Math 425: An Introduction to Partial Differential Equations

Basic Concept: General definition of partial differential equation, Classification, Sources, Solution. Fist order partial differential equation: Linear and quasilinear equations, Lagrange ‘s method for quasilinear equations, Cauchy’s problem. Linear second-order equations : Classification into Elliptic, Hyperbolic and Parabolic types. Solutions by factorization .Cauchy problem. This concept leads to find the general solutions for some PDEs with variable coefficients. Elliptic Equation: Laplace’s equation, properties of harmonic functions, Maximum and Minimum principle harmonic functions, Dirichlet problem, Neuman problem. Separation of  variable method for solving Laplace’s equation on some domains with boundary conditions and using Laplace transform method. Heat equations: Homogeneous and non-homogeneous boundary conditions. Initial and boundary value problems in 1, 2 and 3 dimensions, using eigenfunctions expansion and separation of variable method. Heat transfer in an infinite bar expansion and separation of variable method  Heat transfer in an infinite bar. Representation of solution by using Fourier transform and Laplace transform method on some domains. Wave equation: Mathematical model for a vibrating string, solutions by separation of variable method  and Laplace transform method. D’Alembert’s representation. Problem in two space dimensions. Representation of solution by using Fourier transform and Laplace transform method on some domains.

Text Book: M. A. Al-Gwaiz, Introduction to Partial Differential Equations, Star Printing Press, Riyadh, 2012.

                  R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems, McGraw-Hill, 1968.