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MATC46 - University of Toronto at Scarborough - Department of Computer & Mathematical Sciences

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MATC46

Winter 2019

2.

3.

Use separation of variables to solve the IBVP ? utt(x,t)=uxx(x,t), 0<x<1 t>0

Assignment 2

Midterm test Mar 1st , Friday , 7 - 9pm, IC130

Due Date: Wednesday 13 February. Reading: Lecture notes in weeks 3 & 4.

Problems: 1.

Solve the following IBVPs for a string of unit length, subject to the given conditions.

a) b)

c)

g(x) = sin2πx, c = 1. g(x) =2, c = 1/π.

? utt(x,t)=c2uxx(x,t), 0 x 1 t 0< < > ? u(0,t)=0, u(1,t) 0, t= 0 >

??u(x,0)=f(x),u(x,0) g(x)=0 x 1 ?t

f(x) = sinπxcosπx, g(x) = 0, c = 1/π. f(x) = sinπx + (1/2)sin3πx + 3sin7πx,

? 0, if0x1 ? 3

?1?1?12 f (x) = ? ? x ?,? if x ?30? 3? 3 3

? 1(1?x),if2x1 ?30 3

< <

?? u(0,t), u(1,t), t>0

??u(x,0)= f(x),ut(x,0)=g(x) 0<x<1

When

  1. i) ?α= 0, β = 3, f(x) = 3x, g(x) = 4πsin(πx);

  2. ii) ?α= 2, β = 3, f(x) = 2 + x+ 2sin(πx), g(x) = 0

Use separation of variables to solve the IBVP ? utt(x,t)=uxx(x,t), 0<x<1 t>0

?? ux(0,t), ux(1,t), t>0 ??u(x,0)= f(x),ut(x,0)=g(x) 0<x<1

When

  1. a) ?α= 4, β = 4, f(x) = 4x, g(x) = πcos(πx);

  2. b) ?α= 3, β = 5, f(x) = 3x + x2 + cos(πx), g(x) = 0.

4. Use the D’Alembert formula to solve the IVP ??utt(x,t)=c2uxx(x,t), ?∞<x<∞ t>0

a) b)

??

α = β = 0, L = 1, c = 1, f(x) = e-x.
α = 100, β = 100, f(x) = 50x(1 – x), L = 1, c = 1.

?

When i) ii)

iii) iv)

u(x,0) = f (x) ut (x,0) = g(x)

c = 1, f(x) = sinx, g(x) = 0;
c = 2, f(x) = sinx, g(x) = 4;
c = 3, f(x) = sinx, g(x) = 3cosx;

c = 1, f(x) = e-x, g(x) = 2 . 1+ x2

5. Solve IBVP
??u = c2 ?2u 0 x L, t 0

??t ?x2
?u(0,t), u(L,t) β =t 0

??u(x,0)= f(x) 0xL