Conditional Discretization of PDEs
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We will discuss two of the conditional symmetries, ˆ
Q 1 (3), corresponding to the
case when the coefficient of ∂ t in the vector field is equal to 1, and Q 4 (6), when the
same coefficient is equal to 0.
2 Case ˆ
Q 1
Let us discuss here the symmetry reduction and discretization provided by the vector
field ˆ
Q 1 .
2.1 Symmetry Reduced Equations and Solutions
The symmetry variables, that is, the invariants of the vector field, are
v =
x(u − 1)
u
, y = x
2 u − 3
u − 1
+ 18t, v = v(y)
(7)
for u = 0 and u = 1 Then (2) reduces to the ODE
vv yy − 2v
2
y = 0,
whose solution is
v =
c 1
y + c 2
,
where c 1 and c 2 are two integration constants. Then from (7) a solution of (2),
different from the trivial constant u = 1, is
u(x, t) =
x
3x 2 + 18t + c 2
x(x 2 + 18t + c 2 ) − c 1
.
2.2 Construction of the Discretized Equation
To be able to construct the conditionally invariant discretization of (2) we need at
first to construct the invariant lattice.
The relevant necessary invariants in the point nm are
I 1 = x nm −
x nm
u nm
, I 2 = x
2
nm
u nm − 3
u nm − 1
+ 18t nm .