Investigations in Differential and Integral Equations and some Applications

Abstract

Integral equations are encountered in various elds of science and numerous applications (in elasticity, plasticity, heat and mass transfer, oscillation theory, uid dynamics, ltration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, medicine, etc.). Exact (closed-form) solutions of integral equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Lots of equations of physics, chemistry, and biology contain functions or parameters which are obtained from experiments and hence are not strictly xed. Therefore, it is expedient to choose the structure of these functions so that it would be easier to analyze and solve the equation. As a possible selection criterion, one may adopt the requirement that the model integral equation admits a solution in a closed form. Exact solutions can be used to verify the consistency and estimate errors of various numerical, asymptotic, and approximate methods. Through out this thesis we come across many subjects that handle the linear integral and nonlinear integro-di¤erential equations by using the modern mathematical methods, and some of the powerful traditional methods. This thesis will be powerful tool for those who are concerned by applied mathematics, physical science, and engineering, attempts are made so that it presents both analytical and numerical approaches in a clear and systematic fashion to make this thesis accessible to those who work in these elds. Also there will be a part that is devoted to thoroughly examine the nonlinear integral equations and its applications. Mathematical physics models, such as di¤raction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of nonlinear integral equations. Because it is not always possible to nd exact solutions to problems of physical science that are posed, much work is devoted to obtaining qualitative approximations that highlight the structure of the solution and we worked to search for new methods to become closer or to obtain the exact solution.