Mixed-type differential equation
A partial differential equation which is of varying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-linear) differential equation of the second order with two unknown variables,
and with coefficients defined in the domain is an equation of mixed type if the discriminant of the characteristic form
takes the value zero in but is not identically zero there.
The curve defined by the equation is called the parabolic line of equation (1) or the line of degeneracy (change) of type.
If does not change sign when the point crosses the parabolic line in , then equation (1) is a degenerate equation of elliptic-parabolic or hyperbolic-parabolic type (see Degenerate partial differential equation).
Under certain smoothness conditions on , , , and , there is a non-singular real transformation of the independent variables sending equation (1) (in the case where the sign of alternates in the neighbourhood of a chosen point of where ) to one of the following canonical forms (the notation for the independent variables is preserved):
The equations (2) and (3) are equations of mixed (elliptic-hyperbolic) type in any domain containing a segment of the line of degeneracy .
The domain of definition of an equation of mixed type is sometimes called a mixed domain, and boundary value problems in mixed domains are called mixed boundary value problems. The part () of a mixed domain where the equation is of elliptic (hyperbolic) type is called the domain of ellipticity (hyperbolicity).
Many problems of an applied nature reduce to finding specific solutions of equations of mixed type; in particular, problems of plane transonic flow of a compressible medium, and problems in the theory of envelopes.
An equation (1) of mixed type is called an equation of the first kind (second kind) if the characteristic form () everywhere on the parabolic line. The Chaplygin equation
where is a continuously-differentiable monotone function such that when , is a typical example of an equation of mixed type of the first kind. When , equation (4) is usually called the Tricomi equation.
An important model for an equation of mixed type (with a discontinuous coefficient in front of one of the higher derivatives) is the equation of Lavrent'ev–Bitsadze
One of the basic boundary value problems for equations of mixed type (of the first kind) is the Tricomi problem, which is as follows for an equation of the form (2). Let be a finite simply-connected domain in the Euclidean plane with independent variables and , bounded by a simple Jordan curve with end points , lying in the half-plane , and by the parts and of the characteristics of equation (2) going through the point , . The Tricomi problem is to find a solution of equation (2) that is continuous in the closure of and takes given values on the curve .
In the theory of the Tricomi problem, an essential role is played by the Bitsadze extremum principle, which, in the case of equation (5), states that a solution of equation (5) in the class which vanishes on the characteristic : , , in the closure of the domain of ellipticity , attains its extremum on the curve .
This principle, which guarantees the uniqueness and stability of a solution of the Tricomi problem (and also provides a basic estimate to prove existence by the alternating method), can be generalized to a very wide class of linear and quasi-linear equations of mixed type. In particular, it applies to Chaplygin equations (and Tricomi equations) whenever is twice continuously differentiable and for . The Bitsadze extremum principle also holds for the equation
The solution of the Tricomi problem for equation (6) in the corresponding mixed domain can be written in explicit form when the elliptic part of the boundary of this domain coincides with the so-called normal contour :
In the general case, under specific conditions on the curve and on the class in which the solutions are sought, the Tricomi problem for equation (6) reduces to an equivalent singular integral equation which is (by virtue of the uniqueness condition) unconditionally solvable. The method of integral equations can also be applied to prove existence of a solution of the Tricomi problem, and of other mixed problems, for more general equations of the form
with power degeneracy of order .
Function-theoretical methods and functional analysis, in particular the use of a priori estimates, have made it possible to extend significantly the class of equations of mixed type and mixed domains for which existence and uniqueness of a (generalized) solution can be proved, both for the Tricomi problems and for various other mixed problems.
An important generalization of the Tricomi problem is the general mixed Bitsadze problem, which, in the case of equation (5), can be posed as follows. Let be a simply-connected mixed domain bounded by a simple Jordan curve lying in the half-plane with end points , and (smooth) monotone curves and through these points, meeting at a point , . It is assumed that and lie in the domain bounded by the characteristics , and the interval : on the -axis. Let and denote the points of intersection of the characteristics and with the curves and , where is any fixed point of the semi-interval , and let and denote the parts of and lying between , and , respectively. The general mixed Bitsadze problem consists of finding a regular solution (when , ) of equation (5) in which is continuous in , has continuous first derivatives in when , and satisfies given boundary conditions on , and . The uniqueness and existence of a solution of this problem, both for equation (5) and for more general equations, can be proved under certain geometric conditions on the boundary of , especially on the curve . The general mixed Bitsadze problem can be regarded as completely solved in the special case when coincides with the characteristic through the point . An important consequence of the fact that the general mixed Bitsadze problem is correctly posed, in the case of equation (5) for example, is that the Dirichlet problem for mixed domains of the form is incorrectly posed, whatever the size and form of the domain of hyperbolicity.
For a fairly large class of linear equations
it is known that the coefficient has a substantial influence on the correctness of posing the Dirichlet problem in corresponding mixed domains of the form .
Another type of mixed problem is the Frankl problem. Let be a simply-connected domain with the following boundary: the interval : of the line , a smooth curve with end points at and and lying in the quadrant , the interval : of the line , and the characteristic through and of the equation of mixed type under consideration (e.g. equation (4)). The Frankl problem consists of finding a solution of the equation of mixed type in , given the value of on and the conditions
on . This problem has been investigated chiefly for model equations of mixed type and has been completely solved for equation (5) in the case where the curve : is such that , where is the arc length of measured from the point .
Basic boundary value problems have been formulated for equations of mixed type of the first kind and, adapted with appropriate modifications, for equations of mixed type of the second kind. These modifications are necessary because the Dirichlet problem for elliptic equations with characteristic degeneracy is not always correctly posed.
In the formulation of the boundary value problems for equation (1) in mixed domains, a new aspect is introduced if the line of change of type is also a line of degeneracy of the order of equation, which occurs, for example, in the case of the equation
where is a natural number and is a constant such that .
For equations (5), (6), (7), there are, in addition to the above, a number of essentially new boundary value problems. These are chiefly characterized by the fact that the entire boundary of (where the Tricomi problem is posed) carries the following boundary conditions: the Dirichlet conditions, for example, on and on , with some non-local condition pointwise connecting the values of the desired solution or a (fractional) derivative of it of a certain order. In particular, these problems include a simple example of a correctly-posed self-adjoint mixed boundary value problem.
Boundary value problems have also been studied for equations (and systems) of mixed type in domains containing in their interiors several lines of degeneracy of type, or one single closed parabolic line.
Analogues of the Tricomi problem have been studied for certain classes of equations and systems of mixed type in two independent variables, and for equations of higher order.
Significant difficulties arise in the search for well-posed problems for equations of mixed type with many variables. Nevertheless, several important results have been obtained also in this direction. For the equation
which is a simple model of an equation of mixed type having as a time-like plane of degeneracy of type, the following problem is known to be correctly posed. Let be a finite simply-connected three-dimensional domain, bounded by a piecewise smooth surface and by the characteristic surfaces
of equation (8). One has to find a continuously differentiable function in , satisfying equation (8) in for , that vanishes on and on one of the characteristic surfaces , . Existence of a weak solution and uniqueness of a strong solution for this problem have been proved for the more general equation
where is the Laplace operator in the variables .
For the equation
with part of the space-like hyperplane of degeneracy both of type and of order contained in the mixed domain , boundary value problems of a special form have been studied. Here the part of lying in the half-space carries data , and the part lying in the half-space (the characteristic conoid of equation (9)) carries certain integral averages of .
Other model equations of mixed type in bounded and unbounded three-dimensional domains have been studied, including the equations
There is also a uniqueness criterion of the solution of the Dirichlet problem for a large class of self-adjoint equations of mixed type in cylindrical domains.
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