偏微分方程

  1. Fourier Analysis (1)convolution (2)fundamental solution and Green functions  Wave equations   Heat equations    [Kernel]  [Green function method Harnack]
  2. Dispersion  Dispersion and Solitons (Fourier transform)  [非線性色散方程(陶哲軒)]
  3. non-linear PDE  (1)Burges equation (2)conservation laws
  4. [KdV equation] and Solitons [淺談孤立子]
  5. Pax Pair and Spectal Theory(isospectral deformation)
  6. Inverse Scattering Transform
  7. Sturm-Liouville theory/Schrondinger Operator
  8. Darboux Transformation/Backlund Transformation
  9. Hamiltonian Structure
  10. Functional Analysis  (1)Sobolev space (2)weak solution (3)energy estimate)

[偏微分方程的方法 丘]   [幾何分析  丘]  [隨機PDE]   [非線性 PDE的最近發展]   [Connes的非交換幾何  Riemann Zeta]

一個PDE可積性的判別條件 (1)Lax Pair存在(充要條件) (2)無限守恆律(必要條件)  (3)逆散射變換(求解方法)

重要例子 (1)KdV方程  (2)非線性薛丁格方程(光孤子)  (3)sine-Gordon方程  (4)KP方程  (5)Hirota方程  (6)自對偶Yang-Mills方程

Dirac方程的可積性要分幾個層面解釋 略過

AI參與尋找可積的PDE:(1)OptPDE尋找Lax Pair (Max Tegmark)  [AIIntegrablePDE]

  1. [Harmonic function(單位圓盤上的Dirichlet問題)] [Hyperbolic plane]  [Harnack不等式(Li-Yau]
    [Harmonic equations] [定理(單連通上的全純函數)   [mean value theorem]  [共形變換下的Dirichlet問題]  [習作]  [EX01]
  2. [Dirac delta function Arnold]    Dirac delta function  
  3. sine-Gordon
  4. Poincare不等式

Topics: (1) [Monge-Ampere equation]  (2)Navier-Stokes equation

習作

  1. Poisson formula in half-plane
  1. Introduction  to PDE  1.1 Boundary and Initial Value Problems   1.2 Classification  
    What is the [Dirichlet condition] [Neumann condition]
  1. Transport Equations   2.1 Derivation of General Solution    2.2 Particular Solution  [Ex01] [Ex02]
  1. First order semilinear equations    3.1 The Method of Characteristics  [Ex01] 3.2 [First Order Equation]    Second Order PDE
    [Ex01]
  1. Wave Equations   4.1 Derivation of the General Solution    (1)Factoring the Operator (2) Change of Variables
     4.2 Derivation of Particular Solution   [Ex01]  [Ex02]
    [sine-Gordon equation]  [Hodge wave equation]   [Wave equation(2)]   [Conservation of Energy]
  1. Properties of the Wave Equation on R 5.1 Causality    5.2 Wave eq with One Boundary Condition     5.3 General Boundary Condition
  1. Heat Equation(Diffusion Equation)  6.1 Fundametal Solution     6.2 Particular Solution to the Homogeneous(Inhomogeneous) IVP

6.3 Properties of the Heat Equation    6.4 Well-Posed  and Symmetry   6.5 Maximum Principle   [Ex02] [Ex03]

  1. Eigenvalue Problems  7.1 Common Eigenvalue Problems  872 Orthogonality of Eigenfunctions
  2. Fourier Series
  3. Separation of Variables      Separation of Variables
  4. Distributiona and Transform  (1)Distributions  (2)Green functions  (3)Fourier transforms  (4)Source functions  (5)Laplace transform techniques
  5. Wave in space
  6. Nonlinear PDEs (1)Shock equation    Burger equation  [Ex01]  Shock  [Ex01]  (2)Solitons  (3)Calculus of Variations  (4)Bifurcation Theory  (5)Water Waves
  7. Cauchy-Kowalevski theorem
  8. Elliptic functions
  1. Introduction to Partial Differential Equations       Peter J. Olver
  2. An Introduction to PDE                                      Walter A. Strauss
  3. A Basic Course in PDE    Qing Han(韓青Notre Dame大學)  [ResearchGate]
  4. Partial Differential Equations                              Christopher C. Tisdell (有YouTube講解)
  5. Partial Differential Equations in Action              Gianmaria Verzini       有Exercises and Solved Problems
  6. Methods for PDE                                                Marcelo R.Ebert     Michael Reissig    [ResearchGate]
  7. Justin Ko  [Justin Ko]
  8. Lectures on PDE                                                 Vladimir I. Arnold
  9. Methods of Mathematical Physics(II)                 R.Courant and D.Hilbert
  10. Geometry of PDE                                               Paul Bracken   [ResearchGate]
  11. Partial Differential Equations with Mathematica and Maple      Ioannis P. Stavroulakis   Stepan A Tersian
  12. Advanced Calculus for Applications                  Peter D. Lax   Maria Shea Terrell  Ch 9
  13. 教科書/參考書書評  (2)PDE   Lawrence C. Evans   (3)PDE   Fritz John   (4)Elliptic PDE of second order   David Gilbarg,Neil S. Trudinger
  14. >The Monge-Ampere equation and its Applications(付費)   Alessio Figalli
  1. (1)有朋自遠方來(1) 陳恕行  (2) Neil Trudinger---汪徐家(Xu-Jia Wang)   Peter Lax (2)]
  2. Nonlinear PDE國際會議(2023年)---紀念Neil Trudinger (1942--) 從與會學者提出的文章可以看出當代nonlinear PDE研究的方向
    (1)幾何分析與PDE的深度融合 (2)非局部算子的興起 (3)應用驅動的數學問題 (4)自由邊界問題與正則性理論  (5)從定性分析到分類與剛性 (6)跨領域方法
  3. What is the geometric PDE ?