PARTIAL DIFFERENTIAL EQUATIONS, SPRING 2019 MATH 363-001
HOMEWORK 7: NUMERICAL METHODS FOR PDES COURANT INSTITUTE, NYU
DUE: FRIDAY MAY 10, 11:00 AM, IN RECITATION CLASS
You must submit a short writeup (in PDF format) documenting what you did, that is, an explanation of what the code does and answers to the questions posted below. Make sure to include figures of your results in the writeup (use PDF format with legend for the graphs with labeled axis). Also attach your MATLAB codes to the homework.
(1) [20 pts] Find the exact solution to the Initial- Boundary- Value problem:
ut =νuxx, x∈(0,1), t>0,
u(x,0) = sin2πx, u(0,t)=0, t≥0, u(1,t)=0, t≥0.
x ∈ [0,1],
Write the scheme and a code (forward in time, center in space) to approximate the solution of this prob- lem for ν = 1/6. Take ∆x = 0.1 and compare your results with the exact solution at t = 0.01, 0.1, 1, 10 with ∆t = 0.01.
(2) [20 pts] Find the exact solution to the Initial- Boundary- Value problem:
ut =νuxx, x∈(0,1), t>0, u(x,0) = cos πx, x ∈ [0,1],
2 ∂u(0,t)=0, t≥0,
∂x
u(1,t)=0, t≥0.
Write the scheme and a code (backward in time, center in space) to approximate the solution of this prob- lem for ν = 1/6. Take ∆x = 0.1 and compare your results with the exact solution at t = 0.01, 0.1, 1, 10 with ∆t = 0.01.
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