Bitsadze-Lavrent'ev problem
Tricomi–Bitsadze–Lavrent'ev problem
The problem of finding a function which satisfies
![]() | (a1) |
in a mixed domain that is simply connected and bounded by a Jordan (non-self-intersecting) "elliptic" arc (for
) with end-points
and
and by the "real" characteristics (for
)
![]() |
of the Bitsadze–Lavrent'ev equation (a1), which satisfy the characteristic equation
![]() |
and meet at the point , and which assumes prescribed continuous boundary values
![]() | (a2) |
where is the arc length reckoned from the point
and
![]() |
Consider the aforementioned domain (denoted by ). Then a function
is a regular solution of the Bitsadze–Lavrent'ev problem if it satisfies the following conditions:
1) is continuous in
,
;
2) are continuous in
(except, possibly, at the points
and
, where they may have poles of order less than
, i.e., they may tend to infinity with order less than
as
and
);
3) ,
are continuous in
(except possibly on
, where they need not exist);
4) satisfies (a1) at all points
(i.e.,
without
);
5) satisfies the boundary conditions (a2).
Consider the normal curve (of Bitsadze–Lavrent'ev)
![]() |
Note that it is the upper semi-circle and can also be given by (the upper part of)
![]() |
where . The curve
contains
in its interior.
The idea of A.V. Bitsadze and M.A. Lavrent'ev for finding regular solutions of the above problem is as follows. First, solve the problem N (in ,
). That is, find a regular solution of equation (a1) satisfying the boundary conditions:
on
;
on
, where
is continuous for
,
, and may tend to infinity of order less than
as
and
.
Secondly, solve the Cauchy–Goursat problem (in ,
). That is, find a regular solution of (a1) satisfying the boundary conditions:
on
;
on
, where
is continuous for
,
, and may tend to infinity of order less that
as
and
.
Finally, take into account the boundary condition
![]() |
Therefore, one has a Goursat problem (in ,
) for (a1) with boundary conditions:
on
;
on
.
Several extensions and generalizations of the above boundary value problem of mixed type have been established [a3], [a4], [a5], [a6], [a7], [a8]. These problems are important in fluid mechanics (aerodynamics and hydrodynamics, [a1], [a2]).
References
[a1] | A.V. Bitsadze, "Equations of mixed type" , Macmillan (1964) (In Russian) |
[a2] | C. Ferrari, F.G. Tricomi, "Transonic aerodynamics" , Acad. Press (1968) (Translated from Italian) |
[a3] | J.M. Rassias, "Mixed type equations" , 90 , Teubner (1986) |
[a4] | J.M. Rassias, "Lecture notes on mixed type partial differential equations" , World Sci. (1990) |
[a5] | J.M. Rassias, "The Bitsadze–Lavrentjev problem" Bull. Soc. Roy. Sci. Liège , 48 (1979) pp. 424–425 |
[a6] | J.M. Rassias, "The bi-hyperbolic Bitsadze–Lavrentjev–Rassias problem in three-dimensional Euclidean space" C.R. Acad. Sci. Bulg. Sci. , 39 (1986) pp. 29–32 |
[a7] | J.M. Rassias, "The mixed Bitsadze–Lavrentjev–Tricomi boundary value problem" , Texte zur Mathematik , 90 , Teubner (1986) pp. 6–21 |
[a8] | J.M. Rassias, "The well posed Tricomi–Bitsadze–Lavrentjev problem in the Euclidean plane" Atti. Accad. Sci. Torino , 124 (1990) pp. 73–83 |
Bitsadze-Lavrent'ev problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bitsadze-Lavrent%27ev_problem&oldid=46077