Partial Differential Equations 3150Final Exam

Exam Date: Thursday, May 2, 2013

Instructions: This exam is timed for 120 minutes. You will be given extra time to complete

the exam. No calculators, notes, tables or books. Problems use oniy Chapters 1, 2, 3, 4, 7 of the

textbook. No answer check is expected. Details count 3/4, answers count 1/4.

[001. (CH3. Heat Conduction in a Bar)

Consider the heat conduction problem in a laterally insulated bar of length 10 with one

and the other end at 80 Celsius The imtial ternpeiature along the bai is

1\

/1= 0<x<10, t>0,

= •50, t>0,= 80, t>0,

A (a) [20%) Find the steady-state temperature u(x).

A (b) [40%] Solve the bar problem with zero Celsius temperatures at both ends, but f(x)

kreplaced by f(x) —u1(). Call the answeru2(z,t).

(c) [20%] Compute the numerical values of the Fourier coefficients of u2(x, t). Then display

the answer n(x, t) = xi (x) + u2(a t) with the numerical values inserted into the series.

‘ (d) [20%] Display an answer check for the solution u(x, t) u1(x) + 712(Z, t).

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Partial Differential Equations 3150Final Exam

Exam Date: Thursday, May 2, 2013 / 2c

Instructions: This exam is timed for 120 minutes. You will be given extra time to complete

the exam. No calculators, notes, tables or books. Problems use only Chapters 1, 2, 3, 4, 7 of the

textbook. No answer check is expected. Details count 3/4, answers count 1/4. I oQ

OO1. (CR3. Heat Conduction in a Bar)

Consider the heat conduction problem in a laterally insulated bar of lenh 10 with one 1 0 0end at 50 Celsius and the other end at 80 Celsius. The initial temperature along the bar is

f(x) =3x.

?Jt -ii, 0 <x < 10, t> 0,

u(0,t) = 50, t>0,u(10,t) = 80, t >0,

x(x,0) = f(x), 0 <x < 10.

,A (a) [20%] Find the steady-state temperature u1(x).

A- (b) [40%1 Solve the bar problem with zero Celsius temperatures at both ends, hut f(2)

replaced by f(x) — ui(x). Call the answer

(c) [20%] Compute the numerical values of the Fourier coefficients ofi2(x, t). Then display

the answer u(x, t) = x1 (x) + u2(x, t) with the numerical values inserted into the series.

fr (d) [20%.] Display an answer check for the solution ii(x,t) = u(x) +u2(x,f).

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(a) [30J Find and display the nonzero terms in the Fourier series expansion of f(x), formed

as the even 2ir-periodic extension of the function fo (x) — sin2 (x) + 4 cos(2x) on 0 < x < x.

(b) 15°%1 Compute the Fourier sine series coefficients b for the function g(x), defined as

the period 2 odd extension of the function 90(x) = 1 on 0 x < 1. Draw a representativegraph for the partial Fourier sum for five terms of the infinite series.

[20j Define ho(x)= { and let Ji(x) be the 4x odd periodic

extension of ho(x) to the whole real line. Compute the sum f(—5.25x) + f(1.5x).

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3150 Final Exam S2013

3. (CH3. Finite String: Fourier Series Solution)

A (a) [50%] Display the series formula, complete with derivation details, for the solution/‘ u(x, t) of the finite string problem

0 < x < 2, t > 0,u(0,t) = 0,z(2,t) 0, t>0,u(x,0) = f(x), 0<x<2,ut(x,0) = g(x), 0<x<2.

Symbols f and p should not appear explicitly in the series for u(x,t). Expected in theformula for u(x, t) are product solutions times constants.

(b) [25%] Display explicit formulas for the Fourier coefficients which contains the symbols

f(x), g(x).

(c) [25%] Evaluate the Fourier coefficients when f(x) = 100 and g(x) 0.

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4. (CH3. Poisson Problem)

Solve for u(x, y) in the Poisson problem

I u + ‘u sin(irx) sin(2iry), 0 <r < 1, 0 < y < 1,

___u(O,y) = 4sin(3xy), 0<y<l,

7 CI 2(€’) - u(x, y) = 0 on the other 3 boundary edges.

Product solution derivations are not expected. Expected solution details:

(1) Decomposition into two problems, it = ui + zL2. Draw figures.

(2) Eigenfunctions mn and eigenvalues A, of the Helmholtz equation1(z, y)+yy(X, Y) =

—)(x, y) with zero boundary conditions on the rectangle.

(3) Poisson problem defined for x1 (x, y). Solution formula for u1 (a, y). Fourier coefficient

formula.

(4) Dirichlet problem defined foru2(, y). Product solutions. Solution formula for 212(1, y).Fourier coefficient formula.

_. (5) Display the answer u =u + it2.

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Ut(,t) = u1(x,t), —oo<x<oo, t>O,

u(x,0) = f(x), —oo < .x < ,

150 O<<1,f(x) = 100 —1<r<0

0 otherwise

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Fourier transform theory definitions to solve the problem. The answer is expressed in terms

of the error function.

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5. (CH7. Heat Equation and Gauss’ Heat Kernel)Solve the insulated rod heat conduction problem

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3150 Final Exam S2013

5. (CH7. Heat Eqmiation and Gauss’ Heat Kernel)Solve the insulated rod heat conduction problem

—cc <x < cc, t >0,

u(x,0) = f(x), —cc <x < cxc,

150 0<r<1,

f@) 100 —1 <x < 00 otherwise

Hint: Use the heat kernel g the error function erf(x) = e_Z2dz, and

Fourier transform theory definitions to solve the problem. The answer is expressed in termsof the error function.

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