Math 1470 Midterm Take-home Exam
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Math 1470 Midterm Take-home Exam
Due June 4, 2023 at 11:59 pm.
Instructions
● There are six problems in test. Only FIVE will be graded. Please indicated clearly which ones you want to be graded, or else I will pick the first five. Work independently.
● You may use your notes and textbook, but no searching online.
● You may not talk to, or communicate with by any method, any person other than me about the content of this exam.
● Indicate clearly your answers to the problems, and write legibly. Solutions with- out sufficient details will not be granted full credit.
● When you finish, please have your answer scanned/pictured and converted to a single .pdf file, and submitted to Canvas.
● Please include a photo of your student ID along with your answers when upload- ing to Canvas.
Good luck!
Problem 1. (20 points)
(1) Consider the initial value problem
Find the solution to the above problem.
(2) Consider the initial-boundary value problem
Sketch the characteristic curves and find the solution.
Problem 2. (20 points)
An infinite bar with thermal diffusivity equal to 4 is initially at zero degrees. A con- centrated unit heat source is then continually applied at the origin. The corresponding PDE is
ut = 4uxx + (x); u(0; x) = 0; x ∈ R;
where isa Dirac delta function characterized by'a(b) f (y) (y · c) dy = f (c) if a < c < b
and zero else where (for arbitrary, continuous function f).
(1) Write down an integral formula for the solution. Simplify the integral involving , but do NOT evaluate the integral.
(2) Is the temperature of the bar at the origin (i) always increasing, (ii) always de- creasing, (iii) constant, (iv) first increasing, then decreasing, or (v) periodically oscillating? Justify your answer by writing an explicit formula for the tempera- ture.
Problem 3. (20 points)
Consider a semi-infinite bar with unit thermal diffusivity with the left end kept at temperature 1. Suppose that initially the temperature is 0. Write down the initial- boundary value problem that governs the temperature in the bar, and solve it. Express your solution in terms of the error function Erf(x) = 
x e −p2 dp.
Problem 4. (20 points)
Solve the non homogeneous initial-value problem
for u(t; x). Simplify your answer. Is u(0; t) periodic in t?
Problem 5. (20 points)
Consider the transport equation on a finite interval
ut · 2ux = 0; t > 0; x ∈ (a; b):
with boundary conditions u(t; a) = u(t; b) = 0 and initial data u(0; x) = (x · a)(x · b).
(1) Prove that E(t) = lab 
u2 (t; x) dx is constant. (10 points)
(2) Find the total mass M (t) = la b u(t; x) dx. (10 points)
Problem 6. (20 points)
Let u(t; x) be a solution to the heat equation on [0; L] subject to homogeneous Dirichlet boundary conditions. Let
M (t) = max{u(t; x) : 0 ≤ x ≤ L}:
Prove that M (t) is a non-increasing function when t > 0.
2023-06-24