Hello exalted ones. I am working on a set of differential equations for my research and there is one that is becoming mortal.(adsbygoogle = window.adsbygoogle || []).push({});

I am solving a mechanical system whose behavior eq. is that of a one dimensional wave PDE. Namely:

[tex]u_{tt}=a^{2}u_{xx}[/tex]

For which I would derive two parametrized equations in terms of eigenvalues defined by my boundary conditions. Now my problem is that "a" is not constant, but actually a function of both time and space. Plainly:

[tex]a(x,t)=\frac{(E_{m} x+E_{0}) e^{\frac{C_{0} x^{2}}{T(t)}}}{C_{1}}[/tex]

So I have an e^ in function of both variables. I've almost given up trying to look for a closed-form solution.

As boundary conditions go (let's call them North, South, East, West), North is variable but known (input), South is always zero (fixed end). The first derivative of North in terms of x is also zero. Time increases from West to East and the displacement from South to North.

Would you advise me to pursue a numerical solution? What would be your advice on the matter?

JC

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# Variable coefficient Wave Equation

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