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We are given a parabolic p.d.e of the following general form: $$ \frac{\partial f}{\partial t}+\frac{1}{2} f^n \frac{\partial^2 f}{\partial x^2}=0 $$ Is there a way to transform this p.d.e. to the standard heat equation, $$ \frac{\partial g}{\partial t}+\frac{1}{2} \frac{\partial^2 g}{\partial x^2}=0 $$ If not what are the general methods for solving this type of p.d.e.?

Ted Black
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    Travelling wave ansatz $f(x,t) = g(z(x,t)) = g(x-ct)$? Then you get $$\frac{-cg'}{g^{2}} + \frac{1}{2} g'' = 0 \implies \frac{c}{g} + \frac{1}{2} g' = K_{1}$$ which shouldn't be too hard to solve. However, this ansatz might not work depending on the boundary and initial conditions. – Matthew Cassell Dec 10 '21 at 20:35
  • Many thanks @mattos that's a very promising approach. – Ted Black Dec 10 '21 at 21:59

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