Eigenfunctions of the Tricomi problem with an inclined type change line

We construct the eigenfunctions of the Tricomi problem for the case in which the type change line of the elliptic-hyperbolic equation is inclined and forms an arbitrary angle α with the x-axis. These eigenfunctions form a basis in the elliptic domain. In addition, we find an integral constraint on the inclined type change line.     

Unique solvability of an analog of the Tricomi problem with nonlocal integral conjugation condition

Using an alternating method of Schwartz type, we prove the unique solvability of the elliptic-hyperbolic equation in the class of generalized solutions of an analog of the Tricomi problem with nonlocal integral conjugation condition for the case of an arbitrary approach of the elliptic boundary of the domain to the line of type change with the exception of the case of tangency.     

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.     

An analog of the Tricomi problem with a nonlocal integral conjugate condition

We prove the unique solvability of an analog of the Tricomi problem for an elliptic-hyperbolic equation with a nonlocal integral conjugate condition on the characteristic line.     

Tricomi problem for a nonlinear equation of mixed type with functional delay and advance

We study a boundary value problem for a nonlinear equation of mixed type with the Lavrent’ev–Bitsadze operator in the principal part and with functional delay and advance in lower-order terms. The general solution of the equation is constructed. The problem is uniquely solvable.     

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)).     

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.     

An instance of the cutting stock problem for which the rounding property does not hold

We say that an instance of the cutting stock problem has the integer rounding property if its optimal value is the least integer greater than or equal to the optimal value of its linear programming relaxation. In this note we give an instance of the cutting stock problem for which the rounding property does not hold, and show that it is NP-hard to decide whether the rounding holds or not.     

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)     

Initial-boundary value problem for a parabolic-hyperbolic equation with power-law degeneration on the type change line

We consider the first boundary value problem for equations of mixed type in a rectangular domain. A criterion for the solution uniqueness is proved by the spectral expansion method. The solution is constructed in the form of a series in the eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution with respect to the initial function is proved.     

Tricomi problem for <Emphasis Type="Italic">q</Emphasis>-difference analog of anticipatory-retarding equation of mixed type

We investigate a boundary-value problem for mixed-type equation with the Lavrent’ev–Bitsadze operator in the main term and q-difference deviations of the argument in the lowest terms. We construct a general solution to the equation and prove a uniqueness theorem without restrictions on the deviation value. Then we show that the problem is uniquely solvable and find integral representations of the solution in the elliptic and hyperbolic domains.     

Gellerstedt problem with nonclassical matching conditions for the solution gradient on the type change line with data on internal characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation. An initial function is posed in the ellipticity domain of the equation on the boundary of the unit half-circle with center the origin. Zero conditions are posed on characteristics in the hyperbolicity domain of the equation. “Frankl-type conditions” are posed on the type change line of the equation. We show that the problem is either conditionally solvable or uniquely solvable. We obtain a closed-form solvability condition in the case of conditional solvability. We derive integral representations of the solution in all cases.     

A combination of the Tricomi problem and a boundary-value problem with a shift for the Gellerstedt equation

We study a problem whose statement combines the Tricomi problem and the problem with a shift considered by V. I. Zhegalov and A. M. Nakhushev for the Gellerstedt equation with a singular coefficient. We prove its solvability by the method of integral equations, and we do the uniqueness of the solution with the help of the extremum principle.     

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.     

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.     

Gellerstedt problem with a generalized Frankl matching condition on the type change line with data on external characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with nonclassical matching conditions for the gradient of the solution (in the sense of Frankl) on the type change line of the equation. We prove that the inhomogeneous Gellerstedt problem with data on the external characteristics of the equation is solvable either uniquely or modulo a nontrivial solution of the homogeneous problem. We obtain integral representations of the solution of the problem in both the elliptic and the hyperbolic parts of the domain. The solution proves to be regular.     

Error bound on the eigenvalue of the difference analog of the spectral problem in a cylindrical coordinate system

We consider the eigenvalue problem for the operator defined in a rectangle whose vertical left-hand side coincides with the z-axis. A difference scheme is constructed by an integrointerpolation method. An error bound is obtained for the simple eigenvalue of the difference analog in weighted generalized spaces W ₂ ² and W ₂ ³ .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 29–35, 1986.