Differential Equations - Existence & Uniqueness

This statement is true by the Uniqueness Theorem.

$\u_{tt}=c^2u_{xx}, 0<x<1, t>0 \\text{Initial Conditions: }u(x,0)=f(x), u_t(x,0)=g(x), 0\leq x\leq l \\text{Boundary Conditions: }u(0,t)=0, u(l,t)=0, t\geq 0$

is twice differential equation in terms of and .

Now, consider the energy integral

$E(t)=\frac{1}{2}\int_0^l(c^2u^2_x+u^2_t)\ dx$

After performing integration by parts results in the following,

$\int_0^lc^2u_xu_{xt}\ dx=[c^2u_xu_t]_0^l-\int_0^lc^2u_tu_{xx}\ dx$

From here, the initial conditions and boundary conditions are applied.

$\\frac{dE}{dt}=\int_0^l u_t(u_{tt}-c^2u_{xx})\ dx \\\frac{dE}{dt}=0 \\E(t)=C$

Therefore,

which proves the wave equations has only one solution and thus is unique.

Existence & Uniqueness

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