ASSIGNED EXERCISES OF MATH544: PDE set 4 March, 2001

INITIAL AND BOUNDARY VALUE PROBLEMS OF SOME HYPERBOLIC TYPE OF EQUATIONS Ronald B Guenther and John W Lee, Partial Differential Equations of Mathematical Physics and Integral Equations

1. Homogeneous wave equations with initial conditions: Let f(x) and g(x) be some arbitrary functions and

u_{tt} − c^{2} u_{xx} = 0,
u(x, 0) = f(x), u_{t}(x, 0) = g(x)

Theorem 1. Let f(x) have a second derivative and g(x) have a first derivative for −∞ < x < ∞. Then the initila value problem given above is wellposed. Its solution is given by

u(x, t) = \frac{1}{2} [f(x + ct) + f(x − ct)] + \frac{1}{2c} ∫_{x−ct}^{x+ct} g(ξ) dξ

2. Homogeneous wave equation with initial and boundary value problem. Let

u_{tt} − c^{2} u_{xx} = 0,
u_{x,0} = f(x), u_{t}(x, 0) = g(x), 0 ≤ x ≤ L
u(0, t) = u(L, t) = 0, t ≥ 0

Theorem 2. Let f(x) = have a continuous fourth derivative and g(x) have a continuous third derivative for 0 ≤ x ≤ L, and f(0) = f(L) = 0 , f′′(0) = f′′(L) = 0, and g(0) = g(L) = 0. Then the initial boundary value problem above has a solution given by

u(x, t) = ∑_{n=1}^{∞} [A_{n} cos(λ_{n}ct) + B_{n} sin(λ_{n}ct)] sin(λ_{n}x),

A_{n} = \frac{2}{L} ∫_{0}^{L} f(x) sin(\frac{nπx}{L})dx,

B_{n} = \frac{2}{λ_{n}cL} ∫_{0}^{L} g(x) sin(\frac{nπx}{L})dx.

where λ_{n} = \frac{nπ}{L}.

Comment . Uniform convergence requires

|λ_{n}^{2}A_{n}| ≤ \frac{C}{n^{2}} , |λ_{n}^{2}B_{n}| ≤ \frac{C}{n^{2}}

where C = max{C_{1}, C_{2}}, here

C_{1} = \frac{2l}{π^{2}} ∫_{0}^{L} |f^{4}(x)|dx,

C_{2} = \frac{2L}{cπ^{2}} [|g′′(0)| + |g′′(L)| + ∫_{0}^{L} |g′′′(0)|dx]

3. Theorem 3. Let the initila data f and g in the previous problem satisfy the hypothesis of Theorem 2 (above). Then this problem is well posed.

Hint. Let (f_{1}, g_{1}) and (f_{2}, g_{2}) are the data for the solutions u_{1} and u_{2} respectively. Then prove that

|u_{1}(x, t) − u_{2}(x, t)| ≤ D max{||f_{1} − f_{2}||_{2}, ||g_{1} − g_{2}||_{1}}

where D is a constant and

||f_{1} − f_{2}||_{2} = max{||f_{1} − f_{2}||, ||f′_{1} − f′_{2}||, ||f′′_{1} − f′′_{2}||}
||g_{1} − g_{2}||_{1} = max{||g_{1} − g_{2}||, ||g′_{1} − g′_{2}||}

Here the norm is defined as ||h(x)|| = max|h(x)|, for all x ∈ [0, L]. The way one proves this theorem: Let u = u_{1} − u_{2}, f = f_{1} − f_{2}, g = g_{1} − g_{2}. Use the compatibility conditions and integration by parts to find

|λ_{n}B_{n}| ≤ \frac{C_{1}}{n^{2}} , |A_{n}| ≤ \frac{C_{2}}{n^{2}}

where A_{n} and B_{n} are Fourier coefficients based on f = f_{1} −f_{2} and g = g_{1} −g_{2} and

C_{1} = \frac{2L}{cπ^{2}} ∫_{0}^{L} |g′(x)|dx, C_{2} = \frac{2L}{π^{2}} ∫_{0}^{L} |f′′| dx

Using the solution given in the problem 2 above one can find the estimate given above at the beginning of this problem.

4. Inhomogeneous Initial and Boundary value problem: (k¿0)

u_{tt} + ku_{t} − c^{2}u_{xx} = F (x, t), 0 ≤ x ≤ L, t > 0
u(x, 0) = f(x), u_{t}(x, 0) = g(x), 0 ≤ x ≤ L
u(0, t) = r(t), u(L, t) = s(t), t ≥ 0

Theorem 4. There exists at most one solution to the above inhomogeneous initial boundary value problem.

Let the data f = 0, g = 0, r = 0, s = 0. Then we have

Theorem 5. Let F (x, t) have continous third order partial deriveatives for 0 ≤ x ≤ L and t ≥ 0. Assume that F (0, t) = F (L, t) = F_{xx}(0, t) = F_{xx}(L, t) = 0 for all t ≥ 0. Then the above inhomegenous initial and boundary value problem has a unique solution given by

u(x, t) = ∑_{0}^{∞} u_{n}(t) sin(λ_{n}x)

u_{n}(t) = \frac{1}{cλ_{n}} ∫_{0}^{t} F_{n}(τ) G(t, τ)dτ

whereG(t, τ) is the Green’s function corresponding to the equation u_{n}(t)

u′′_{n} + ku′_{n} − c^{2}λ_{n}^{2} u_{n} = F_{n}(t)

We assumed F (x, t) = ∑_{0}^{∞} F_{n}(t) sin(λ_{n}x) with

F_{n}(t) = \frac{2}{L} ∫_{0}^{L} F (x, t) sin(λ_{n}x)dx.