Existence of solutions to higher order partial differential equations

In this thesis, we study the existence and uniqueness of the initial and boundary value problem for a class of higher order semi-linear hyperbolic and parabolic partial differential equations. Based on the priori estimates of the solution, the existence of the weak solution in the form of Fourier series under suitable conditions is proved. For this purpose Picard's successive approximation method is used. Also, the uniqueness of the weak solution is proved. In the introductory chapter of the thesis, it is underscored that partial differential equations are frequently encountered and play a fundamental role in various fields of science, particularly in physics, engineering, and related disciplines. Thereafter, a concise overview of the historical development and existing literature pertaining to the subject matter is presented. In the second chapter, titled Previous Studies, a summary of the existing literature concerning the subject and equations under consideration is provided, with a particular focus on earlier works related to the equation studied in this thesis. In the third chapter, references and sources are provided for several fundamental concepts, definitions, inequalities, function spaces, lemmas, and theorems that will be utilized in the subsequent chapters. The fourth chapter, entitled Research Findings, constitutes the original contribution of this thesis. It is structured in two parts. The first part addresses the initial-boundary value problem for a higher-order, semi-linear hyperbolic type equation with a damping term, considered within a bounded domain. In this context, the existence and uniqueness of solutions to the proposed problem are rigorously analyzed. The second part is devoted to the initial-boundary value problem for a higher-order, semi-linear pseudo-parabolic type equation, also defined on a bounded domain. A similar analysis is carried out to establish the existence and uniqueness of solutions for this class of problems. For both cases, the existence of a weak solution expressed in the form of a Fourier series is demonstrated under suitable conditions, employing the method of Picard successive approximations. In the fifth chapter, the conclusions drawn from the thesis and the recommendations are presented.