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6.2: Inhomogeneous Heat Equation

( \newcommand{\kernel}{\mathrm{null}\,}\)

Here we consider the initial value problem for u=u(x,t), uC(Rn×R+),

utu=f(x,t)  in xRn, t0,u(x,0)=ϕ(x),

where ϕ and f are given. From

$$ \widehat{u_t-\triangle u}=\widehat{f(x,t)} \]

we obtain an initial value problem for an ordinary differential equation:

dˆudt+|ξ|2ˆu=ˆf(ξ,t)ˆu(ξ,0)=ˆϕ(ξ).

The solution is given by

$$\widehat{u}(\xi,t)=e^{-|\xi|^2 t}\widehat{\phi}(\xi)+\int_0^t\ e^{-|\xi|^2(t-\tau)}\widehat{f}(\xi,\tau)\ d\tau.\]

Applying the inverse Fourier transform and a calculation as in the proof of Theorem 5.1, step (vi), we get}

u(x,t)=(2π)n/2Rn eixξ(e|ξ|2tˆϕ(ξ)  +t0 e|ξ|2(tτ)ˆf(ξ,τ) dτ) dξ.

From the above calculation for the homogeneous problem and calculation as in the proof of Theorem 5.1, step (vi), we obtain the formula

u(x,t)=1(2πt)nRn ϕ(y)e|yx|2/(4t) dy+t0Rn f(y,τ)1(2π(tτ))n e|yx|2/(4(tτ)) dy dτ.

This function u(x,t) is a solution of the above inhomogeneous initial value problem provided

$$\phi\in C(\mathbb{R}^n),\ \ \sup_{\mathbb{R}^n}|\phi(x)|<\infty\]

and if

$$f\in C(\mathbb{R}^n\times[0,\infty)),\ \ M(\tau):=\sup_{\mathbb{R}^n}|f(y,\tau)|<\infty,\ 0\le\tau<\infty.\]

Contributors and Attributions


This page titled 6.2: Inhomogeneous Heat Equation is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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