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15.1 : Partial derivatives

15.2 : Poisson or Laplace Equation

15.3 : Heat Equation

15.4 : Steady state

Back to Table of Contents# 15 Partial Differential Equations

## 15.1 Partial derivatives

## 15.2 Poisson or Laplace Equation

## 15.3 Heat Equation

## 15.4 Steady state

Back to Table of Contents
15.2 : Poisson or Laplace Equation

15.3 : Heat Equation

15.4 : Steady state

Back to Table of Contents

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.

crumb trail: > pde > Partial derivatives

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: \[ u_x(x,y) = \lim_{h\rightarrow0}\frac{u(x+h,y)-u(x,y)}h,\quad u_y(x,y) = \lim_{h\rightarrow0}\frac{u(x,y+h)-u(x,y)}h \]

crumb trail: > pde > Poisson or Laplace Equation

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*
.

crumb trail: > pde > Heat Equation

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 $u_t=\alpha u_{xx}$.

crumb trail: > pde > Steady state

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 $-u_{xx}=q(x)$.