Exercise 1: Classification of partial differential equations
Date of exercise: 8th May 2019
Last update: 16th May 2019 by Lukas Babor
1.1 Classification of PDEs
Consider an example of a partial differential equation
The aim of this chapter is to introduce a classification which would describe the properties of the equation and its solution.
Basic classification
- Order
- Number of independent variables
- Linearity
- Constant/variable coefficients
- Homogenity
With this criteria we can describe our example as: Linear, non-homogeneous ( 
) , second-order PDE with d independent variables and constant coefficients ( 
).
) , second-order PDE with d independent variables and constant coefficients ( ).In addition, second order PDEs and some systems of PDEs can be divided into three types: elliptic, parabolic and hyperbolic. The type of equation determines certain properties of the solution and it imposes restrictions on boundary conditions and discretization methods which can be used to solve it numerically. The next section illustrates each type on examples. It is followed by some methods to determine the type of an equation.
1.2 Properties of different types of PDEs
Elliptic
- Boundary conditions must be prescribed all along the boundary
- Local change of the boundary data influences the solution everywhere
- No propagation of waves
- No real characteristics
Parabolic
- Parabolic equations arise by adding time dependence
to equation which was originally elliptic. - Waves can propagate in one specific direction only, corresponding to time-like variable
- Equivalent system of n first-order equations has 1 to
real characteristics - Boundary conditions are required in space, initial condition in time
- Local change in boundary conditions affects immediately the solution at all positions in space, but only in future times
Hyperbolic
- Waves can reflect from boundaries
- Equivalent system of n first-order equations has n real characteristics.
- Boundary conditions are required in space, initial condition in time
- Information about local change of boundary condition travels through space with finite speed.
Types of boundary conditions
- Dirichlet:
- Newton:
- Robin:
- periodic:
1.3 Methods to determine the type of PDE
Second order PDE
Second order PDEs describe a wide range of physical phenomena including fluid dynamics and heat transfer. It is convenient to classify them in terms of the coefficients multiplying the derivatives. Replacing 
by 
we can write the characteristic equation of the left hand side as
by we can write the characteristic equation of the left hand side as
The PDE is:
- elliptic if the matrix
is (positive or negative) definite, i.e. its eigenvalues are non-zero and all have the same sign. - hyperbolic if the matrix has non-zero eigenvalues and all but one have the same sign.
- parabolic if the matrix is semi-definite (i.e. exactly one eigenvalue is zero, while the others have the same sign) and
Second order PDE with two independent variables
Note that in case of two independent variables we can write the second-order terms as
The equation can then be classified as
- elliptic for
- hyperbolic for
- parabolic for
Remark: In case of variable coefficients, the equation can change its type in different parts of the domain.
Remark: In more than two independent variables, it is often useful to determine how the solution behaves on a certain two-dimensional sub-space. One can "freeze" 
independent variables on some fixed values (i.e. remove derivatives with respect to the "frozen" variables) and determine the type of the resulting reduced equation with two independent variables.
independent variables on some fixed values (i.e. remove derivatives with respect to the "frozen" variables) and determine the type of the resulting reduced equation with two independent variables. Systems of first-order PDEs with two independent variables
The general form can be written in Einstein's summation notation,
or in matrix-vector form,
where 
is the vector of unknowns. The characteristics can be computed from the matrices 
and 
by solving
is the vector of unknowns. The characteristics can be computed from the matrices and by solving
for either 
or 
. We can conclude that the PDE is
or . We can conclude that the PDE is- elliptic for no real solutions
- hyperbolic for n real solutions
- parabolic for 1 to real and no complex solutions
where n is the number of equations. Note that there are cases at which this method cannot determine the type of the system.
Systems of PDEs with three (or more) independent variables
The characteristics are always one dimension lower than the number of independent variables. That means that with three independent variables the characteristics are surfaces. The solution of the equation
now provides the normal vectors 
. One can divide the above equation by 
, 
or 
, provided it does not destroy any solutions. The classification is then the same as in the case of two independent variables.
. One can divide the above equation by , or , provided it does not destroy any solutions. The classification is then the same as in the case of two independent variables.Higher order PDEs
It is always possible to transform a higer order equation to a system of first-order equations by introducing auxiliary dependent variables. The methods above can then be used to investigate the type of the equation.
Attention: Transformation to the system of first-order equation can create spurious solutions - the solution of the original higher-order equation solves the first-order system, but not all solutions of the first-order system solve the original higher-order equation. Therefore, the type of the system can differ from the original equation. Second order PDEs should therefore be prefferentially analysed in terms of the coefficient matrix.
1.4 Method of characteristics
Characteristics of an equation with d independent variables are 
dimensional objects (curves for 
, surfaces for 
), such that the propagation of the solution along these objects can be described by ODE (i.e. partial derivatives can be replaced by total differentials). The number of real characteristics determines the type of an equation.
dimensional objects (curves for , surfaces for ), such that the propagation of the solution along these objects can be described by ODE (i.e. partial derivatives can be replaced by total differentials). The number of real characteristics determines the type of an equation. First order equations
Consider a first-order PDE with two independent variables
which we want to replace by an ODE
where the total derivative is defined as
on the characteristic curve 
. Dividing the original PDE by a we obtain the same form as the definition of the total derivative
. Dividing the original PDE by a we obtain the same form as the definition of the total derivative
thus obtaining 
and 
. After integration we obtain the characteristic
and . After integration we obtain the characteristic
along which the solution evolves as
The extension of this example to the case of multiple independent variables should be evident (see Appendix A). Finding the characteristics of higher order equations or systems of equations follows similar arguments.
Second order equations
Let us now consider the case of second-order PDE with two independent variables.
First we introduce auxiliary unknowns 
, 
with total differentials
, with total differentials
Note that this corresponds to converting the second-order equation into a system of first-order equations. Next we want to rewrite the original equation in terms of total derivatives 
. We try to expand
. We try to expand
and compare it to the original equation. We obtain
which after multiplication by or , leads to
or , leads to
thus giving us the criteria
no real characteristics → elliptic
one real characteristic → parabolic
two real characteristics → hyperbolic
Systems of first order equations with two independent variables
Now we want to represent the linear combination of all lines of the system
with linear combination of total differentials
Matching all partial derivatives we obtain
Multiplying the first set of equations by 
and the second set by 
and subtracting them leads to
and the second set by and subtracting them leads to
and from the solvability condition we obtain
which can be solved for or . Integration then leads to characteristics 
or 
.
or . Integration then leads to characteristics or .1.5 Examples
Second order PDE: Time-dependent diffusion
Considering one spatial variable 
and taking time as the second independent variable 
, we can rewrite the equation as
and taking time as the second independent variable , we can rewrite the equation as
with
a = [ 1 0
0 0 ];
b = [ 0
1 ];
eigs(a)
rank([a b])
As one eigenvalue is zero and the rank of the composite matrix 
equals the number of independent variables, we conclude that the equation is parabolic.
equals the number of independent variables, we conclude that the equation is parabolic.Task: Repeat this example for higher number of spatial dimensions.
Recommended literature
Fletcher, C. (1998), 'Partial Differential Equations' In Computational Techniques for Fluid Dynamics 1: Fundamental and General Techniques, Springer, Berlin.
Appendix A: Characteristics of first order PDE with multiple independent variables
Consider a generic first-order PDE
which we want to replace by an ODE, e.g.
where the total derivative is defined as
Rearranging the original PDE to the form
we obtain 
. Note that in case of 
the characteristic is a curve evolving in time.
. Note that in case of the characteristic is a curve evolving in time.