in complete agreement with the known analytical solutions of simple equations. The methodology of the present work is expected to be useful for solving singular integral equations of the first kind, involving partly singular and partly regular kernels, as well as equations of the second kind involving similar kernels, with appropriate adjustments regarding the endpoint behaviours of the unknown function.

50 citations

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02 May 2011

TL;DR: In this paper, Cauchy principal value integrals are used to solve singular integral equations involving closed contours in Riemann-Hilbert problems, and the Galerkin method is used for singular integral equation solutions.

Abstract: Preface Introduction Basic Definitions Occurrence of singular integral equations Some Elementary Methods of Solution of Singular Integral Equations Abel integral equation and its generalization Integral equations with logarithmic type of singularities Integral equations with Cauchy type kernels Application to boundary value problems in elasticity and fluid mechanics Riemann-Hilbert Problems and Their Uses in Singular Integral Equations Cauchy principal value integrals Some basic results in complex variable theory Solution of singular integral equations involving closed contours Riemann Hilbert problems Generalised Abel integral equations Singular integral equations with logarithmic kernels Singular integral equation with logarithmic kernel in disjoint intervals Special Methods of Solution of Singular Integral Equations Integral equations with logarithmically singular kernels Integral equations with Cauchy type kernels Use of Poincare'-Bertrand formula Solution of singular integral equation involving two intervals Hypersingular Integral Equations Definitions Occurrence of hypersingular integral equations Solution of simple hypersingular integral equation Solution of hypersingular integral equation of the second kind Singular Integro-differential Equations A class of singular integro-differential equations A special type of singular integro-differential equation Numerical solution of a special singular integro-differential equation Approximate method based on polynomial approximation Approximate method based on Bernstein polynomial basis Galerkin Method and its Application Galerkin method Use of single-term Galerkin approximation Galerkin method for singular integral equations Numerical Methods The general numerical procedure for Cauchy singular integral equation A special numerical technique to solve singular integrals equations of first kind with Cauchy kernel Numerical solution of hypersingular integral equation using simple polynomial expansion Numerical solution of simple hypersingular integral equation using Bernstein polynomials as basis Numerical solution of some classes of logarithmically singular integral equations using Bernstein polynomials Numerical solution of an integral equation of some special type Numerical solution of a system of generalized Abel integral equations Some Special Types of Coupled Singular Integral Equations of Carleman Type and their Solutions The Carleman singular integral equation Solution of the coupled integral equations for large l Solution of the coupled integral equations for any l Bibliography Subject index

49 citations

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TL;DR: In this paper, the problem of scattering of obliquely incident surface water waves by small undulation on a sea-bed is considered for solution by assuming that the bed is composed of porous material of a specific type.

Abstract: The problem of scattering of obliquely incident surface water waves by small undulation on a sea-bed is considered for solution by assuming that the bed is composed of porous material of a specific type. The reflection and transmission coefficients are determined approximately after assuming a perturbation analysis in conjunction with the Fourier transform technique, in terms of the first order of smallness of an undulation parameter $\varepsilon$ introduced in the description of the sea-bed. The special case of a patch of sinusoidal ripples on the bed is handled in detail and numerical results are presented graphically.

49 citations

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1,969 citations

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TL;DR: In this paper, the authors pointed out that the distinction between "finite" and "infinite" is one which does not require definition, and that the authors' view is not the only accepted view.

Abstract: THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

740 citations

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01 Jan 1966

TL;DR: Boundary value problems in physics and engineering were studied in this article, where Chorlton et al. considered boundary value problems with respect to physics, engineering, and computer vision.

Abstract: Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

733 citations

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555 citations

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TL;DR: It is shown that far from being a curiosity, the trapezoidal rule is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

Abstract: It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

481 citations