Can we rewrite the wave equation with damping (aka. telegrapher's equation) $$u_{tt} - c^2 u_{xx} + a u_t = 0 $$ as a first order diagonal system of two equations?
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4What are your ideas about it? – Cesareo Dec 28 '21 at 16:21
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Have you tried this approach, where does it get you? – EditPiAf Jan 04 '22 at 14:17
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First assume there were two first order equations $$f(u,u_x)=u_t$$ $$g(u,u_t)=u_x$$ so that $$\frac{\partial f(u,u_x)}{\partial x}=\frac{\partial g(u,u_t)}{\partial t}$$ would become $$c^2 u_{xx} = u_{tt} + a u_t$$ So you can say \begin{align} f(u,u_x)&=c^2u_x+C_1x+C_2\\[2ex] g(u,u_t)&=u_t+au+C_1t+C_3 \end{align} And the system of first order equations can be \begin{cases} c^2u_x-u_t+C_1x+C_2&=0\\[2ex] -u_x+u_t+au+C_1t+C_3&=0 \end{cases} And then you can make those diagonal from there.
Kay K.
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