MATH-4500 Spring 2023
PDEs of Math/Phys Problem Set 5 D.W. Schwendeman
Notes
1. DUE DATE: Monday, March 20, by 10am (start of class).
2. Please show all work for the problems. Illegible or undecipherable solutions will not be graded.
3. Figures, if any, should be displayed clearly and properly labelled.
Problems
1. Consider the regular Sturm-Liouville eigenvalue problem(
xφ′
)′
+ λxφ = 0, 1 < x < 2, φ(1) = φ(2) = 0
(a) Use the Rayleigh quotient to show that λ > 0 . (Hint: first show that λ ≥ 0 , and then conside
the case λ = 0 separately.)
(b) Use the Rayleigh quotient and a test function of the form u(x) = a+ bx−x2 , with suitably chosen
constants a and b , to estimate the value of the smallest eigenvalue, λ1 .
(c) Extra credit. Consider a new test function v(x) = (1 + cx)u(x) , where c is a constant and u(x)
is the test function in part (b). Note that c = 0 gives the same estimate for λ1 in part (b). However,
if c 6= 0 , then v(x) may give an improved estimate for λ1 using the Rayleigh quotient. Find the
value for c that gives the best estimate. (Notes: The alge
a involved for this problem is messy and
I used Maple to help solve it. Also, you could read ahead in section 7.7.7 of the textbook to work out
an exact formula to find λ1 so that you can compare your estimates with the exact value.)
2. The vertical displacement of a vi
ating rectangular mem
ane satisfies
utt = c
2(uxx + uyy), 0 ≤ x ≤ L, 0 ≤ y ≤ H, t ≥ 0
with boundary conditions
u(0, y, t) = u(L, y, t) = uy(x, 0, t) = uy(x,H, t) = 0
and initial conditions
u(x, y, 0) = 0, ut(x, y, 0) = g(x, y)
Use separation of variables to find the solution u(x, y, t) . Determine the natural frequencies of vi
a-
tion. (Hint: start with u(x, y, t) = φ(x, y)T (t) .)
3. Consider the heat flow in a three-dimensional region R with smooth boundary S and unit outward
normal n . The temperature u(x, t) satisfies
ut = k∇2u+Q(x), x ∈ R, t > 0,
where k > 0 is a constant thermal diffusivity and Q(x) is a heat source, both assumed to be known.
The initial and boundary conditions are
u(x, 0) = f(x), x ∈ R
n · ∇u = g(x), x ∈ S
1
(a) Define the total thermal energy in the region as E(t) =
∫
R
u(x, t) dV . Integrate the PDE, and use
the divergence theorem, to determine a formula for ddtE(t) .
(b) Determine a condition on Q(x) and g(x) such that a steady state solution exists. (You need not
find this solution.) Give a
ief physical interpretation of the condition you derive.
(c) Assuming the condition in part (b) holds, determine E(t) .
4. Determine expressions for the eigenvalues λ and eigenfunctions φ(r) for the following Sturm-Liouville
eigenvalue problems.
(a)
(
φ′
)′
+
(
λr − 4
)
φ = 0, 0 < r < 3, φ(0) bounded and φ′(3) = 0
(b)
(
φ′
)′
+
(
λr − 1
)
φ = 0, 1 < r < 2, φ′(1) = 0 and φ(2) = 0
Extra credit. Use Maple’s BesselJ and BesselY functions (or Matlab’s besselj and bessely) to de-
termine numerical values for the smallest three eigenvalues for each problem, and plot the co
esponding
three eigenfunctions.
2
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APPLIED PARTIAL
DIFFERENTIAL EQUATIONS
with Fourier Series and
Boundary Value Problems
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APPLIED PARTIAL
DIFFERENTIAL EQUATIONS
with Fourier Series and
Boundary Value Problems
Fifth Edition
Richard Haberman
Southern Methodist University
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Li
ary of Congress Cataloging-in-Publication Data
Haberman, Richard,
Applied partial differential equations : with Fourier series and
oundary value problems / Richard Haberman. – 5th ed.
p. cm.
ISBN-13: XXXXXXXXXXalk. paper)
ISBN-10: XXXXXXXXXXalk. paper)
1. Differential equations, Partial. 2. Fourier series. 3. Boundary value
problems. I. Title.
QA377.H27 2013
515’.353–dc23
XXXXXXXXXX
XXXXXXXXXX CW XXXXXXXXXX
ISBN-10: XXXXXXXXXX
ISBN-13: XXXXXXXXXX
To Liz, Ken, and Vicki
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Contents
Preface xvii
1 Heat Equation 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX1
1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod XXXXXXXXXX
1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . XXXXXXXXXX11
1.4 Equili
ium Temperature Distribution . . . . . . . . . XXXXXXXXXX14
1.4.1 Prescribed Temperature . . . . . . . . . . . XXXXXXXXXX14
1.4.2 Insulated Boundaries . . . . . . . . . . . . . XXXXXXXXXX16
1.5 Derivation of the Heat Equation in Two or Three Dimensions XXXXXXXXXX
Appendix to 1.5: Review of Gradient and a Derivation of Fourier’s Law
of Heat Conduction . . . . . . . . . . . . . . . . . . . XXXXXXXXXX30
2 Method of Separation of Variables 32
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX32
2.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX32
2.3 Heat Equation with Zero Temperatures at Finite Ends XXXXXXXXXX35
2.3.1 Introduction . . . . . . . . . . . . . . . . . . XXXXXXXXXX35
2.3.2 Separation of Variables . . . . . . . . . . . . XXXXXXXXXX35
2.3.3 Time-Dependent Ordinary Differential Equation XXXXXXXXXX
2.3.4 Boundary Value (Eigenvalue) Problem . . . XXXXXXXXXX38
2.3.5 Product Solutions and the Principle of Superposition XXXXXXXXXX
2.3.6 Orthogonality of Sines . . . . . . . . . . . . XXXXXXXXXX46
2.3.7 Formulation, Solution, and Interpretation of an Example XXXXXXXXXX
2.3.8 Summary . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX50
Appendix to 2.3: Orthogonality of Functions . . . . . . . . . XXXXXXXXXX54
2.4 Worked Examples with the Heat Equation (Other Boundary
Value Problems) . . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX55
2.4.1 Heat Conduction in a Rod with Insulated Ends XXXXXXXXXX
2.4.2 Heat Conduction in a Thin Insulated Circular Ring XXXXXXXXXX
2.4.3 Summary of Boundary Value Problems . . . XXXXXXXXXX64
2.5 Laplace’s Equation: Solutions and Qualitative Properties XXXXXXXXXX
2.5.1 Laplace’s Equation Inside a Rectangle . . . XXXXXXXXXX67
2.5.2 Laplace’s Equation Inside a Circular Disk . XXXXXXXXXX72
2.5.3 Fluid Flow Outside a Circular Cylinder (Lift XXXXXXXXXX76
2.5.4 Qualitative Properties of Laplace’s Equation XXXXXXXXXX79
vii
viii Contents
3 Fourier Series 86
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . XXXXXXXXXX86
3.2 Statement of Convergence Theorem . . . . . . . . . . . . . . . . . .