This is a homework problem that I can almost finish. I just can’t invert the Laplace transform at the end. $$u_{xx}=u_{tt}, u(t=0)=u_t(t=0)=0, u(x=0)=\sin\omega t, u(x=2)=0.$$ Taking the Laplace transform with respect to $t$ and using the zero initial conditions, I obtain $U” = s^2U$, where $s$ is the transform variable and $U$ is the transform […]

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave equation, based–surprisingly–upon the initial-value problem. $\quad$ Suppose $u$ is a bounded, smooth solution of the initial-value problem: \begin{cases} u_{tt}-\Delta u = 0 […]

Show that the solution of the equation $\frac { \partial^2u}{\partial t^2}= \frac { \partial^2u}{\partial x_1^2}+ \frac { \partial^2u}{\partial x_2^2}+ \frac { \partial^2u}{\partial x_3^2}$ Subject to $u (x,0)=f (x)$ and $\frac {\partial u}{\partial t}(x,0)=g (x )$ , is given by $u (x,t)=\frac {1}{|S (x,t)|}\int _{S (x,t)}[tg (y)+f (y)+\nabla f (y ). (y-x )]d\sigma ( y)$. Where […]

Is it possible to find an explicit solution to the wave equation: $$ \partial_t^2u-c^2 \partial_x^2 u=0 \\ u(x,0)=f(x), \ u(cx,x)=g(x) $$ or do we need information about a derivative of $u$ as well?

Problem Use the general solution to solve the signalling problem with homogeneous wave equation on the half line, homogeneous IC and nonhomogeneous Neumann boundary conditions. Where c>0 is a constant, and h is continuous function. The solution u should be continuous. $u_{tt} – c^2 u_{xx} = 0, \quad\quad \quad \quad 0 < x < \infty,\quad […]

The D’Alembert solution of the wave equation $$u_{tt}-c^2u_{xx}=0, x \in \mathbb{R}, t>0$$ $$u(x,0)=\phi(x), x \in \mathbb{R}$$ $$u_t(x,0)=\psi(x), x \in \mathbb{R}$$ is the following: $$u(x,t)=\frac{1}{2}\begin{bmatrix} \phi(x-ct)+\phi(x+ct) \end{bmatrix}+\frac{1}{2c} \int_{x-ct}^{x+ct} \psi(\tau)d \tau$$ I am looking at the proof of this solution, which is the following: The general solution of the wave equation is given from: $\displaystyle{u(x,t)=f(x+ct)+g(x-ct)} \ \ […]

Small-amplitude water waves travel on the free surface $y = \eta(x, t)$ of an incompressible inviscid fluid of uniform depth $h$. Derive the linearised boundary conditions $$\text{ at }y=0\quad \frac{\partial\varphi}{\partial t}=\frac{\partial\eta}{\partial t},\quad \frac{\partial\varphi}{\partial t}+g\eta=0 $$ and write down the boundary condition satisfied by the velocity potential $\varphi$ at the rigid boundary $y = −h$. ￼￼￼Show […]

Usually second-order linear PDE’s are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE’s: Elliptic – Laplace’s equation $\nabla^2 u = 0$. Parabolic – the heat equation $u_t = \nabla^2 u$. Hyperbolic – the wave equation $u_{tt} […]

Right now I’m studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)…$) I know how to solve it when the boundary conditions are $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ with d’Alambert’s formula. But I don’t know how to solve it in any other case. Does anyone know of any book, notes, etc. […]

Suppose $u$ is a solution of the two-dimensional wave equation $$u_{tt}-c^2 \Delta u=f(x)$$ with initial values $u(0),u_t(0)$ that have support on the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point $(x_1,x_2)=(2,3)$? My attempt: I think it depends on the point farthest from $(2,3)$ in the disc, […]

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