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I have following problem: I want to construct a Greens Function for solving following inhomogeneous PDE: $$ \left(-D(t) \left( \frac{\partial ^2 }{\partial x^2} + \frac{\partial ^2 }{\partial y^2} \right) + \frac{\partial}{\partial t} \right)u(x,y,t) = f(x,y,t)$$ where G fulfills $ \left(-D(t) \left( \frac{\partial ^2 }{\partial x^2} + \frac{\partial ^2 }{\partial y^2} \right) + \frac{\partial}{\partial t} \right)G = \delta(\vec r)\delta(t)$ and the solution then yields $ u(x,y,t) = \int \int \mathrm d r^2 \mathrm d t \,G(\vec r, t) f(\vec r, t) $

I've tried several things to construct it from the problem, where D is a constant - but without success.

Maybe you have any ideas?

physicus
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1 Answers1

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Try separating your coefficient $D(t)$ as a steady-state component and a time-varying one, such as $D(t)=D_0+D'(t)$. That way, you can write your PDE as $$ \left[-D_0\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2} {\partial y^2}\right)+\frac{\partial}{\partial t}\right]u(x,y,t)=f(x,y,t)+D'(t)\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2} {\partial y^2}\right)u(x,y,t). $$ The Green's function of the LHS is well-known. What you have now, though, is an implicit integral equation, or a Fredholm equation to solve.

Joey Dumont
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