# Partial Differential Equations

$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% %%%% This text file is part of the source of %%%% `Introduction to High-Performance Scientific Computing' %%%% by Victor Eijkhout, copyright 2012-2020 %%%% %%%% mathjax.tex : macros to facility mathjax use in html version %%%% %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand\inv{^{-1}}\newcommand\invt{^{-t}} \newcommand\bbP{\mathbb{P}} \newcommand\bbR{\mathbb{R}} \newcommand\defined{ \mathrel{\lower 5pt \hbox{{\equiv\atop\mathrm{\scriptstyle D}}}}} \newcommand\macro[1]{\langle#1\rangle} \newcommand\dtdxx{\frac{\alpha\Delta t}{\Delta x^2}}$ 15.1 : Partial derivatives
15.2 : Poisson or Laplace Equation
15.3 : Heat Equation

# 15 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.

## 15.1 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 derivates} indicate the rate of change if only one variable changes and the other stays constant.

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

## 15.2 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 $\frac{\delta Q}{\delta t}=c\Delta x\frac{\delta u}{\delta t}=0$ 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 $0= \left.\frac{\delta u}{\delta x}\right|_{x+\Delta x}- \left.\frac{\delta u}{\delta x}\right|_{x}$ 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 .

## 15.3 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 $\frac{\delta Q}{\delta t}=c\Delta x\frac{\delta u}{\delta t}$ 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 $\frac{\delta Q}{\delta t}= \left.\frac{\delta u}{\delta x}\right|_{x+\Delta x}- \left.\frac{\delta u}{\delta x}\right|_{x}$ 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: $\lim_{t\rightarrow\infty} u(x,t)=u_{\mathrm{steady state}}(x).$ This solution satisfies a BVP , which can be found by setting $u_t\equiv\nobreak0$. For instance, for the heat equation $u_t=u_{xx}+q(x)$ the steady state solution satisfies $-u_{xx}=q(x)$.