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''Laplace method''

A method in the theory of partial differential equations enabling one, in some cases, to find the general solution of a linear partial differential equation of hyperbolic type,

$$ \tag{1 }
Lu  \equiv \ 
u _ {xy} +
a (x, y)
u _ {x} +
b (x, y)
u _ {y} +
c (x, y)
u =
$$

$$ 
= \ 
f (x, y),
$$

by constructing a sequence of equations

$$ 
L _ {i} u  \equiv \ 
u _ {xy} +
a _ {i} (x, y)
u _ {x} +
b _ {i} (x, y)
u _ {y} +
c _ {i} (x, y)
u  = \   ( 2 _ {i} )
$$

$$ 
=  f _ {i} (x, y) ,
$$

$  i = \pm  1, \pm  2 \dots $
such that the solution of (1) is expressed in terms of the solutions of the latter.

Equation (1) can be written in one of the following forms:

$$ 
v _ {x} + bv - hu  =  f,\ \ 
w _ {y} + aw - ku  =  f,
$$

where

$$ 
h  =  a _ {x} + ab - c,\ \ 
k  =  b _ {y} + ab - c,
$$

$$ 
v  =  u _ {y} + au,\  w  =  u _ {x} + bu.
$$

The functions  $  h $
and  $  k $
are called invariants of (1).

When  $  h = 0 $, 
solving (1) reduces to the integration of ordinary differential equations, and its solution has the form:

$$ 
u  =  e ^ 
{- \int\limits a  dy }
\left [
X + \int\limits
\left \{
Y + \int\limits
fe ^ {\int\limits b  dx } \ 
dx \right \}
e ^ {\int\limits a  dy - b  dx } \ 
dy \right ] ,
$$

where  $  X $
and  $  Y $
are arbitrary functions of  $  x $
and  $  y $, 
respectively. Similarly, if  $  k = 0 $, 
the solution of (1) can be written in the form

$$ 
u  =  e ^ 
{- \int\limits b  dx }
\left [ Y + \int\limits
\left \{ X + \int\limits fe ^ {\int\limits a  dy } \ 
dy \right \}
e ^ {\int\limits b  dx - a  dy } \ 
dx \right ] .
$$

In case  $  h \neq 0 $, 
the solution  $  u $
to (1) can be obtained from the solution  $  u _ {1} $
of  $  (2 _ {1} ) $
whose coefficients  $  a _ {1} , b _ {1} , c _ {1} $
and right-hand side  $  f _ {1} $
have the form

$$ 
a _ {1}  = \ 
a - (  \mathop{\rm ln}  h) _ {y} ,\ \ 
b _ {1}  =  b,\ \ 
c _ {1}  = \ 
c - a _ {x} + b _ {y} - b (  \mathop{\rm ln}  h) _ {y} ,
$$

$$ 
f _ {1}  =  f _ {0} \cdot (a - (  \mathop{\rm ln}  h) _ {y} ) + f _ {y} ,
$$

by means of the formula

$$ 
u  = \ 

\frac{u _ {1x }  + bu _ {1} - f }{h}
 .
$$

If  $  h _ {1} = 0 $, 
the solution  $  u _ {1} $
of  $  (2 _ {1} ) $
is obtained by the above method; if  $  h _ {1} \neq 0 $, 
the process is further continued by constructing equations  $  (2 _ {i} ) $
for  $  i = 2, 3 , .  .  . $; 
the solution of (1) is expressed by means of quadratures in terms of the solutions of this sequence of equations. For the case  $  k \neq 0 $, 
a chain of equations  $  (2 _ {i} ) $
can similarly be constructed for  $  i = - 1, - 2 , .  .  . $. 
If at some stage  $  h _ {i} $(
or  $  k _ {i} $) 
vanishes, then the general solution to (1) is obtained in quadratures.

The cascade method can be used to pass from a given equation to an equation for which some other known analytic or numerical method of solution is more easily applied; for obtaining families of equations whose solutions are known and whose coefficients closely approximate those of the equations encountered in important applied problems; and finally for obtaining fundamental operators in perturbation theory of operators.

The cascade method was discovered by P. Laplace [[#References|[1]]] in 1773 and developed by G. Darboux [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Laplace,  , ''Oeuvres complètes'' , '''9''' , Paris  (1893)  pp. 5–68</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)  pp. 1–18</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.G. Tricomi,   "Lectures on partial differential equations" , Moscow  (1957)  (In Russian; translated from Italian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Babich,   et al.,   "Linear equations of mathematical physics" , Moscow  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.A. Dombrovskii,   "The method of adiabatic approximation in the theory of plane gas flows" , Moscow  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T.V. Chekmarev,   "Generalized model systems of mixed-type equations"  ''Izv. Vuzov. Mat.'' , '''11'''  (1972)  pp. 72–79  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.I. Pashkovskii,   "Related operators and boundary-value problems for elliptic equations"  ''Differential Eq.'' , '''12''' :  1  (1977)  pp. 81–88  ''Differentsial'nye Uravneniya'' , '''12'''  (1976)  pp. 118–128</TD></TR></table>