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Consider the heat equation solution $h(x,t)$ where $x \in [0,1]$ and $t \geq 0$ with initial condition $h(x,t=0)=f(x)$ and Neumann boundary $\frac{d}{dx} h(x=0,t)=0$ and $\frac{d}{dx} h(x=1,t)=0$. Suppose $f(x)=x^n$ where $n \in \mathbb{N}$. Obviously, the solution $h$ can be expressed as an infinite Cosine series. However, I was wondering if for this particular choice of $f$, whether $h(x,t)$ has a closed form solution? If yes, then, great, what is the expression? :)

If not, then here is the next question: is there a good approximation for $\hat{h}(x,t) \approx h(x,t)$ such that $\hat{h}(x,t)$ is a polynomial of degree at most $n$ with respect to variable $x$ (does not necessarily need to be polynomial in $t$)?

Thanks,

Golabi

Golabi
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  • If you're happy with a spline, rather than a polynomial, you can write the heat equation in weak form over the Hilbert space of splines and bring to bear the finite elements machinery, which will give you a good spline approximation to the solution. – user7530 Feb 13 '15 at 04:32
  • Also, what do you consider a "good approximation"? Are you interested in $t$ large? $t$ small? – user7530 Feb 13 '15 at 04:33
  • Thanks! Could you provide a reference for spline approximation of the solution of the heat equation? – Golabi Feb 13 '15 at 09:55

1 Answers1

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I assume the constant in your heat equation is $1$, so it's $$\dfrac{\partial h}{\partial t} = \dfrac{\partial^2 h}{\partial x^2}$$ If I'm not mistaken, your solution can be written as an integral of a Jacobi theta function:

$$h(x,t) = \int_0^1 s^n \vartheta_3\left(\dfrac{\pi (x-s)}{2}, e^{-\pi^2 t}\right)\; ds $$

although that probably won't help you much. It's basically equivalent to your cosine series.

For large $t$, there's not much point in approximating the solution using polynomials: the first few terms of your cosine series will give you a very good approximation, because $\exp(-n^2 \pi^2 t)$ goes to $0$ very rapidly unless $n$ is small.

Robert Israel
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  • Yes, constant of the heat equation is 1 as you guessed. Truncated cosine is accurate only for large $t$ and accuracy drops rapidly when $t$ approaches $0$. Ideally, I would be interested in an approximation that has moderate accuracy for all $t$. I was hoping that the monomial form of the initial condition $f(x)$ somewhow becomes helpful for finding a moderately good approximation for all $t$. – Golabi Feb 13 '15 at 10:01