I was playing around with the heat equation in one dimension and tried to guess what the solution to homogenous boundary conditions and a sine wave as initial condition on the interval $0<x<\pi$ would look like for the slightly different equation :$$\frac{\partial u}{\partial t} = \frac{1}{\frac{\partial^2 u}{\partial x^2}}$$Can anyone help me use a method to find an analytic solution ? I have tried to visualize the solution nummerically on Matlab but it looks to be very unstable and after a very short time the real look of the solution is messed by the nummerical limitations of using a finite difference method (forward in time and centered in space). I was wondering if there is anyone who can give me some reference on any document that analyses this equation. Thank you!
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