1. Solve for u(x, t): 1 3ux+4u 0, u(x, 0) = 1 + x²
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 49E
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PDE Review for Midterm
1. Solve for u(x, t):
3ux+4u 0, u(x, 0) = 1 + x²
2. For which value of a is u(x, y) = x3 + axy2 harmonic in R².?
3. Find the function that is harmonic in the unit disk and agrees with f(0) = sin²(6) on the
boundary.
4. Write a formula for the function that is harmonic in the unit disk and agrees with [sin(0)|on the
boundary.
5. Find the function u that is harmonic in the annulus 4 <|x| <6, u = 0 on |x| = 4 and u =
1 on x 6 (a) in R³ (b) in R²
6. Explain why, in ten words or less, for u = 4r6 cos(60) and B = any finite box in R²
maxlu(x, y) max lu(x, y)|
B
дв
7. Find a function u(x, y) that is harmonic in R2 and that equals f(0) = 4 sin(30) + 5 cos(60) on
the unit circle.
8. Find a function u(x, y) that is harmonic outside the unit disk, equals f(0) = 4 sin(30) + 5
cos(60) on the unit circle and tends to zero as r→ co.
9. Find the Fourier cosine series of f(x) = 1 on [0,1]
10. Find some functions which are harmonic in the strip S = {(x, y):0 < x < 1, y R}, vanish on
as, and agree with f(x) = 3 sin(4x) + 5 sin (6πx) on the segment {(x, 0):0 <x<1}
11. Let D={x ER:0 <x<1}.
(a) Find the solution of the heat equation uxxu in D x [0, ∞) such that u(x, 0) = f(x) as
in #10 and u(0, t) = u(1, t) = 0. How fast does u(x, t) → 0 as t→ co?
(b) Find the solution of the heat equation Uxue in D X [0, ∞o) such that u(x, 0) =
cos(7x) and ux(0, t) = ux(1, t) = 0. Find the limit of u(x, t) as t→ co.
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PDE Review for Midterm
1. Solve for u(x, t):
3ux+4u 0, u(x, 0) = 1 + x²
2. For which value of a is u(x, y) = x3 + axy2 harmonic in R².?
3. Find the function that is harmonic in the unit disk and agrees with f(0) = sin²(6) on the
boundary.
4. Write a formula for the function that is harmonic in the unit disk and agrees with [sin(0)|on the
boundary.
5. Find the function u that is harmonic in the annulus 4 <|x| <6, u = 0 on |x| = 4 and u =
1 on x 6 (a) in R³ (b) in R²
6. Explain why, in ten words or less, for u = 4r6 cos(60) and B = any finite box in R²
maxlu(x, y) max lu(x, y)|
B
дв
7. Find a function u(x, y) that is harmonic in R2 and that equals f(0) = 4 sin(30) + 5 cos(60) on
the unit circle.
8. Find a function u(x, y) that is harmonic outside the unit disk, equals f(0) = 4 sin(30) + 5
cos(60) on the unit circle and tends to zero as r→ co.
9. Find the Fourier cosine series of f(x) = 1 on [0,1]
10. Find some functions which are harmonic in the strip S = {(x, y):0 < x < 1, y R}, vanish on
as, and agree with f(x) = 3 sin(4x) + 5 sin (6πx) on the segment {(x, 0):0 <x<1}
11. Let D={x ER:0 <x<1}.
(a) Find the solution of the heat equation uxxu in D x [0, ∞) such that u(x, 0) = f(x) as
in #10 and u(0, t) = u(1, t) = 0. How fast does u(x, t) → 0 as t→ co?
(b) Find the solution of the heat equation Uxue in D X [0, ∞o) such that u(x, 0) =
cos(7x) and ux(0, t) = ux(1, t) = 0. Find the limit of u(x, t) as t→ co.
5
сл
田
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