# MATH400-101 : Applied Partial Differential Equations (First term 2016/2017)

### Lecture I: Monday 9:00am--10:00am, LSK-460.

### Lecture II: Wednesday 9:00am--10:00am, LSK-460.

### Lecture III: Friday 9:00am--10:00am, LSK-460.

### Office Hours: Every Monday, Wednesday, Friday, 4:30pm-5:20pm, LSK 303B.

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# Lecture Notes For MATH400

# Downloads For MATH400

# Updates For MATH 400

### Sept. 7: Equation of Characteristics. Intial Data and Initial Data Curve. Simple example $ u_x=0$.

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### Sept 9: Solving first order PDE, Case 1: $a u_x+ b u_y=0$. up to Example 6 of Lecture Note 1.

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### Sept 12: Solving first order PDE, case 2: $ a(x, y)u_x+ b(x, y) u_y= c(x, y, u)$. Up to page 20 of Lecture Note 1.

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### Sept 14: Solving firts order PDE, case 2: $ a(x,y) u_x+ b(x, y) u_y= c(x,y, u)$. Second Method using change of variables. Completed Lecture Note One.

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### Sept 16: Solving first order PDE, case 3: $ u_t+ c(u) u_x=0$. General solutions. Brekaing time $t_B$. Lecture Note 2.

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### Sept. 19: Solving first order PDE, case 3: $u_t + c(u) u_x=0$. Expansion Fan Solution. Lecture Note 2.

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### Sept. 21: Solving first order PDE, case 3: $u_t + c(u) u_x=0$. Shock Curve. Lecture Note 2.

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### Sept. 23: Case 4: $F(x,y, u, u_x, u_y)=0$. Lecture Note 3.

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### Sept. 26: Case 4: $F(x,y, u, u_x, u_y)=0$. Two Examples. Lecture Note 3.

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### Sept. 28: Second order PDEs. derivation of wave and heat equation. PDE problems. Up to 1.4 of the book. Lecture Note 4.

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### Sept. 30: PDE problems. Well-posedness. Classification of 2nd order PDEs.

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### Oct. 3: Second order PDEs: change of variables. Solutions to Wave equation. Start on Lecture Note 5.

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### Oct. 5: d'Alembert's formula. Geometric meaning of solutions of wave equation.

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### Oct. 7: d'Alembert's formula. general hyperbolic equations. Principle of Causality. Domain of Dependence and Domain of Influence. Energy.

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### Oct. 12: Wave equation with source. Well-posedness. Finished Lecture Note 5.

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### Oct. 14: Wave equations in half lines and bounded intervals. Reflection method and extension method.

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### Oct. 17: Lecture Note 7 on Diffusion. Derivation of solution formula on the whole line.

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### Oct. 19: Lecture Note 8. Solutions Formula for Diffusion Equation with Sources.

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### Oct. 21: Lecture Note 8. Well-posedness of Diffusion equation. Energy Method. Comparison of Wave and Diffusion. Diffusion equation on half line with Dirichlet or Neumann BC.

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### Oct. 24: Diffusion in a bounded interval. Method of Separation of Variables. Lecture Note 9.

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### Oct. 26: Method of Separation of Variables. Neumann. Wave. Robin.

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### Oct. 28: Midterm Test.

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### Oct. 31: Robin BCs. Lecture Note 9 and 9.5.

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### Nov. 2: Robin BCs. Periodic BCs, General S-L. Lecture Note 10.

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### Nov. 4: Sturm-Liouville eigenvalue problems. Transformations and Properties. Example 1. Lecture Note 10.

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### Nov. 7: Sturm-Liouville. Bessel function od order zero. Finished Lecture Note 10.

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### Nov. 9: Method of Separation of Variables Applied to Inhomogeneous Problems. Lecture Note 11.

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### Nov. 14: Method of Shifting Data. Laplace Equation. Uniqueness by Energy Method. Rectangle. Lecture Note 12.

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### Nov. 16: Method of separation of variables for rectangle and cubes. Polar coordinates.

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### Nov. 18: Method of Separation of Variables to Laplace Equation on a disk.

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### Nov. 21: Poisson's formula. Mean-Value Property. Maximum Principle. Annulus

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### Nov. 22: Laplace on Annulus, Wedges.

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### Nov. 25: Laplace in Sectors. Bessel functions.

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### Nov. 28: Bessel functions. Diffsuion and Wave in higher dimensions.

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### Nov. 30: Bessel functions of order $n$. Diffusion in a disk. Diffusion with inhomogeneous.

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### Dec. 2: Bessel functions of order $n$. (Lecture Note 15). Summary.

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# Announcements For MATH 400

### Office Hours: Every Monday, Wednesday, Friday, 4:30pm-5:20pm, LSK 303B. Starting Date: September 8

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### Problem 3 in assignment 2: change $ u(0-, t)=3$ to $ u(0-, t)=4$.

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### Midterm: Oct. 28, in class. Coverage: will be announced.

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### No office hour on Sept. 28.

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### Coverage on Midterm: Lectures up to Oct. 19 (First order, Wave, Diffusion on the whole line).

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### The fourth HW4 can be handed in by 5:30pm, Oct. 31.