Linear and Quasi-Linear PDEs
First-order PDEs are usually classified as linear, quasi-linear, or nonlinear. The first two types are discussed in this tutorial.
A first-order PDE for an unknown function is said to be linear if it can be expressed in the form
The PDE is said to be quasilinear if it can be expressed in the form
A PDE which is neither linear nor quasi-linear is said to be nonlinear.
For convenience, the symbols , , and are used throughout this tutorial to denote the unknown function and its partial derivatives.
Here is a linear homogeneous first-order PDE with constant coefficients.
The equation is linear because the left-hand side is a linear polynomial in , , and . Since there is no term free of , , or , the PDE is also homogeneous.
As mentioned earlier, the general solution contains an arbitrary function C
of the argument
This verifies that the solution is correct.
Particular solutions of the homogeneous PDE are obtained by specifying the function C
Here is a plot of the surface for this particular solution.
The transport equation is a good example of a linear first-order homogeneous PDE with constant coefficients.
In this transport equation,
Note that the solution to the transport equation is constant on any straight line of the form in the plane. These straight lines are called the base characteristic curves. The equation defines a plane in three dimensions. The intersections of these planes with the solution surface are called characteristic curves. Since the characteristic curves are solutions to a system of ODEs, the problem of solving the PDE is reduced to that of solving a system of ODEs for , , and , where is a parameter along the characteristic curves. These ODEs are called characteristic ODEs.
The solution to an inhomogeneous PDE has two components: the general solution to the homogeneous PDE and a particular solution to the inhomogeneous PDE.
This is a linear inhomogeneous PDE of the first order.
The first part of the solution,
, is the particular solution to the inhomogeneous PDE. The rest of the solution is the general solution to the homogeneous equation.
Here is a linear homogeneous PDE with variable coefficients.
This verifies the solution.
Here is a linear inhomogeneous PDE with variable coefficients.
The solution is once again composed of the general solution to the homogeneous PDE and a particular solution, Sin[x]
, to the inhomogeneous PDE.
Now consider some examples of first-order quasi-linear PDEs.
This PDE is quasi-linear because of the term
on the right-hand side.
This verifies the solution.
It can be written using the notation introduced earlier.
The term makes this equation quasi-linear.
This solves the equation.
This verifies the solution to Burgers' equation.
A practical consequence of quasi-linearity is the appearance of shocks and steepening and breaking of solutions. Thus, although the procedures for finding general solutions to linear and quasi-linear PDEs are quite similar, there are sharp differences in the nature of the solutions.