Studies on integral equations and their applications to direct and inverse problems

Abstract

newlineThe thesis is devoted to the study of various aspects of integral equations and their applications to boundary value problems of Mathematical Physics. newlineSome important aspects of integral equations of both first and second kind arising in studies of scattering of surface water waves by vertical barriers have been examined for the existence and uniqueness of their solutions in the presence of some known real parameter, newlinecalled the wave number. newlineA class of new quadrature formulae has been developed to evaluate, numerically, integrals having weak singularities in the range of integration, either end, both ends, or at an newlineinterior point. Particular examples have been dealt with and the results are compared with newlinethe exact ones wherever possible. Applications to weakly singular integral equations of newlinethe Abel type have been dealt with. The presently developed quadrature formulae are also newlinefound to be very efficient in the circumstances when the integrands are regular functions in newlinetheir ranges of integration. newlineThe famous Dirichlet problem involving Laplace s equation in two dimensions has been newlinere-examined in the light of their solutions by the aid of integral representations of Dirac s newlineDelta functions and new formulae for solutions are derived which match exactly with the newlinecorresponding results obtained by the use of integral equations in the case of circles and newlinehalf planes. The new formula developed devoids the root of integral equations used by previous workers. The general solution formula derived here is useful for regions enclosed by newlinesmooth closed curves of all types in an approximate manner. The final results are compared with the ones obtained by previous workers in the case of an ellipse.