Boundary value problem for a mixed-type equation with a difference-differential operator

The Tricomi problem for a mixed-type equation with retarded argument in an unbounded domain is considered. The unique solvability of the problem is proved without restrictions on the delay magnitude. The existence of a solution follows from the solvability of a difference equation. Closed-form integral representations for the solutions are derived.     

Boundary value problem for a mixed type equation with an advanced-retarded argument

We consider the Tricomi problem for the Lavrent??ev-Bitsadze equation with a mixed deviation of the argument. The uniqueness theorem for the problem is proved under constraints on the deviation of the argument. The existence of a solution is related to the solvability of a difference equation. We obtain integral representations of solutions in closed form.     

Solvability of the Tricomi problem for second order equations of mixed type with degenerate curve on the sides of an angle

In [1]–[6], the author posed and discussed the Tricomi problem of second order mixed equations, but he only consider some special mixed equations. In [3], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equation with nonsmooth degenerate line. The present paper deals with the Tricomi problem for general second order mixed equations with degenerate curve on the sides of an angle. I first give the formulation of the above problem, and then prove the solvability of the Tricomi problem for the mixed equations with degenerate curve on the sides of an angle, by using the existence of solutions of the mixed problem for the degenerate elliptic equations (see [11]). Here I mention that the used method in this paper is different to those in other papers or books, because I introduce the new notation (2.1) below, such that the second order equation of mixed type can be reduced to the first order complex equation of mixed type with singular coefficients, hence I can use the advantage of complex analytic method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Tricomi Problem for a Quasilinear Mixed Type Equation

In this paper, the Tricomi problem and the generalized Tricomi problem for a quasilinear mixed type equation are studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to these problems is proved. The method developed in this paper can be used to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

Uniqueness of Quasi-Regular Solutions for a Bi-Parabolic Elliptic Bi-Hyperbolic Tricomi Problem

The Tricomi equation $ yu_{xx} + u_{yy} = 0 $ was established in 1923 by Tricomi who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945 Frankl established a generalization of these problems for the well-known Chaplygin equation $ K(\,y)u_{xx} + u_{yy} = 0 $ subject to the Frankl condition 1 + 2( K / K ')' > 0, y <0. In 1953 and 1955 Protter generalized these problems even further by improving the above Frankl condition. In 1977 we generalized these results in R n ( n > 2). In 1986 Kracht and Kreyszig discussed the Tricomi equation and transition problems. In 1993 Semerdjieva considered the hyperbolic equation $ K_1 (\,y)u_{xx} + (K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ for y<0. In this paper we establish uniqueness of quasi-regular solutions for the Tricomi problem concerning the more general mixed type partial differential equation $ K_1 (\,y)(M_2 {\rm (}x{\rm )}u_x )_x + M_1 (x)(K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ which is parabolic on both lines x = 0; y = 0, elliptic in the first quadrant x > 0, y > 0 and hyperbolic in both quadrants x< 0, y > 0; x > 0, y< 0. In 1999 we proved existence of weak solutions for a particular Tricomi problem. These results are interesting in fluid mechanics.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

General Tricomi-Rassias problem and oblique derivative problem for generalized Chaplygin equations

Many authors have discussed the Tricomi problem for some second order equations of mixed type, which has important applications in gas dynamics. In particular, Bers proposed the Tricomi problem for Chaplygin equations in multiply connected domains [L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958]. And Rassias proposed the exterior Tricomi problem for mixed equations in a doubly connected domain and proved the uniqueness of solutions for the problem [J.M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, Singapore, 1990]. In the present paper, we discuss the general Tricomi-Rassias problem for generalized Chaplygin equations. This is one general oblique derivative problem that includes the exterior Tricomi problem as a special case. We first give the representation of solutions of the general Tricomi-Rassias problem, and then prove the uniqueness and existence of solutions for the problem by a new method. In this paper, we shall also discuss another general oblique derivative problem for generalized Chaplygin equations.     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.     

The Exterior Tricomi Problem for Generalized Mixed Equations with Parabolic Degeneracy

This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy. Firstly the representation of solutions of the problem for the equations is given, and then the uniqueness and existence of solutions are proved by a new method.     

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

On a Maximum Principle and Uniqueness Theorems for a System of First Order Equations of Mixed Type

A maximum principle for a system of first order equations of mixed type is established. The uniqueness theorems of solutions t.o the generalized Tricomi type problem and to the Frankl's problem are proved by the method of auxiliary functions.     

The Tricomi problem for second order linear equations of mixed type with parabolic degeneracy

In Bers, 1958, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York: Wiley); Bitsadze, 1988, Some Classes of Partial Differential Equations (New York: Gordon and Breach); Rassias, 1990, Lecture Notes on Mixed Type Partial Differential Equations (Singapore: World Scientific); Salakhitdinov and Islomov, 1987, The Tricomi problem for the general linear equation of mixed type with a nonsmooth line of degeneracy. Soviet Math. Dokl., 34, 133–136; Smirnov, 1978, Equations of Mixed type (Providence, RI: American Mathematical Society), the authors posed and discussed the Tricomi problem of second order equations of mixed type with parabolic degeneracy, which possesses important application to gas dynamics. The present article deals with the Tricomi problem for general second order equations of mixed type with parabolic degeneracy. Firstly the formulation of the problem for the equations is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension. In this article, we use the complex method, namely the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used (see Wen, 2002, Linear and Quasilinear Equations of Hyperbolic and Mixed Types (London: Taylor and Francis)).     

About the Tricomi problem for the mixed parabolic–hyperbolic type equation with complex spectral parameter

The unique solvability of the Tricomi problem for the parabolic–hyperbolic equation with complex spectral parameter is proved. Uniqueness of the solution is shown by the method of energy integral and existence by the method of integral equations.     

A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION

In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.     

A three-dimensional analog of the Tricomi problem for a parabolic-hyperbolic equation

For a parabolic-hyperbolic equation, we study the three-dimensional analog of the Tricomi problem with a noncharacteritic plane on which the type of the equation changes. The uniqueness of the solution to the problem is proved by the method of a priori estimates, and the existence of a solution is reduced to the existence of a solution to a Volterra integral equation of the second kind.     

Problem with lacking Tricomi condition on a characteristic and an analog of the Frankl condition on a segment of the degeneration line for a class of mixed type equations

For the Gellerstedt equation with a singular coefficient, we consider a boundary value problem that differs from the Tricomi problem in that the boundary characteristic AC is arbitrarily divided into two parts AC ₀ and C ₀ C and the Tricomi condition is posed on the first of them, while the second part C ₀ C is free of boundary conditions. The lacking Tricomi condition is equivalently replaced by an analog of the Frankl condition on a segment of the degeneration line. The well-posedness of this problem is proved.     

First Order System of Equations of Mixed Type and Second Order Equation of Mixed Type

A system of first order equations of mixed type, which may be reduced to a general second order equation of mixed type, is considered. Uniqueness of solution to the generalized Tricomi problem is proved by the method of auxiliary function. Existence of H¹ strong solution is based on a characteristic problem and is proved by the Fredholm's alternative properties.     

TRICOMI PROBLEM FOR A MIXED EQUATION OF SECOND ORDER WITH DISCONTINUOUS COEFFICIENTS

This paper is devoted to the Tricomi problem for a mixed type equation of second order. The coeffcients are assumed to be discontinuous on the line where the type is changed. The unique existence of the solution to the problem is proved if the domain is small enough. Correspondingly, some estimates on the solution is also established.     

Boundary value problem for an advanced-retarded equation of mixed type with a nonsmooth degeneration line

We consider the Tricomi problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part, with advanced-retarded arguments, and with parallel degeneration lines. We prove the uniqueness theorem under restrictions on the values of the argument deviations. The problem is uniquely solvable. We find integral representations of solutions in closed form.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.