1.1 Basic Concepts

1.1.1 The prevalence of partial differential equations

Partial differential equations are equations involving derivatives with respect to more than one independent variable.

Partial differential equations are the basic equations in many areas of science and engineering. Some examples:

Fluid mechanics

The basic equations that govern the inviscid flow of simple idealized substances are called the Euler equations. The basic equations that govern the viscous flow of simple substances like air and water under normal conditions are called the Navier-Stokes equations. Both are partial differential equations. Nonlinear ones, unfortunately. However, they do become linear in many special cases of great interest.

Heat transfer

The equations of heat conduction and convection are partial differential equations. Radiation may be described by Maxwell’s equations, which are also partial differential equations. However, often radiation can be more simply described by so-called boundary integral methods. In the most basic cases, the partial differential equations describing convection are linear.

Solid mechanics

The equations that govern simple solids are partial differential equations. Often the interest is in steady problems. The equations for simple relatively stiff solids are linear.

Dynamics

The dynamics of flexible solids is governed by partial differential equations. The equations for simple relatively stiff solids are linear.

Electromagnetics and optics

The Maxwell equations that govern basic electromagnetic phenomena are partial differential equations. They are linear in vacuum. Simplified partial differential equations govern the special cases of electrostatics and magnetostatics.

Geometry

Many geometrical issues such as minimal surfaces and developable surfaces are governed by partial differential equations. The equation for minimal surfaces may be reduced to a very simple linear one.

That did not even touch on such areas as biology and economics that are also awash in partial differential equations.

1.1.2 Definitions

Here is a list of some basic definitions used for partial differential equations:

Partial differential equations

are equations that involve derivatives with respect to more than one independent variable. The simplest partial differential equation that you can write is:

$$u_t = u_x$$

In that case $u(x,t)$ would be unknown, or dependent variable. That is the function you want to find. The independent variables would be $x$ and $t$. People usually think of a spatial coordinate when they use $x$, and time when they use $t$.

It may be noted that equations that involve only derivatives with respect to one independent variable are called “ordinary differential equations.” Equations that also involve integrals are called “integro-differential equations.”

Partial derivatives

are derivatives with respect to one independent variable, keeping the other variables constant. The figure below illustrates the definition of partial derivatives:

$$\left(u_x\right)_P \equiv \left(\frac{\partial u}{\partial... ...ight)_P = \lim_{\Delta t \to 0} \frac{u_R - u_P}{\Delta t}$$

A simple numerical approximation of a partial derivative will just eliminate the limit process. In that case $\Delta{x}$ and $\Delta{t}$ are merely taken to be small compared to the typical scale of the problem.

Order

The order of a partial differential equation is the order of the highest derivative. Generally speaking, the highest derivatives are most responsible for the nature of the solutions.

Degree

The degree of a partial differential equation is the highest power to which the dependent variable appears in the equation.

Consider for example the Burgers’ equation

$$u_t + u u_x = 0$$

It is the simplest nonlinear model for the equations of fluid mechanics, and for other systems involving shock formation. It is of order 1 and degree 2.

Linear

partial differential equations are equations of first degree. The terms that are of the first degree in the unknown are called the homogeneous part. The remaining terms that do not involve the unknown are called the inhomogeneous part.

Consider the following Poisson equation governing steady heat conduction in a plate with external heat addition:

$$- \kappa (u_{xx} + u_{yy}) = x^2y^3 +\sin(x)\cos(y)$$

Here $u$ is the temperature, $\kappa$ the heat conduction coefficient, and the right hand side represents the external heat added per unit area. This equation is first degree (in $u$.) The left hand side, linear in $u$ is the homogeneous part and the right hand side, independent of $u$, is the inhomogenous part.

The following equation describes steady heat conduction in a plate without external heat addition, but with a heat conduction coefficient that depends on temperature:

$$\kappa(u) (u_{xx}+u_{yy}) + \kappa'(u) u_x^2 + \kappa'(u) u_x^2 = 0$$

This equation is nonlinear. It is of infinite degree, not second degree, because the Taylor series of $\kappa$ intruduces all powers of $u$. However, this equation is still linear in terms of the highest, second order, derivatives $u_{xx}$ and $u_{yy}$. (Not counting lower order derivatives.) Therefore it is called “quasi-linear.”

The domain $\Omega$

is usually taken to be the spatial region in which the partial differential equation applies. For unsteady heat conduction in a sphere, the domain $\Omega$ is the sphere and time is an additional coordinate.

The boundary $\delta \Omega$

is where the domain stops. Boundary points are immediately adjacent to both points inside the domain and points outside it. The boundary is also often indicated by $S$ (for surface). Typically, $\Omega$ indicates the domain without the boundary points and $\overline{\Omega}$ the domain including the boundary points.

For unsteady heat conduction in a sphere of radius $a$, the boundary is all points with spherical coordinate $r$ equal to $a$. It is the surface of the sphere. For this example $\Omega$ is all points $r<a$ and $\overline{\Omega}$ is all points $r\le{a}$.

Boundary conditions

are conditions on the solution that apply at points on the boundary.

Initial conditions

are conditions on the solution that apply at the starting time. Almost always, the starting time is taken to be the zero of time.

Singularities

You cannot do much in partial differential equations without having to deal with singularities. There might be singularities in the solution, in the initial and boundary conditions, or in the shape of the boundary. Some typical singularities, in a rough order from relatively mild to more severe, are

• Locations where all derivatives exist, but the function does not have a Taylor series with a nonzero radius of convergence. For example, that happens for the function $e^{-1/x^2}$ at $x=0$.
• Locations where higher order derivatives have singularities. For example, the function $\vert x\vert^{3/2}$ has an infinite second order derivative at $x$=0. That means it has infinite radius of curvature at $x=0$. In general, the lower the order of the derivative involved, the stronger the singularity.
• Kinks: Locations where the first derivative jumps from one finite value to another. For example, the function $\vert x\vert=\sqrt{x^2}$ has a kink at $x=0$.
• Cusps: where the function itself is still continuous, but the derivative jumps from $-\infty$ to $\infty$ or vice-versa. For example, $\sqrt[3]{x^2}$ has a cusp at $x=0$
• Jumps: where the function jumps from one finite value to another. For example, the Heaviside function $H(x)$, which is 0 for negative $x$ and 1 for positive $x$, has a jump at $x=0$.
• Spikes: where the function is infinite at a single point. An important example is the Dirac delta function $\delta(x)$, which is the derivative of the Heaviside function. The delta function is a single infinite spike at $x=0$ and the area under the spike is 1. If you take derivatives of the delta function, you get increasingly singular functions. For example, the first derivative is the more singular “dipole.”
• Poles: where the function goes to infinity. For example, the function $1/x$ has a “simple pole” at $x=0$. The higher the negative power of $x$, the more singular the function. For example $1/x^2$ is more singular than $1/x$.

1.1.3 Typical boundary conditions

To get a meaningful solution to a partial differential equation, you will need initial and/or boundary conditions. Initial conditions are normally straightforward. Boundary conditions vary a lot, however.

There are some very simple types of boundary condition that you must know by heart:

Dirichlet

The value of the unknown $u$ itself is given on the boundary:

$$u = f \qquad\mbox{on}\qquad \delta \Omega$$

Here $f$ is some given function of the position on the surface.

Neumann

The derivative $\partial{u}/\partial{n}$ of the unknown in the direction normal to the boundary is given. Here $n$ is the coordinate normal to the boundary:

$$\frac{\partial u}{\partial n} = f \qquad\mbox{on}\qquad \delta \Omega$$

If the boundary is oblique, the following formula can be used to relate the derivative in the direction normal to the boundary to the partial derivatives of $u$ in Cartesian coordinates:

$$\frac{\partial u}{\partial n} = \vec n \cdot \nabla u \q... ...partial}{\partial y} + {\hat k}\frac{\partial}{\partial z}$$

Here $\vec{n}$ is a unit vector that is normal to the boundary at the considered point.

Note that if $u$ itself is prescribed on the boundary, it already implies the value of derivative along the boundary. That is why only the derivative normal to the boundary is included in this list of the most basic boundary conditions.

Mixed

A linear combination of $u$ and $\partial{u}/\partial{n}$ is given on the boundary. This is also called a “radiation” or “Robin” boundary condition.

$$\alpha u + \beta \frac{\partial u}{\partial n} = f \qquad\mbox{on}\qquad \delta \Omega$$

Here $\alpha$, $\beta$, and $f$ are all given functions of the position on the surface. Such boundary conditions are often used in simple wave propagation problems to indicate that no waves enter the domain from outside.