| This thesis emphasizes the application of the theory of functional linear partial delay differential equations (LPDDE). In particular, we are interested in studying the theoretical concepts of delay partial differential equations in contrast to basic partial differential equations that arise in heat, wave, telegraph and viscoelastic equations. We first establish the concepts and theory of functional differential equations with constant delay, which is then extended to continuous delay partial differential equations.; Functional differential equations (FDE) incorporating a delay are useful in modelling various complex phenomena, in which a change in some quantity is dependent upon how that quantity is affected by a previously known mechanisms. Such mathematical models are often used in physio-chemical and biological systems, in which the rate of change of the system depends upon its past history.; The present work is an attempt to extend this theory to LPDDE for the classes of test equations such as heat, wave, and telegraph equations. Techniques such as Fourier transform and separation of variables are used to find a solution to these equations. Analytical solution for the above equations in the case of continuous delay can have discontinuities. Therefore, an approximate solution is sought by solving simpler equations with piecewise constant argument (EPCA), showing that both solutions converge uniformly at any time.; To validate the theoretical concept of EPCA, an example of the heat equation with and without delay is presented in chapter 2. Also in chapter 3, an approximate solution of the heat equation with multiple arguments is solved by mixed continuous-constant delay equation, known as neutral type. The telegraph equation is studied for constant and continuous delay in chapter 4, and the influence of time-delay on the stability of the solution is also investigated.; In conclusion, we have shown that the solution to linear partial delay differential equations (LPDDE) can be approximated by solving their respective piecewise constant delay. In this case the solution is represented by linear difference equations, which can be tracked successively over the grid points. Therefore, the results in this dissertation support the idea of extending FDE to LPDDE. Many complex applications (control theory, biomedical problems, etc) in nature can be described by these kinds of equations where many researchers are concerned about knowing the exact or approximate solution over the intervals of interest. |
