The course is an introduction to the theory of partial differential equations and also provides methods of solution of boundary and initial value problems of Mathematical Physics. Students are taught how to identify different types of partial differential equations and to obtain the equations that are satisfied by given functions. Analytical methods of solutions of first order equations having certain standard forms are given. The same is done for second-order equations. Second-order equations of particular importance are examined in great detail. Students are shown how some of these equations arise in the study of the propagation of waves, heat transfer and other phenomena. Various methods of solution of these equations are considered, taking into account any initial and boundary value problems relevant to a given situation in the real world. Throughout the course, the emphasis is on methods of solution.
|Idea of a partial differential equation (PDE) and boundary and initial conditions. Classification of PDE's. Well posed problems and uniqueness of solutions.||1|
|General methods for first order PDE's. Quasi-linear equations and characteristics. Charpit's and Jacobi's method.||11|
|Classification of second order PDE's and relation to characteristics. Appropriate boundary and initial conditions. Operator methods for reducible PDE's. Riemann's method for hyperbolic equations as an introduction to Green's function. Monge's Method.||12|
|Separation of variables methods, and related eigenfunction expansions, with applications.||8|
|Integral transform methods, particularly the Fourier and Laplace transforms.||8|
|Green's function methods.||8|
|(Separation of variables methods and integral transform methods will be with particular reference to the wave equation, the diffusion equation and the Laplace and Helmholtz equations.)|