Partial Differential Equations

Specific Objectives of course:

Partial Differential Equations (PDEs) are at the heart of applied mathematics and many other scientific disciplines. The course aims at developing understanding about fundamental concepts of PDEs theory, identification and classification of their different types, how they arise in applications, and analytical methods for solving them. Special emphasis would be on wave, heat and Laplace equations.

Course Outline:

First order PDEs:

Introduction, formation of PDEs, solutions of PDEs of first order, The Cauchy’s problem for quasilinear first order PDEs, First order nonlinear equations, Special types of first order equations.

Second order PDEs:

Basic concepts and definitions, Mathematical problems, Linear operators, Superposition, Mathematical models: The classical  equations,  the  vibrating  string,  the  vibrating  membrane, conduction of  heat  solids,  canonical forms  and  variable,  PDEs  of second order in two independent variables with constant and variable coefficients,  Cauchy’s  problem  for  second  order   PDEs  in   two independent variables

Methods of separation of variables:

Solutions of elliptic, parabolic and hyperbolic PDEs in Cartesian and cylindrical coordinates

Laplace transform:

Introduction and properties of Laplace transform, transforms of elementary functions, periodic functions, error function and Dirac delta function, inverse Laplace transform, convolution theorem, solution of PDEs by Laplace transform, Diffusion and wave equations

Fourier transforms:

Fourier integral representation, Fourier sine and cosine representation, Fourier transform pair, transform of elementary functions and Dirac delta function, finite Fourier transforms, solutions of heat, wave and Laplace equations by Fourier transforms.