I have done a reduction of this equation: $t^{\beta} u_{t} - u_{xx} = f(x,t)$ to an ODE and this integral seems to be impossible:
$$\int_{0}^{T} e^{\frac{-2 \lambda t^{1- \beta}}{1- \beta}} dt$$
I've tried Wolfram, but it doesn't work either.
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It's already well known that $\int e^{x^2} dx$ doesn't have an elementary form, so this integral definitely won't. – Kaynex May 29 '17 at 13:50
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… except in special case like $\beta=1$ (trivial) or $\beta=\frac12$ (easy). Perhaps there are other special cases, but the general case is hopeless. – Harald Hanche-Olsen May 29 '17 at 13:52
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What do you mean by 'impossible integral'? – Yuriy S May 29 '17 at 14:04