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How to use NDSolve to solve 1+1 D heat equation $u_t=u_{xx}$ with $ -\infty<x<\infty$ and $0\leq t\leq T$?

NDSolve[{Derivative[1, 0][u][t, x] == Derivative[0, 2][u][ t, x], 
u[0, x] == Sin[x]}, u, {t, 0, 10}, {x, -Infinity, Infinity}]

The error code is: NDSolve::ndnl: Endpoint -∞ in {x, -∞, ∞} is not a real number.

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How to use NDSolve to solve 1+1 D heat equation ut=uxx with −∞<x<∞ and 0≤t≤T?

For infinite spatial domain, do not give boundary conditions. No need for numerical solution, since exact solution exist

ClearAll["Global`*"];
pde = D[u[x, t], t] == D[u[x, t], {x, 2}];
ic = u[x, 0] == Sin[x];
sol = DSolve[{pde, ic}, u[x, t], {x, t}, Assumptions -> t > 0]

$$ \left\{\left\{u(x,t)\to e^{-t} \sin (x)\right\}\right\} $$

Now you have solution for all time. You can plot it for any period you want.

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  • $\begingroup$ Thanks for the answer. But the pde I really want to solve is a nonlinear pde. DSolve can not solve it analytically. I was thinking to solve it numerically using NDSolve. The ranges are the same: $x\in (-\infty,\infty), t\in[0,10])$. $\endgroup$
    – mike
    Commented Apr 14, 2020 at 4:14
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    $\begingroup$ @mike then you could post separate question on that, but it will be better to use the PDE you actually want solved and not another one to reduce confusion. $\endgroup$
    – Nasser
    Commented Apr 14, 2020 at 6:03
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    $\begingroup$ @mike You can only have numerical solution for finite numbers, you can't have numerical solution with infinities. $\endgroup$
    – m0nhawk
    Commented Apr 14, 2020 at 6:03

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