91Ó°ÊÓ

By using Laplace transforms determine the effect of the initial conditions in terms of the Green's function for the initial value problem, $$ \alpha \frac{d^{2} y}{d t^{2}}+\beta \frac{d y}{d t}+\gamma y=0 $$ subject to \(y(0)=y_{0}\) and \(\frac{d y}{d t}(0)=v_{0}\).

Determine the Laplace transform of the Green's function for the wave equation if the boundary conditions are (a) \(u(0, t)=a(t) \quad\) and \(\quad \frac{\partial u}{\partial x}(L, t)=b(t)\) (b) \(\frac{\partial u}{\partial x}(0, t)=a(t) \quad\) and \(\quad \frac{\partial u}{\partial x}(L, t)-b(t)\)

Solve for \(u(x, t)\) using Laplace transforms $$ \begin{aligned} \frac{\partial u}{\partial t} &=k \frac{\partial^{2} u}{\partial x^{2}} \quad-\infty

Solve for \(u(x, t)\) using Laplace transforms: $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ subject to \(u(x, 0)=f(x), u(0, t)=0\), and \(u(L, t)=0\). By what other method(s) can this representation of the solution be obtained?

Derive the convolution theorem for Laplace transforms without using the Dirac delta function. [Hint: Introduce the variable \(z=\bar{t}+\overline{\bar{t}}\) in order to evaluate the double integral (12.2.14c).]