Cascade method

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 [1] in 1773 and developed by G. Darboux [2].

References

[1]	P.S. Laplace, , Oeuvres complètes , 9 , Paris (1893) pp. 5–68
[2]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18
[3]	F.G. Tricomi, "Lectures on partial differential equations" , Moscow (1957) (In Russian; translated from Italian)
[4]	V.M. Babich, et al., "Linear equations of mathematical physics" , Moscow (1964) (In Russian)
[5]	G.A. Dombrovskii, "The method of adiabatic approximation in the theory of plane gas flows" , Moscow (1964) (In Russian)
[6]	T.V. Chekmarev, "Generalized model systems of mixed-type equations" Izv. Vuzov. Mat. , 11 (1972) pp. 72–79 (In Russian)
[7]	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