MA4332 Partial Differential Equations

This course provides an introduction to the theory of partial differential equations. It includes the following topics: classification of second order equations; initial value and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations; existence and uniqueness of linear elliptic and parabolic PDEs; nonlinear parabolic and elliptic PDEs; Hamilton-Jacobi equations; systems of conservation laws and nonlinear wave equations; transform methods and Green's functions.

Prerequisite

MA3132, and MA3232 strongly recommended

Lecture Hours

4

Lab Hours

0

Course Learning Outcomes

After successfully completing this course, the student will be able to:

• Classify second order partial differential equation.
• Understand the concept of linearity and well-posedness.
• Solve the heat equation and know the following concepts: initial-boundary value problems; solution by separation of variables; Sturm-Liouville problem; well-posedness of IBVP of heat equation; maximum-minimum principle; energy inequality; Fourier integral transform; nonhomogeneous heat equation (Duhamel’s principle); the Laplace transform method; regularity and similarity method; applications to fluid dynamics (Stoke’s equation for slow viscous incompressible flow; Brownian motion).
• Solve the Laplace equation and know the following concepts: Green’s identities; representation theorem; mean value theorem; maximum principle; well-posedness of the Dirichlet problem; existence theory (P-condition); solution of the Dirichlet problem using Green’s function; boundary condition at infinity; solution of exterior Dirichlet problem.
• Solve the Poisson equation and know the following concepts: solution using Green’s function; deviation of the 2-D Poisson equation using the variational approach; Poisson-Boltzmann equation.
• Solve the wave equation and know the following concepts: solutions of the 1-D wave equation; plane waves of the n-dimensional wave equation; spherical waves of the 3-D wave equation; cylindrical waves of the 2-D wave equation; energy methods; domain of dependence; range of influence; solution of IVP of the 1-D equation; solution of IVP of the 3-D wave equation (Kirchhoff’s formula; Huygens’ principle); solution of IVP of the 2-D wave equation (method of descent); nonhomogeneous 1-D wave equation (Duhamel’s principle); derivation of the 1-D wave equation (the Calculus of variations); applications: vibrating string and hanging chain, propagation of light and sound.
• Solve scalar conservation laws and understand the following concepts: characteristics; weak solutions; Rankine-Hugoniot jump condition; shock wave; rarefaction wave; entropy condition; solution of Riemann problem; self-similar solutions.
• Solve Schrodinger’s equation.
• Find series solutions and know the Cauchy-Kovalevsky theorem.