MA 361 - Intermediate Partial Differential Equations
This course offers a rigorous approach to classical partial differential equations. It begins with definitions, properties, and derivations of some basic equations of mathematical physics followed by the topics: solving of first order equations with the method of characteristics; classification of second order equations; the heat equation and wave equation; Fourier series and separation of variables; Green's functions and elliptic theory; examples of the first and second order nonlinear partial differential equations.
Prerequisites: MA 221, MA 227
MA361 is a first course in course in partial differential equations (PDEs) intended for mathematics majors and students from the sciences and engineering programs needing a strong introduction to PDEs. Upon completing the course, the student will be familiar with the modeling assumptions and derivations that lead to PDEs, know and recognize the major classification of PDEs, understand the qualitative differences between the classes of equations, and be competent in solving linear PDEs using classical solution methods. This course should serve as a good vehicle for students to acquire experience with an integrated computational environment that includes tools for solving differential equations, data analysis, and visualization. A background in ODEs and some basic knowledge of linear algebra is required.
- Derivation of the classical models: Be able to derive the advection, diffusion, advection-diffusion equation with source in one and three-spatial dimensions for different physical models. Known the derivation of the Laplace equation in three-spatial dimensions and the wave-equation in one spatial dimension.
- Classification of the PDEs: be able to classify the given second order PDE by using the discriminant of the principal part as a hyperbolic, parabolic or elliptic equation. Understand the geometrical and physical meanings of this classification, how this classification is connected with patterns that the equation predicts.
- Spectral methods for boundary value problems: Know the properties of eigenvalues and eigenvectors of a symmetric matrix operator. Understand the analogy between BVPs and linear algebraic systems. Understand the spectral method for symmetric linear differential operators and the meaning of the boundary conditions. Be able to investigate if the given linear differential operator is symmetric and find its eigenvalues and eigenfunctions.
- Solving BVPs for the diffusion equation using Fourier series: Be able to solve boundary value problems for the diffusion equation with Dirichlet, Neumann and Robin boundary conditions using Fourier series method in statistics or dynamics. Be able to use computer software to perform numerical computations and to visualize approximated solutions given by the partial sums of Fourier series.
- Solving BVPs for the diffusion equation using the Finite element method: Be able to write the given BVP in the weak form and prove the equivalence of weak and strong forms of the BVP. Understand the general ideas of the Galerkin method, be able to derive the energy inner product, the energy norm, and the Gram (stiffness) matrix for the given BVP. Be able to solve one dimensional BVP by using piecewise polynomial finite elements, and use software to perform these calculations and visualize the solution.
- Green’s functions for the diffusion equation: Understand the nature of Green’s functions and be able to find Green’s functions for the simplest BVPs and visualize them using software tool.
- Fourier series methods for the wave equations: Be able to solve BVPs for the one-dimensional homogeneous and inhomogeneous wave equation with different boundary conditions by using Fourier method.
- Finite element methods for the wave equation: Be able to solve BVPs for the wave equation using the finite element method and perform calculations using computer.
Gockenbach, M. S., Partial Differential Equations: Analytical and Numerical Methods, 2002.
Courant, R. and D. Hilbert, Methods of Mathematical Physics, Volume I, 1991.
Strang, G., Introduction to Applied Mathematics, 1986.
S. J. Farlow, Partial Differential Equations for Scientists and Engineers
Richard Haberman, Applied Partial Differential Equations.