MATH529

SP.24: Homework 2

Instructions

This is a first homework assignment on use of the concepts of linearity and separability to solve linear partial differential equations in two variables.

Exercises

From p.725: 2-6

Model solution:  13.2.1 (p.725)

Statement

A rod of length L with initial temperature f(x) is held at left at temperature zero and at right is insulated. State initial boundary value problem for heat equation.

Solution

The IBVP is:
(Subscript[u, t])= k (Subscript[u, x x]) for 0<x<L, t>0
u(x,t=0)=f(x) for 0<=x<= L
u(x=0,t)=0 for t>0
(Subscript[u, x])(x=L,t)=0 for 

Note: Pay careful attention to the definition of the domain of validity of the PDE and the initial and boundary value conditions:
(1) Derivatives cannot be defined at endpoints
(2) Initial conditions can be discontinuous w.r.t. boundary conditions

PDE = (∂_t)u[x,t] == k (∂_(x,x))u[x,t];
IC = u[x,0] == f[x];
BClft = u[0,t] == 0;
BCrgt = (((∂_x)u[x,t]) /. x->L) == 0;

IBVP = {PDE,IC,BClft,BCrgt}

Problems

From pp.727-8:2-8

Model solution:  13.3.1 (p. 727)

Statement

Solve the IBVP: 
(Subscript[u, t])= k (Subscript[u, x x]) for 0<x<L, t>0
u(x,t=0)=100 for 0<=x<= L
u(x=0,t)=0 for t>0
(Subscript[u, x])(x=L,t)=0 for

Solution

This is a heat equation with homogeneous boundary conditions and has a solution of the form

u(x,t) = (∑_(n=1)^∞)(Subscript[A, n]) sin ((π n x)/L)exp[-((k(((π n)/L)))^2) t]

with coefficients (Subscript[A, n]) given by the sine series of the initial condition

(Subscript[A, n])=(2/L)(∫_0^L)f(x)sin ((π n x)/L)dx=(200/L)(∫_0^L) sin ((π n x)/L)dx=-(Subscript[(200/(π n))[cos  ((π n x)/L)], x=0]^(x=L))⇒
(Subscript[A, n])=(200/(π n))(1-(((-1))^n)) 

The initial condition is discontinuous w.r.t. the homogeneous Dirichlet condition, hence the sine series coefficients decay slowly as (n^(-1)) and Gibbs phenomenon is observable in the formulation of the initial condition. However, for t>0 Gibbs oscillations disappear rapidly.

Solution visualization

A[n_,L_]=Assuming[n∈Integers,Integrate[Sin[((π n x)/L)],{x,0,L}]]

u[x_,t_,nt_,k_,L_] := Sum[ A[n,L] Sin[ ((π n x)/L)]Exp[-((k(((π n)/L)))^2) t],{n,1,nt}]

Plot[Evaluate[Table[u[x,t,10,1,1],{t,0,0.1,0.02}]],{x,0,1}]

Project

From p.728:12-13. Three-dimensional plots are constructed with Plot3D. Comment the results you obtain. Interpret the physical significance of the result, making use of the exercises solved above.