Solving Partial Differential Equations with Laplace Transform

[mliuzzolino]

Homework Statement

<br /> <br /> \dfrac{\partial^2 \varphi }{ \partial x^2} - \dfrac{\partial ^2 \varphi }{\partial t^2} = 1<br /> <br />

Initial Conditions:

\varphi (x, 0) = 1; \varphi_t (x, 0) = 1

Boundary Condition:

\varphi (0, t) = 1


On 0 \leq x &lt; \infty, 0 \leq t &lt; \infty

Homework Equations

Let {\Phi} denote the Laplace transform from t to s.

The Attempt at a Solution

Apply Laplace Transform to PDE:

<br /> <br /> \dfrac{\partial^2 \Phi }{ \partial x^2} - s^2\Phi + s\varphi (x, 0) + \varphi _t (x, 0) = \dfrac{1}{s}<br /> <br />

Apply Initial Conditions:

<br /> <br /> \dfrac{\partial^2 \Phi }{ \partial x^2} - s^2\Phi + s + 1 - \dfrac{1}{s} = 0<br /> <br />




<br /> <br /> \dfrac{\partial^2 \Phi }{ \partial x^2} - s^2\Phi + \dfrac{s^2 + s - 1}{s} = 0<br /> <br />


This leads to the eigenvalue being something like:

<br /> <br /> r = \dfrac{s^3 \pm \sqrt{s^6 - 4s(s^2 + s - 1)}}{2s}<br /> <br />

which is always positive for both cases.

The solution in Laplace space should be something like:

<br /> <br /> \Phi (x, s) = A(s) e ^{\dfrac{s^3 + \sqrt{s^6 - 4s(s^2 + s - 1)}}{2s}x} + B(s) e ^{\dfrac{s^3 - \sqrt{s^6 - 4s(s^2 + s - 1)}}{2s}x}<br /> <br />

and since the eigenvalue is always positive, as x goes to infinity the exponential terms will explode, requiring both A(s) and B(s) to be equal to 0.

What am I doing wrong? Any ideas?

 

[christoff]

I would check your final calculation for the solution Φ in Laplace space. It seems to not depend on x, which is weird. If you're still working on this, that is.

 

[Mute]

<br /> <br /> \dfrac{\partial^2 \Phi }{ \partial x^2} - s^2\Phi + \dfrac{s^2 + s - 1}{s} = 0<br /> <br />This leads to the eigenvalue being something like:

<br /> <br /> r = \dfrac{s^3 \pm \sqrt{s^6 - 4s(s^2 + s - 1)}}{2s}<br /> <br />

which is always positive for both cases.

I would double-check this step. It looks like you assumed a solution $\Phi(x,s) = e^{rx}$, and then tried to solve the characteristic equation $r^2 - s^2 r + (s^2 + s -1)/s = 0$, but this equation for r is not correct. Your purely ordinary differential equation in x is not homogeneous - only two of your terms depend on $\Phi$. If you plug in $\Phi(x,s) = e^{rx}$, what happens when you try to divide out the $e^{rx}$ to get the characteristic equation for r?