The focus of this course are the concepts and techniques for solving partial differential equations (PDE) that permeate various scientific disciplines. There will be an emphasis on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, and quantum mechanics.
Prerequisite
Basic theory of one complex variable and ordinary differential equations. These courses would suffice: Complex Variables with Applications (18.04) or Advanced Calculus for Engineers (18.075) or Functions of a Complex Variable (18.112). No prior knowledge of PDE is assumed.
Textbook
There is no required textbook. In my lectures, I will follow primarily the material in a few recommended but not required textbooks. These recommended books are listed in the readings section.
Homework
There will be 6 sets that will be graded, handed out every week and usually due two weeks later. No late homeworks will be accepted. In addition, another 5 similar sets with problems for your own practice will be handed out but not graded nor collected.
Exams
There will be two quizzes. There is no final exam.
Grading Policy
The final grade will be determined from:
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Course Outline
A tentative list of topics to be covered in lectures and homeworks follows:
Introduction
Some terminology; boundary and initial value problems; well- and ill-posed problems.
First-order PDEs
Complete solutions; characteristics; conservation laws; systems of PDEs; introduction to weak solutions: shocks and jump conditions; entropy condition; examples: traffic flow and gas dynamics.
Linear PDEs
Review and classification; the Laplace, wave and diffusion equations; the Klein-Gordon equation; more on characteristics; standard methods: separation of variables, integral transforms, Green's functions; potential scattering; special topics in conformal mapping; dispersion and diffusion; dimensional analysis and self-similarity; regular and singular perturbation theory; asymptotics for complete solutions; eikonal equation; high-frequency expansions; caustics; theory of rainbow and glory; a fun problem: Can one hear the shape of a drum?
More on Nonlinear PDEs
Conversion into linear PDEs; some exactly solvable cases; Burgers' equation; dimensional analysis and similarity; traveling waves; nonlinear diffusion and dispersion; the KdV, nonlinear Schrödinger and Sine-Gordon equations; waves in nonlinear optics, Bose-Einstein condensation, crystals dislocations and Josephson junctions; reaction-diffusion equations; Fisher's equation; singular perturbation: the boundary layer idea; shallow water theory; solitons; introduction to Bäcklund transformations; the Painlevé conjecture.
Variational Methods: Calculus of variations; first and second variation; Euler-Lagrange equation; constraints; examples; water waves; Stokes' problem.
Free-boundary Problems: Formulation; perturbation theory; more on water waves; method of extended gradient; materials surface evolution; tumor growth; other exciting, solved or open, problems.