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Equations of Mathematical Physics (Ⅰ)

Offered in 112-2

  • Serial Number

    29236

  • Course Number

    MATH7419

  • Course Identifier

    221 U5780

  • No Class

  • 3 Credits

  • Compulsory

    GRADUATE INSTITUTE OF MATHEMATICS / Institute of Applied Mathematical Sciences

      Compulsory

    • GRADUATE INSTITUTE OF MATHEMATICS

    • Institute of Applied Mathematical Sciences

  • Van Tien Nguyen

  • Tue 3, 4 / Thu 5

  • 天數302

  • Type 3

  • 30 Student Quota

    NTU 30

  • No Specialization Program

  • English
  • NTU COOL
  • Notes
    The course is conducted in English。

  • NTU Enrollment Status

    Enrolled
    0/30
    Other Depts
    0/0
    Remaining
    0
    Registered
    0
  • Course Description
    This course introduces mathematical methods for application to problems arising from physical and biological processes in terms of Nonlinear Partial Differential Equations (PDEs). It is addressed to graduate students and anyone with an interest in mathematical physics. The content is built around the following topics: first-order quasilinear equations, reaction-diffusion equations, aggregation-diffusion equations, free boundary problems and phase transitions, and wave equations. By PDE methodology, we study and solve these problems within a mathematically rigorous framework. The course also introduces simulation and symbolic calculation from Matlab Toolbox for solving PDEs.
  • Course Objective
    - Derivation of physical models - Mathematical methods of solving different types of PDEs - Simulation and symbolic Matlab Toolbox for solving PDEs.
  • Course Requirement
    Basic knowledge of differential and integral calculus in more than one dimen-sion, along with Linear Algebra, ODEs, and linear PDEs is strongly recommended.
  • Expected weekly study hours after class
    12 hours
  • Office Hour

    Office hours: Thursday 2 - 3:30 pm

  • Designated Reading
    1. Applied Partial Differential Equations by J. Ockendon, S. Howison, A. Lacey, A. Movchan Recommended: 2. Applied Partial Differential Equation: A Visual Approach by P. Markowich 3. Mathematical Concepts of Quantum Mechanics by S. Gustafson and I. M. Sigal 4. Singularities: Formation, Structure, and Propagation by J. Eggers and M. A. Fontelos
  • References
  • Grading
    5%

    Attendance

    If there is a necessary absence, you should notify the instructor.

    25%

    Homework

    There are 6 HW sets, each contributing 5% to the final grade. The lowest one will be discarded.

    30%

    Midterm

    Writing exam. Make sure you complete HWs.

    40%

    Final

    Writing exam or report replacement

  • Adjustment methods for students
    Adjustment MethodDescription
    Teaching methods

    Provide students with flexible ways of attending courses

    Assignment submission methods

    Mutual agreement to present in other ways between students and instructors

    Others

    Negotiated by both teachers and students

  • Course Schedule
    Week 1First-order quasilinear equations: models, Cauchy problem, characteristics, domain of definition, blowup
    Week 2First-order quasilinear equations: Solutions with discontinuities, generalized (weak) solutions, shock waves, energy estimates, and Riemann problem
    Week 3Introduction to second-order scalar equations: models, Cauchy problem, characteristics, canonical forms
    Week 4Fisher-KPP equation: models, traveling wave and stability
    Week 5Predator-prey models, traveling wave and stability
    Week 6Keller-Segel system: derivation, global solutions and long-time asymptotic behaviors.
    Week 7Keller-Segel system: self-similarity, collapsing solutions and blowup behaviors.
    Week 8Nonlinear Fokker-Planck equation. Midterm.
    Week 9One-phase Stefan problem: derivation, existence and uniqueness of solutions, long-time asymptotic behavior.
    Week 10One-phase Stefan problem: dynamics of melting ice balls.
    Week 11Two-phase Stefan problem: derivation, existence and uniqueness of solutions, asymptotic behavior.
    Week 12Semilinear wave equations: derivation, existence and uniqueness of solutions, asymptotic of global solutions
    Week 13Semilinear wave equations: self-similarity, singularity formation
    Week 14Nonlinear Schrödinger equation: derivation, solitary waves, stability
    Week 15Nonlinear Schrödinger equation: self-similarity, singularity formation, stability
    Week 16Final exam