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### Eigenfunction expansion of a heat equation with Legendre polynomials

The equation for $\Phi$ is actually Mathieu's equation. The standard form of the equation is $$\frac{d^2 y}{dx^2} + (a - 2 q \cos(2x)) y = 0.$$ If we take the substitution $v = 2 x$, the resulting ...
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### How to pass partial derivatives with respect to t to the integral - Heat Equation in $\mathbb{R}^{n}$

By the Leibniz integral rule, \begin{multline} \partial_t u = \lim_{s->t^-}\left[\int_{\mathbb R^n}K(x-y, t-s)f(y,s)dy\right] + \int_0^t\int_{\mathbb R^n} \partial_tK(x-y,t-s)f(y,s)dyds \\ = \...
1 vote
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### question about the proof for the Key Lemma for Alexandroff maximum principle

I'll try to speak to your stated questions 1 and 2, assuming you are okay with $D\chi_{\epsilon}$ being negative definite in $\Gamma^{+}$, I've also taken the liberty to include a calculation at the ...
1 vote

### notation when changing variables in Partial Differential Equation

Defining $g$ as you have done is indeed the right thing to do, but for the sake of everyone's sanity, might I suggest that the function $h$ be defined as in the order $h(t,r,C)$ (so compose your $h$ ...

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