Tutorial-12

Department of mathematics
MAL518: Methods of Applied Mathematics
Tutorial 12: The Potential equation
1. Solve the potential equation in the strip 0 < x < a, −∞ < y < ∞, subject to the
boundedness conditions u(x, y) bounded as y → ±∞, u(0, y) = 0, u(a, y) = e−|y| , −∞ <
y < ∞.
2. Find product solutions of this potential equation in the quarter-plane: 0 < x, 0 < y,
with the boundary conditions u(0, y) = 0, u(x, 0) = f (x). Note that u(x, y) must remain
bounded as x → ∞ and y → ∞.
3. Find product solutions of this potential equation in the half-plane: 0 < y, with the
boundary condition u(x, 0) = f (x), −∞ < x < ∞. What boundedness conditions must
u(x, y) must satisfy?
4. Use the formula
1
u(x, y) =
π
Z
∞
f (x0 )
−∞
y2
y
dx0
+ (x − x0 )2
to solve
the potential problem in the upper half-plane, with boundary condition u(x, 0) =
1, 0 < x,
f (x) =
0, x < 0.
5. Show that u(x, y) = x is the solution of the potential equation in a slot under the boundary
conditions f (x) = x, g1 (y) = 0, g2 (y) = a.
6. Solve the potential equation in the disk 0 < r < c if the boundary condition is v(c, θ) =
|θ|, −π < θ < π.

cos(θ), π/2 < θ < π/2,
7. Same as above exercise, with boundary condition v(c, θ) = f (θ) =
0,
otherwise.
8. Solve Laplace’s equation in the quarter-disk 0 < θ < π/2, 0 < r < c, subject to the
boundary conditions v(r, 0) = 0, v(r, π/2) = 0, v(c, θ) = 1.
9. Classify the following equations:
∂2u
(a) ∂x∂y
= 0;
2
2
∂2u
(b) ∂∂xu2 + ∂x∂y
+ ∂∂yu2 = 2x;
2
2
∂2u
(c) ∂∂xu2 − ∂x∂y
+ ∂∂yu2 = 2u;
2
2
∂2u
+ ∂∂yu2 = ∂u
(d) ∂∂xu2 − 2 ∂x∂y
∂y
∂2u
∂2u
∂u
(e) ∂x2 − ∂y2 − ∂y = 0
1
10. In which of the above equations can the variables be separated ?
11. Solve these three problems and compare the solutions:
2
2
(a) ∂∂xu2 + ∂∂yu2 = 0, 0 < x < 1, 0 < y, u(x, 0) = f (x), 0 < x < l, u(0, y) = 0, u(1, y) = 0, 0 <
y;
2
2
(b) ∂∂xu2 = ∂∂yu2 , 0 < x < 1, 0 < y, u(x, 0) = f (x), ∂u
(x, 0) = 0, 0 < x < l, u(0, y) =
∂y
0, u(1, y) = 0, 0 < y;
2
, 0 < x < 1, 0 < y, u(x, 0) = f (x), 0 < x < l, u(0, y) = 0, u(1, y) = 0, 0 < y.
(c) ∂∂xu2 = ∂u
∂y
12. Longitudinal waves in a slender rod may be described by this partial differential equation:
2
4u
∂2u
= ∂∂t2u − ∂x∂2 ∂t
2 . Show how to separate the variables.
∂x2
2

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