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For the planar heat equation with point injection of strength $p(s)$, zero initial data gives $\Psi (t,\mathbf{r})={\int }_0^{t}p(s)e^{-r^2/[4D(t-s)]}/[4\pi D(t-s)]ds$. If injection is $p_0$ until $t_0$ and zero later, then for $t>t_0$ this equals $p_0[\mathrm{Ei}(-b/(t-t_0))-\mathrm{Ei}(-b/t)]/(4\pi D)$, where $b=r^2/(4D)$. The exponential integral expansion gives $p_0\ln [t/(t-t_0)]/(4\pi D)$ when $b/(t-t_0)\ll 1$.

The Cole-Hopf transformation $u=-2\nu {\varphi }_x/\varphi$ converts the viscous Burgers equation into the heat equation ${\varphi }_t=\nu {\varphi }_{xx}$. Initial data $u(x,0)=g(x)$ correspond to $\varphi (x,0)=C\exp [-(2\nu {)}^{-1}{\int }^{x}g(s)ds]$.

In a unit square with prescribed conductive boundary temperatures and impermeable boundaries, the stream function convention $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$ gives linear equations $\Delta \psi =R{\theta }_x$, ${\theta }_t={\psi }_x+\Delta \theta$, with $\psi =\theta =0$ on all sides. The velocity equation follows by taking the vertical-plane curl of Darcy law, and the temperature equation by linearizing the heat equation about $1-z$. Unlike a laterally unbounded Darcy layer, the conducting side boundaries produce a two-dimensional real marginal eigenspace at $R_c=8{\pi }^2$.

A Darcy velocity $\mathbf{u}$ driven by pressure in a stationary porous skeleton dissipates mechanical power $\mu |\mathbf{u}{|}^2/\Pi$ per bulk volume. It follows by dotting Darcy law with the discharge: $-\mathbf{u}\cdot \nabla p=\mu |\mathbf{u}{|}^2/\Pi$. Its contribution to a heat equation can be neglected only when it is small relative to the relevant thermal-energy transfer.

The formal $L^2$ gradient flow of this energy is $u_t=\mathrm{div}({\Phi }^{\prime }(r)\nabla u/r)$, with a Neumann boundary condition for no flux. The radial Hessian eigenvalues of the gradient integrand are ${\Phi }^{\prime \prime }(r)$ and ${\Phi }^{\prime }(r)/r$. Their signs distinguish convex forward diffusion from formal backward diffusion. The heat equation uses $\Phi (r)=r^2/2$, whereas total variation flow uses $\Phi (r)=r$ with a subgradient interpretation at zero.

Diffusion filtering evolves an observed image signal using a diffusion equation. The heat equation gives Gaussian image smoothing, while gradient-dependent or tensor-dependent diffusion can reduce transport across image edges. A image smoothing scale, stopping rule and appropriate boundary conditions are part of the filter.

For the heat equation with zero Dirichlet boundary conditions on $(0,L)$,It is also obtained by the method of images, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The eigenfunction expansion is useful at long times; the image expansion is useful at short times.

A dynamic boundary condition includes a time derivative of the boundary trace. For a half-line heat equation, $u_t(0,t)+\alpha u_x(0,t)+\beta u(0,t)=f(t)$ gives a separate boundary evolution coupled to the interior flux. Its Laplace transform contains the initial trace $u_0(0)$. The sign of $u_x$ differs from that of the outward normal derivative at a left endpoint.

Classical heat equation solutions with identical continuous initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\epsilon {\Phi }_b$, where ${\Phi }_b=(1-4bt{)}^{-n/2}\exp (b|x{|}^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The heat equation maximum principle bounds both signs of the difference by $\epsilon {\Phi }_b$ inside. Increasing the cylinder radius and then sending $\epsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.

For the heat equation on ${\mathbb{R}}^n$ with diffusion coefficient one, the heat kernel is $K_t(x)=(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}$. Its convolution with bounded initial data is smooth for positive time, solves the equation by differentiation under the integral, and returns continuous initial data locally uniformly through the approximate identity property. It is Gaussian filtering with per-coordinate variance $2t$.

For $x>0$ and the heat equation on a half-line, the zero-initial-data solution with prescribed Dirichlet boundary condition $g$ is ${\int }_0^{t}P(x,t-s)g(s)ds$, whereIts mass concentrates at $t=0$ as $x\downarrow 0$, recovering the boundary trace.

An ill-posed problem fails existence, uniqueness or continuous dependence in the specified function spaces. Backward heat equation evolution is unstable because its Fourier modes grow like $e^{t|\xi {|}^2}$, making arbitrarily small high-frequency errors grow without a uniform bound.

Image smoothing suppresses rapid fluctuations attributed to noise. Linear heat equation image smoothing attenuates Fourier frequencies by $e^{-t|\xi {|}^2}$ but also blurs image edges. Nonlinear filtering uses image signal geometry to distinguish fluctuations from meaningful transitions.

In thermal-time units, taking the Prandtl number large removes momentum inertia at leading order. The Stokes flow momentum equation then determines velocity instantaneously from the temperature perturbation, while the nonlinear heat equation retains its time derivative. Small Prandtl number does not generally give this model. An exact contrasting shear in a free-slip layer is $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$: it decays slowly as the Prandtl number tends to zero but is excluded by the instantaneous Stokes equation.

For $\mathcal{C}\gg 1$ and latent-to-sensible heat ratio $S\gg 1$ with $S/\mathcal{C}=O(1)$, put $\Omega =1+S/(r\mathcal{C})$, $\zeta =Vz/\kappa$, and $H=Vh/\kappa$. The steady heat equation becomes ${\theta }^{\prime \prime }+r\Omega {\theta }^{\prime }=0$ in the mush and ${\theta }^{\prime \prime }+r{\theta }^{\prime }=0$ in the liquid. Matching temperature and heat flux at $\zeta =H$, with $\theta (0)=-1$ and liquid temperature ${\theta }_{\infty }>0$ far above, givesThe latent heat modifies the mush's effective thermal decay rate and therefore its thickness.

For the heat equation $u_t=\frac{1}{2}{u}_{xx}$ on $x>0$ with the Neumann boundary condition $u_x(t,0)=0$, the method of images givesThus $u(t,x)={\int }_0^{\infty }{K}_t^N(x,y)f(y)dy$. This is also the transition kernel of Reflected Brownian motion.

The Neumann solution places a planar phase boundary at $h(t)=2\lambda \sqrt{\kappa t}$. Similarity reduction of the heat equation gives temperature profiles in terms of the error function, while the Stefan condition determines $\lambda$ implicitly.

For a smooth solution of the heat equation, backward Euler with the centered second difference has residual divided by $k$ equal to $-(k/2+d^2/12)u_{xxxx}+O(k^2+k{d}^2+d^4)$. Both leading terms have the same sign. Under parabolic mesh refinement their coefficient is $-d^2(r/2+1/12)$, which cannot vanish for a positive diffusion Courant number. Consistency of an update $U^{n+1}-U^n=\alpha (r){\delta }^2{U}^{n+1}$ forces $\alpha (r)=r$ for every fixed positive $r$.