MATH 212 Homework Assignment 7 Questions, Fall 2016

MATH 212: INTRODUCTORY PARTIAL DIFFERENTIAL EQUATIONS ASSIGNMENT 7 SOPHIE MOUFAWAD Exercise 1: The goal of this exercise is to solve the following heat equation with homoge- neous boundary conditions and constant initial condition: $ ’ ’ ’ ’ & ’ ’ ’ ’ % ut “ uxx ` π2 4 u ´ b, t ą 0, b P R upt, 0q “ upt, 1q “ 0 t ě 0 up0, xq “ c, 0 ă x ă 1, c P R. 1. Solve the ODE z2pxq ` π2 4 zpxq “ b. 2. Find the equilibrium solution uEpxq. 3. Define the difference vpt, xq “ upt, xq ´ uEpxq that measures the deviation of the solution upt, xq from equilibrium. (a) Show that vpt, xq satisfies a heat equation of the form vt “ γvxx ` βv, where β, γ P R. (b) Then, adapt the boundary and initial value conditions. 4. There are two ways for solving initial-boundary value problem obtained in part 3. (a) Let wpt, xq “ eαtvpt, xq. Find an α value for which wpt, xq satisfies the heat equation wt “ wxx. Then, change the boundary and initial conditions accord- ingly. And solve the obtained initial boundary value problem. Finally, deduce the solution vpt, xq. (b) Solve for vpt, xq directly by using the method of separation of variables. 5. Deduce the solution upt, xq and find its limit as t tends to infinity. 6. Approximate the solution upt, xq by uEpxq and the first term of vpt, xq. (a) Find an upper bound for the error between upt, xq and its first term approxi- mation, i.e. |upt, xq ´ uEpxq ´ v1pt, xq|, when c “ 4b π2 . (b) After what time will the error from the first term approximation be less than 8ce1{4 πpe3´e2q. Exercise 2: Consider the following heat equation with mixed boundary conditions and initial condition: $ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ % ut “ uxx, t ą 0 upt, 0q “ 0 t ě 0 uxpt, lq “ 0 t ě 0 up0, xq “ fpxq, 0 ă x ă l. 1. Find the solution to the initial-boundary value problem, and discuss its asymptotic behavior as t Ñ 8. 2. Find the equilibrium solution uEpxq. 2