1.1) Time dependent diffusion equation
a) Classification based on coefficients
The generic form of second order PDE reads
Let us select 
. Then the matrix 
and the vector 
for our case are given by
. Then the matrix and the vector for our case are given bysyms alpha
a = [ -alpha 0 0
0 -alpha 0
0 0 0 ];
b = [ 0
0
1 ];
The eigenvalues of
eig(a)
imply that the matrix is semi-definite, and the 
:
:rank([a b])
equals the number of independent variables. Thus we conclude that the PDE is parabolic.
b) Transformation to system of first-order equations
The original equation can be transformed into a system of first order equations
by introducing the auxiliary unknowns 
. In general we cannot say that the system is equivalent to the original equation, because the transformation can create spurious solutions. The solutions of (1) still solve (3), but not vice-versa. Therefore, the type of the first order system can also differ from the original equation. We will see that our example illustrates this problem. Let us write the system (3) in matrix form
. In general we cannot say that the system is equivalent to the original equation, because the transformation can create spurious solutions. The solutions of (1) still solve (3), but not vice-versa. Therefore, the type of the first order system can also differ from the original equation. We will see that our example illustrates this problem. Let us write the system (3) in matrix form
where 
and
andsyms alpha n_x n_y n_t
A = [ 1 0 0
0 0 0
0 0 0 ] ;
B = [ 0 -alpha 0
1 0 0
0 0 -1 ] ;
C = [ 0 0 -alpha
0 0 0
0 1 0 ] ;
We can analyze the system either with Fourier method, which analyzes whether solutions can have the form of a plane wave, or with the method of characteristics. Both methods lead to similar equations in the end. With the method of characteristics the normals to characteristic surfaces are given by
determinant = simplify( det( A'*n_t + B'*n_x + C'*n_y ) )
N_x = solve( determinant , n_x )
i.e. there is one real solution and two complex solutions. The real solution 
is typical for parabolic equations, since it represents the infinite speed of information spreading 
. The second component of this normal vector 
would be obtained by replacing the equation (3b) by
is typical for parabolic equations, since it represents the infinite speed of information spreading . The second component of this normal vector would be obtained by replacing the equation (3b) by
The two complex solutions 
most likely manifest the spurious solutions created by the non-equivalent transformation from (1) to (3). The system (3) is thus hybrid since it does not fall into any category. If we did not know the type of the original equation a priori, we would conclude that the original equation is not hyperbolic because it has for sure less than three real characteristics. One of the messages of this exercise is that second-order PDEs should be preferentially analysed by method a), unless it can be proved that the transformation to first-order system is equivalent. Two common cases where the transformation leads to equivalent first-order systems are considered in the next example. For more details refer to Wesseling (2001).
most likely manifest the spurious solutions created by the non-equivalent transformation from (1) to (3). The system (3) is thus hybrid since it does not fall into any category. If we did not know the type of the original equation a priori, we would conclude that the original equation is not hyperbolic because it has for sure less than three real characteristics. One of the messages of this exercise is that second-order PDEs should be preferentially analysed by method a), unless it can be proved that the transformation to first-order system is equivalent. Two common cases where the transformation leads to equivalent first-order systems are considered in the next example. For more details refer to Wesseling (2001).1.2)
a) Transient diffusion in one dimension
Neglecting the variations in y leads to a second order PDE with two independent variables
so we can determine the type from the discriminant
of the second order differential operator
which is in this case
Thus the equation (5) is parabolic. We can also convert it to a system of first order equations
which in matrix form reads
A = [ 1 0
0 0 ];
B = [ 0 -alpha
1 0 ];
and compute the characteristics by
syms dx dy dt
determinant = simplify( det( A'*dx - B'*dt ) )
Dt = solve( determinant , dt )
We receive the typical result for parabolic equations, , representing the infinite speed of information spreading.
, representing the infinite speed of information spreading.b) Steady diffusion in two dimensions
Neglecting the variation of ϕ in time (corresponding to a steady/equilibrium state) leads to
The discriminant
implies the equation (7) is elliptic.
Using the same auxiliary unknowns as in 1.1b) we can eliminate ϕ from the transformed system to obtain the Cauchy-Riemann system
which is equivalent to the Laplace equation (7). Computation of characteristics
A = [ 1 0
0 -1 ];
B = [ 0 1
1 0 ];
determinant = simplify( det( A'*dy - B'*dx ) )
Dx = solve(determinant, dx)
does not reveal any real solutions, so the system is elliptic as well.
As we saw in 1.1), with more than two independent variables the transformation of both elliptic and parabolic equations creates spurious solutions and the resulting system is therefore no longer equivalent. Thus it is very often advantageous to analyze the type of an equaiton or a system of equations with several independent variables on sub-spaces of only two independent variables.
1.3) Korteweg-de Vries
We introduce the auxiliary unknowns 
such that we can form a system
such that we can form a system
Further we define a vector of unknowns 
and write the system (10) in matrix form as
and write the system (10) in matrix form as
where
A = [ 1 0 0
0 0 0
0 0 0 ];
B = [ 0 0 1
1 0 0
0 1 0 ];
The characteristics
determinant = simplify( det( A'*dx - B'*dt ) )
Dt = solve( determinant, dt )
are again typical for parabolic equations.
1.4) Accoustic waves
First we use the definition of the speed of sound (11c) to eliminate pressurep from (11b) to reduce the problem to a system of two equations with two unknowns. We apply the chain rule
The reduced system is then
or in matrix form
syms u rho a
A = [ 1 0
0 1 ];
B = [ u rho
a^2/rho u ];
The number of real characteristics
determinant = det( A'*dx - B'*dt )
DxDt = simplify( solve(determinant, dx) / dt )
equals the number of equations, so the system is hyperbolic. The result confirms that if 
, the flow is subsonic so the presure disturbance at a point can travel in two directions - upstream and downstream...
, the flow is subsonic so the presure disturbance at a point can travel in two directions - upstream and downstream...