Physical events $\Phi $ generally take place in the three space dimensions $x,y,z$ and the time $t$: $\Phi =\Phi \left(x,y,z,t\right)$. The spatial- and time development are coupled to each other. The differential equations describing the phenomena then contain partial derivatives with respect to the space coordinates and with respect to time, and therefore are referred to as partial differential equations. The functional relationship for a general partial differential equation of second order for a physical quantity $\Phi \left(x,y,z,y\right)$ reads:

$$\begin{array}{c}F\left(\frac{{\partial}^{2}\Phi}{\partial {\alpha}^{2}},\frac{\partial {\Phi}^{2}}{\partial \alpha \partial \beta},\frac{\partial \Phi}{\partial \alpha},\Phi ,x,y,z,t\right)=0\hfill \\ \phantom{\rule{0em}{0ex}}\text{with}\alpha =x\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}y\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}z\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}t\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{and}\beta =x\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}y\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}z\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{or}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}t,\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\beta \ne \alpha \hfill \\ \hfill \end{array}$$

To keep the presentation readable, we have not shown all terms in the brackets, but only one of each type. This type is characterized by one of the variables or, in the case of mixed derivatives by two of them. Thus, in addition to first and second partial derivatives also all mixed second derivatives can appear.

Fortunately the partial differential equations important in physics and engineering are much simpler, then this general form, as the following examples will show. They are however still rather complicated and only allow an analytic solution and simple interpretation in very elementary cases. In the following we want to only cite a few important partial differential equations of physics and want to make you aware of the crucial differences between the boundary value problem / initial value problem for ordinary and for partial differential equations. For further information we refer to the specialist literature.

The simulation examples show special solutions of the corresponding one-dimensional

- diffusion equation for point-like initial impulse (delta impulse),
- Schrödinger equation for a point mass and for different oscillators,
- wave equation for a vibrating string.

$$\begin{array}{c}\text{a)waveequation}\hfill \\ \Phi \left(x,y,z,t\right)\text{describesthedeviationofthephysicalquantityattime}t\text{,}\hfill \\ \text{forexampleofthefieldstrength,thepressureandsoon}\hfill \\ \text{thedifferentialequationreads}\frac{{\partial}^{2}\Phi}{d{\left(ct\right)}^{2}}=\frac{{\partial}^{2}\Phi}{d{x}^{2}}+\frac{{\partial}^{2}\Phi}{d{y}^{2}}+\frac{{\partial}^{2}\Phi}{d{z}^{2}},\text{}\hfill \\ \text{andinonedimension:}\frac{{\partial}^{2}\Phi}{d{t}^{2}}={c}^{2}\frac{{\partial}^{2}\Phi}{d{x}^{2}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \phantom{\rule{0em}{0ex}}\text{thegeneralsolutionsisthen}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\Phi \left(x,t\right)=f\left(x+ct\right)+g\left(x-ct\right).\hfill \\ \hfill \end{array}$$

The very general one-dimensional solution of the wave equation contains two arbitrary functions $f\left(x\right)$ and $g\left(x\right)$, that are propagated along the $x$-axis with velocity $c$ in negative/positive direction without changing their form. This example already demonstrates an important difference to solutions of ordinary differential equations of second order: While there a solution was determined via two initial values ${y}_{0},{y}_{0}^{\prime}$, is now fixed via two initial functions $g(x,0$ and $f\left(x,0\right)$. The ordinary differential equation of second order has as solution a family of functions with two arbitrary number parameters. The solution of this partial differential equation is described via a family of functions with two initial functions. For the one-dimensional case those are defined along the $x$-axis, for example a wave packet, in the simplest case a sine wave of undetermined position or a Gaussian impulse.

In the three-dimensional case initial functions can be defined on a boundary, a surrounding surface or in the volume. Heat-eq

$$\begin{array}{c}\text{b)one-dimensionalheatconductionequation}\hfill \\ \text{thefield}\Phi \left(x,t\right)\text{isherethetemperaturethatdependsonspaceandtimecoordinates}\hfill \\ \text{thedifferentialequationreads}\frac{\partial \Phi}{\partial t}=a\frac{{\partial}^{2}\Phi}{\partial {x}^{2}}\hfill \\ \text{Foritsanalyticalsolutionforadelta-pulseasinitialfunctiononeobtains}\hfill \\ K\left(x,t\right)=\frac{1}{\sqrt{4\pi at}}{e}^{-\frac{{x}^{2}}{4at}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{(seealsosection10.1)}\hfill \\ \hfill \end{array}$$

The heat conduction equation, also called diffusion equation, describes equilibration processes in time (here along a line, the $x$-axis). The special solution $K\left(x,t\right)$ in the example starts with the delta impulse as initial function. That means: the total heat is first concentrated in the point $x=0$. This amount is then spread over time as Gaussian distribution, while the integral (the amount of heat) stays the same. Thus the temperature maximum at $x=0$ decreases accordingly. Schrödinger

$$\begin{array}{c}\text{c)Schr\xf6dingerequation}\hfill \\ \text{Theprobabilityamplitudeorwavefunctionis}\psi \left(x,y,z,t\right);\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \phantom{\rule{0em}{0ex}}\text{andthepotentialis}\phantom{\rule{0em}{0ex}}V\left(x,y,z.t\right);\hfill \\ \frac{ih}{2\pi}\frac{\partial \psi}{dt}=-{\left(\frac{h}{2\pi}\right)}^{2}\frac{1}{2m}\left(\frac{{\partial}^{2}\psi}{d{x}^{2}}+\frac{{\partial}^{2}\psi}{d{y}^{2}}+\frac{{\partial}^{2}\psi}{d{z}^{2}}\right)+V\psi \hfill \\ \hfill \end{array}$$

The form of the Schrödinger equation given above is valid in the non-relativistic case for a particle of mass $m$ in a potential $V$. It describes the relationship between time and space development of its complex wave function $\psi $. Maxwell

$$\begin{array}{c}\text{d)Maxwell}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{equations}\text{fortheelectromagneticfields}\mathbf{E},\mathbf{D},\mathbf{B},\mathbf{H}\hfill \\ 1.\hfill \end{array})\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}div\mathbf{D}=\rho \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\nabla \cdot \mathbf{D}=\rho ,2.)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}div\phantom{\rule{0em}{0ex}}\mathbf{B}=0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\nabla \cdot \mathbf{B}=0,\hfill \\ 3.)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}curl\phantom{\rule{0em}{0ex}}\mathbf{E}+\frac{\partial \mathbf{B}}{dt}=\phantom{\rule{0em}{0ex}}0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{dt}=0,\hfill \\ 4.)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}curl\phantom{\rule{0em}{0ex}}\mathbf{H}=\mathbf{j}+\frac{\partial \mathbf{D}}{dt}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\nabla \times \mathbf{H}=\mathbf{j}+\frac{\partial \mathbf{D}}{dt}.\hfill \\ \hfill $$

The Maxwell equations, that are very important in practice describe the interaction between the magnetic and electric fields (2 and 3) and their connection with the charge density $\rho $ and current density $\mathbf{j}$. There the first equation means, that charges are the sources of the electric fields, from which field lines emanate and where they end. The second equation means, that magnetic sources (monopoles) do not exist and therefore magnetic field lines are always closed.

On the left the traditional notation and on the right the formally quite unified notation with the nabla operator are given.

The electrical flux density $\mathbf{D}$ is connected to the electrical field strength $\mathbf{E}$ via the material properties electrical permeability of the vacuum ${\epsilon}_{0}$ and electric polarization $\mathbf{P}$:

$\mathbf{D}={\epsilon}_{0}\mathbf{E}+\mathbf{P}$

The magnetic flux density $\mathbf{B}$ is connected to the magnetic field strength $\mathbf{H}$ via the material properties magnetic permeability of the vacuum ${\mu}_{0}$ and magnetic polarization $\mathbf{J}$ (written in caps as opposed to the current density $\mathbf{j}$):

$\mathbf{B}={\mu}_{0}\mathbf{H}+\mathbf{J}$

Since $\mathbf{D},\mathbf{B},\mathbf{E}$ and $\mathbf{H}$ are vectors, we have to deal with a system of coupled partial differential equations for all field components, which therefore has a wealth of solutions. Therefore the mathematical solution is also very complex.

Numerical solution methods are therefore even more important for partial differential equations than for ordinary differential equations. While one starts for ordinary differential equations from one or more initial values and iteratively proceeds from point to point for the independent variable, one has to cover the whole space of variables with a grid of computation points. For a two-dimensional problem one then deals with a plane grid and for a three-dimensional one with a three-dimensional space grid. One starts from one point of the initial function, calculates the neighboring points using suitable procedures, which together constitute the initial values for the next step, always while taking into account the connections provided by the differential equations. In technical applications and engineering one refers in this connection to the method of finite elements.

For the visualization one simplifies the conditions radically. Already in section 8.5.8 we had simulated the movement of an electron in a three-dimensional homogeneous electromagnetic field, that is stationary, i.e. is constant as a function of time.