Partial Differential Equations

COURSE OVERVIEW
Partial differential equations is a subject that touches on many branches of pure mathematics and applied mathematics. The focus on this course is on the techniques to prove the existence local and global solutions, existence of traveling waves and stability. There will be a discussion about the deduction of some mathematical models arising in physics and mathematical biology.

SYLLABUS
1. DIFFUSION PROCESSES ( 12 hours)
1.1.Diffusion and Random Motion
1.2. Similarity methods
1.3.Reaction-diffusion: Fisher´s equation
1.4. Traveling wave solutions
1.5. Stability of traveling waves
1.6. Nagumo´s equation
1.7. Advection-Diffusion: Burger´s equation
2. REACTION-DIFFUSION SYSTEMS ( 18 hours)
2.1. Reaction-Diffusion models
2.2. Traveling waves
2.3.Tranport
2.4 Existence of solutions
2.5 Maximum principles and comparison theorems
2.6 Energy estimates and asymptotic behavior
2.7 Pattern formation
3. ELLIPTIC EQUATIONS (18 hours)
3.1. Laplace equation
3.2.Maximum principle
3.3 Existence theorem
3.4 Eigenvalue problems
3.5 Stability and bifurcation
 

BIBLIOGRAFÍA

· J.W. Jordan. Nonlinear ordinary differential equations. Oxford.

· David Logan. Applied mathematics. Tercera edición. Wiley- Interscience

· David Logan. Introduction to nonlinear partial differential equations. 2nd edition. Wiley Interscience

· Evans, L. Partial differential equations. AMS

· Farlow, S. Partial differential equations for scientists and engineers. Dover