2025 Fall PDE1 Course Webpage

English Version|中文版

Basic Info

Instructor: Junyan ZHANG, yx3x@ustc.edu.cn

    Teaching assistants: There are three TAs who are all outstanding junior undergraduate students.
  • Fu ZHOU, email: zf010309@mail.ustc.edu.cn (student ID ≤ PB24000310)
  • Yuanyi ZHANG, email:wz10231027@mail.ustc.edu.cn (student ID between PB24000323 and PB24010484)
  • Yuanpeng TU, email:typ23000101@mail.ustc.edu.cn (student ID ≥ PB24010485)

Time & Location: 5204@East campus, Week 10~18, 1(3,4), 3(1,2),4(8,9).

Prerequisites

Mathematical Analsysis (especially multi-variable calculus and Fourier series), ODE.

Course contents

Textbook: My PDE1-Lecture Notes (in Chinese). Last updated: 01/26/2026.

Our course covers Ch 1.1~Ch 5.2. Chapter 1.4 and 2.6 are not required.

    References:
  • [1] Lawrence C. Evans. Partial Differential Equations (2nd edition), Graduate Studies in Mathematics 19, AMS, 2010. (Chapter 1~4)
  • [2] 姜礼尚、陈亚浙、刘西垣、易法槐.《数学物理方程讲义》第三版，高等教育出版社，2007.
  • [3] 周蜀林.《偏微分方程》，北京大学出版社, 2008.
  • [4] Sung-Jin Oh. Lectures Notes for MATH 222A, UC Berkeley.
  • [5] Elias M. Stein, Rami Shakarchi. Fourier Analysis: An Introduction. Princeton Lectures on Analysis I. (Chapter 5, 6)

Course contents.

    This course covers various methods to solve the classical solutions to linear PDEs, including:
  • Method of charateristics: transport equations, Burgers equation, wave equations.
  • Fourier transform and energy method: multi-dimensional wave equations, finite propagation speed, heat equations, *an introduction to .
  • Separation of variables: wave and heat equations in a finite interval, harmonic function in/outside a disk, Sturm-Liouville, *
  • Maximum principle: weak max principle for heat equations, mean-value properties of harmonic functions and various corollaries (strong max principle, Harnack's inequality, gradient estimates, Hopf lemma, analyticity, removable singularities).
  • Green's function: potential equations.
  • Miscellaneous (not required): oscillatory integrals(stationary phase) and decay estimates of Schrodinger and wave equations, Variational principle for the principal eigenvalues of (-Δ).
Exams and Grades

Your total score = max{30%HW+70%Final, 30%HW+25%Mid-Test+45%Final, 100%Final}.

Mid-Test: 16:00-17:20, 12/18/2025 @5204 and 5202, east campus. Here is the solution to mid-test.

Final Exam: 14:30-17:00, 01/16/2026 @5202 and 5201 east campus. Here is the solution to final exam.

HW and Tutorials

HW: NO COPYING, otherwise you'll receive 0 score.

    Rules for HW:
  • HWs are published on this webpage and QQ group.
  • Late HW will be deducted 20%-40% scores.
  • Hand-written HW, scanned PDF and TeX-complied/Markdown HW are accepted.

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Week Date Course Contents Homework Due
  Week 10     11.10     Transport equations (Ch. 1.1)     [1.1] 2, 3     HW1, TeX code.
  Due: 11/27/2025.
  HW1-Solution
  11.12     Transport equations and Burgers (Ch. 1.2), 1D wave equation (Ch. 2.1.1-2.1.2)     None  
  11.13     1D wave equation (Ch. 2.1.3-2.1.5)     [2.1] 1, 2, 3, 4  
  Week 11     11.17     Multi-dimensional wave equations (Ch. 2.1.5-2.2.1)     None  
  11.19     Multi-dimensional wave equations (Ch. 2.2.2-2.2.3)     [2.2] 3, Q1(bonus) 
  11.20     Finite propagation speed, Energy method(Ch. 2.3)     [2.3] 3  
  Week 12     11.24     Bootstrap method(exe 2.3.4, not required),
  Fourier transform(Ch. 2.4 & Appendix C.1)  		   [2.4] 2, 3, 4     HW2, TeX code.
  Due: 12/08/2025.
  HW2-Solution  
  11.26     Properties of heat equations(Ch. 2.4.2-2.4.3)、Plancherel theorem(Appendix C.1)     [2.4] 8, Q2; [C.1] Q2  
  11.27     Fourier method and asymptotic energy equi-partition of wave equations (Ch. 2.5)     Bonus: [2.5] 1、[C.1] Q7  
  Week 13     12.1     *An introduction to oscillatory integrals (stationary phase) (Ch. 2.6, not required)     None  
  12.3     Separation of variables (Ch. 3.1.1-3.1.2)     [3.1] 2, 5, 6     HW3, TeX code.
  Due: 12/22/2025.
  HW3-Solution  
  12.4     Tutorial 1 (courtesy to Yuanyi ZHANG)     None  
  Week 14     12.8     Separation of variables (Ch. 3.1.3, 3.2)     [3.2] 3, 5  
  12.10     *Variational principle for the principal eigenvalues of (-Δ)(Ch. 3.4.2, not required),
  Separation of variables: harmonic functions (Ch. 3.3)  		   [3.4] 6  
  12.11     Tutorial 2 (courtesy to Fu ZHOU)     None  
  Week 15     12.15     Potential equations in \R^d (Ch. 5.1)     [3.3] 1, 4  
  12.17     Method of Green's functions (Ch. 5.2.1-5.2.2)     None     HW4, TeX code.
  Due: 01/05/2026.
  HW4-Solution  
  12.18     Mid-test 16:00-17:20     None  
  Week 16     12.22     Method of Green's functions (Ch. 5.2.3),
  Maximum Principle for heat equations (Ch. 4.1.1)  		   [5.2] 3, [4.1] 3, 4, Q1(Bonus)  
  12.24     Maximum Principle for heat equations (Ch. 4.1.2) ,
  Mean-value property of harmonic functions
  and the strong maximum principle (Ch. 4.2.1-4.2.2)  		   [4.2] 3, 5(bonus)，[5.2] 1  
  12.25     Tutorial 3 (courtesy to Yuanpeng TU). Hopf lemma (Ch. 4.2.2)     None  
  Week 17     12.29     Harnack's inequality, gradient estimates (Ch. 4.2.3-4.2.4)     [4.2] 6, 8  
  12.31     Smoothness and real-analyticity of harmonic functions,
  removable singularities (Ch. 4.2.5-4.2.6),
  *Interior gradient estimates, logarithmic gradient estimates (Ch. 4.3, not required)  		   [4.3] 5  
  1.1     New-Year Holiday     None     Late HW Due:
  01/07/2026  
  Week 18     1.5     Final review     None  
  1.7     Tutorial 4     None  
  1.8     Q & A     None