Introduction

Classification

Order of PDE

The order is defined to be the order of the highest derivative in the equation. If the highest derivative is of order $k$, then the equation is said to be of order $k$.

A first-order partial differential equation involves two or more independent variables (say, $x$ and $y$ ) and a function $u(x, y)$ and its first partial derivatives, $u_{x}$ and $u_{y}$. Recall that the subscripts ' $x$ ' and ' $y$ ' denote partial differentiation with respect to $x$ and $y$, respectively. For example, Eq. (2.2) is a first-order equation:

Our objective is to find the most general form of a differentiable function, $u(x, y)$ which satisfies the PDE in some region of interest of the $x, y$ variables. This is the most general form of the solution which normally contains an arbitrary function. Compare this with the general solution of a first-order ordinary differential equation (ODE) which contains an arbitrary constant.

A second-order partial differential equation involves two or more independent variables (say, $x$ and $y$ ) and a function $u(x, y)$, its first partial derivatives, $u_{x}$ and $u_{y}$ and second partial derivatives, $u_{x x}, u_{x y}$ and $u_{y y}$. Equation (2.3) is a second-order equation:

The general solution to Eq. (2.3) contains two arbitrary functions.

Linear & nonlinear PDEs

An equation is called linear if in Eq. (2.1), $F$ is a linear function of the unknown function $u$ and its derivatives. For example, the equation for $u(x, y)$,

is a linear equation, while the equation,

is a nonlinear equation. Nonlinear equations are often further classified according to the type of the nonlinearity. The following two PDEs are both nonlinear:

While Eq. (2.7) is nonlinear, it is said to be linear as a function of the highest-order derivative. Such a nonlinearity is called quasilinear. In Eq. (2.6), the nonlinearity is only in the unknown function, $u$. Such equations are often called semilinear.

Systems of equations

A single PDE with just one unknown function is called a scalar equation. In contrast, a set of $m$ equations with $l$ unknown functions is called a system of $m$ equations (like, for example, Eqs. (2.16) defined below).

Examples in mathematical modelling

Before we start our discussion of solution techniques for PDEs, we give a short list of examples which occur in physical applications.

Equations representing transport phenomena can take the forms:

Such equations can be used to model a substance (mass or energy) flowing in a region of space. Both equations in (2.8) are first-order and linear.

The nonlinear transport equation or shock-wave equation is:

Such equations play a crucial role in the modelling of gas dynamics, traffic flow, chromatography and chemical reactions (see also Subsec. 2.2.4). Equation (2.9) is also known as the Riemann equation or the (inviscid) Burgers' equation.

The one-dimensional wave equation is given by:

where $c$ is a constant. Equation (2.10) may be used to describe sound waves in a pipe or vibrations of a string. The two-dimensional version is expressed as follows,

The diffusion equation

is a conservation equation used to model heat or mass transfer. Equation (2.12) is the $1 \mathrm{D}$ version of the diffusion equation and it is second-order and linear. The $2 \mathrm{D}$ version is given by:

In the absence of temporal effects, i.e when $u_{t}=0$, Eq. (2.13) reduces to Laplace's equation,

whose solutions are described by harmonic functions.

The Shrödinger equation is a linear PDE:

where $\Psi$ is the complex wave function.

As a final example, consider the Navier-Stokes equations for a vector $\boldsymbol{u}$ where $\boldsymbol{u}=(u, v, w)$ are the $x-, y-$ and $z$ - components of the vector. You will be introduced to these PDEs in Fluid Mechanics II. The Navier-Stokes equations are Newton's second law of motion for a fluid, based on the conservation of momentum. The equations are given by:

where $\boldsymbol{f}$ models external forces. Equation (2.16) is considered a system of PDEs since $\boldsymbol{u}$ is a vector and they are used to model the motion of a Newtonian fluid. Note that we have 4 unknown quantities in (2.16): these are the 3 scalar velocities (i.e. $u, v$ and $w$ ) and the pressure, $P$. We therefore need an additional equation: this is given by the continuity equation. In the case of an incompressible fluid, i.e. the density, $\rho$ is constant, the continuity equation takes the form:

Prerequisites

To understand PDEs, we need to know all the basic facts about partial derivatives and multiple integrals. These include use of the chain rule, directional derivatives, change of variables in a double integral, Green's theorem and the divergence theorem for the integrals of derivatives. Further, knowledge of Fourier series expansions is needed for the solution of some linear PDES. It is useful to keep the following in mind when dealing with PDEs:

• We assume that all derivatives exist and are continuous. Hence, mixed derivatives are equal, e.g. $u_{x y}=u_{y x}$.
• Derivatives are local. To calculate the derivative $u_{x}(x, y)$ at $x=x_{0}$, $y=y_{0}$ we just need to know the values of $u\left(x, y_{0}\right)$ for $x$ near $x_{0}$ since the derivative is given by the limit $x \rightarrow x_{0}$. Finally, in solving certain classes of PDEs, we often reduce them to ODEs. A review of ODEs covered in Year I Mathematics is recommended prior to studying PDEs.