Partial Differential Equations

$\newcommand\inv{^{-1}}\newcommand\invt{^{-t}} \newcommand\bbP{\mathbb{P}} \newcommand\bbR{\mathbb{R}} \newcommand\defined{ \mathrel{\lower 5pt \hbox{{\equiv\atop\mathrm{D}}}}}$ 18.1 : Partial derivatives
18.2 : Poisson or Laplace Equation
18.3 : Heat Equation

18 Partial Differential Equations

Partial Differential Equations are the source of a large fraction of HPC problems. Here is a quick derivation of two of the most important ones.

18.1 Partial derivatives

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Derivatives of a function $u(x)$ are a measure of the rate of change. Partial derivatives to the same, but for a function $u(x,y)$ of two variables. Notated $u_x$ and $u_y$, these \indexterm{partial derivates} indicate the rate of change if only one variable changes and the other stays constant.

Formally, we define $u_x,u_y$ by: \begin{equation} u_x(x,y) = \lim_{h\rightarrow0}\frac{u(x+h,y)-u(x,y)}h,\quad u_y(x,y) = \lim_{h\rightarrow0}\frac{u(x,y+h)-u(x,y)}h \end{equation}

18.2 Poisson or Laplace Equation

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Let $T$ be the temperature of a material, then its heat energy is proportional to it. A segment of length $\Delta x$ has heat energy $Q=c\Delta x\cdot u$. If the heat energy in that segment is constant \begin{equation} \frac{\delta Q}{\delta t}=c\Delta x\frac{\delta u}{\delta t}=0 \end{equation} but it is also the difference between inflow and outflow of the segment. Since flow is proportional to temperature differences, that is, to $u_x$, we see that also \begin{equation} 0= \left.\frac{\delta u}{\delta x}\right|_{x+\Delta x}- \left.\frac{\delta u}{\delta x}\right|_{x} \end{equation} In the limit of $\Delta x\downarrow0$ this gives $u_{xx}=0$, which is called the Laplace equation . If we have a source term, for instance corresponding to externally applied heat, the equation becomes $u_{xx}=f$, which is called the Poisson equation .

18.3 Heat Equation

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Let $T$ be the temperature of a material, then its heat energy is proportional to it. A segment of length $\Delta x$ has heat energy $Q=c\Delta x\cdot u$. The rate of change in heat energy in that segment is \begin{equation} \frac{\delta Q}{\delta t}=c\Delta x\frac{\delta u}{\delta t} \end{equation} but it is also the difference between inflow and outflow of the segment. Since flow is proportional to temperature differences, that is, to $u_x$, we see that also \begin{equation} \frac{\delta Q}{\delta t}= \left.\frac{\delta u}{\delta x}\right|_{x+\Delta x}- \left.\frac{\delta u}{\delta x}\right|_{x} \end{equation} In the limit of $\Delta x\downarrow0$ this gives $u_t=\alpha u_{xx}$.

The solution of an IBVP is a function $u(x,t)$. In cases where the forcing function and the boundary conditions do not depend on time, the solution will converge in time, to a function called the steady state solution: \begin{equation} \lim_{t\rightarrow\infty} u(x,t)=u_{\mathrm{steady state}}(x). \end{equation} This solution satisfies a BVP , which can be found by setting $u_t\equiv0$. For instance, for the heat equation \begin{equation} u_t=u_{xx}+q(x) \end{equation} the steady state solution satisfies $-u_{xx}=q(x)$.