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In this paper, we study an analogue of Tricomi problem for a parabolichyperbolic equation of the first type. To solve the problem for the heat equation, we use properties of the spectral problem with a spectral parameter in the boundary conditions. We show the basicity of the spectral problem with the spectral parameter in the boundary conditions. The conditions for the correctness of the formulated Tricomi problem are found. [ABSTRACT FROM AUTHOR]

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

In this paper, we study an analogue of Tricomi problem for a parabolic–hyperbolic equation. One boundary condition in the parabolic part of the domain contains the third derivative of the space variable. A spectral problem with a spectral parameter in the boundary condition arises, when solving this problem. The system of eigenfunctions of this problem (after removing any one eigenfunction) forms a Riesz basis. The conditions for the correctness of the formulated Tricomi problem are found. [ABSTRACT FROM AUTHOR]

In this paper, an analog of the Tricomi problem for a mixed-type equation with a fractional derivative is studied. In one part of the domain, the considered equation is a subdiffusion equation with a fractional derivative of order in the sense of Riemann–Liouville, and in the other part, it is a wave equation. Two types of inverse problems are considered. Assuming parameter to be unknown, the corresponding inverse problem is studied and an additional condition is found that provides a unique definition of the sought-for parameter. Then, assuming the boundary value of the solution to be unknown, an additional condition is defined that ensures the existence and uniqueness of the unknown value of the solution. [ABSTRACT FROM AUTHOR]

The solvability of the spatial Tricomi problem for a class of multidimensional hyperbolic-elliptic equations is proved. [ABSTRACT FROM AUTHOR]

Для нагруженного уравнения гиперболо-параболического типа исследуется однозначная разрешимость аналога задача Трикоми. Нагрузка определена в фиксированных точках области искомых решений, в том числе и во внутренних точках. Найдены условия существования и единственности регулярного решения задачи.

In this paper, a theorem on a priori estimates of solutions of the Tricomi problem for a second-order, mixed-type equation with the Gellerstedt operator in the hyperbolicity domain is proved. The a priori estimate obtained implies the uniqueness of the regular solution of the problem considered. [ABSTRACT FROM AUTHOR]

We study the Tricomi problem for the functional-differential mixed-compound equation $LQu(x,y)=0$ in the class of twice continuously differentiable solutions. Here $L$ is a differential-difference operator of mixed parabolic-elliptic type with Riemann–Liouville fractional derivative and linear shift by $y$. The $Q$ operator includes multiple functional delays and advances $a_1(x)$ and $a_2(x)$ by $x$. The functional shifts $a_1(x)$ and $a_2(x)$ are the orientation preserving mutually inverse diffeomorphisms. The integration domain is $D=D^+\cup D^-\cup I$. The “parabolicity” domain $D^+$ is the set of $(x,y)$ such that $x_0

As part of this scientific work, we study a displacement boundary value problem for a third - order parabolichyperbolic type equation with a third - order parabolic equation backward in time and a wave equation in the domain of hyperbolicity. As one of the boundary conditions we have a linear combination including variable coefficients of the sought function on the characteristic lines AC and BC . The present paper reports following results: inequality between characteristics of AC and BC lines limiting the hyperbolic part Ω1 of the domain Ω as carriers of data for the Tricomi problem as 0≤ x ≤2 π , as a matter of fact, the solvability of the Tricomi problem with data on the characteristic line BC does not imply the solvability of the Tricomi problem with data on the AC ; necessary and sufficient conditions for the existence and uniqueness of a regular solution to the problem under study are found. Under certain conditions for the given functions, the solution to the problem under study is written out explicitly. It is shown that under violation of the necessary conditions established in this paper the homogeneous problem has innumerable linearly independent solutions, while the set of solutions to the corresponding inhomogeneous problem can exist only with additional conditions

In mathematical modeling of continuous media with memory, we deal with equations that describe a new type of wave motion, something between ordinary wave diffusion and classical wave propagation. There are fractional differential equations, which are the basis for the most mathematical models describing a wide class of physical and chemical processes in the fractal geometry of the Nature. The paper presents a new moisture transfer equation with a fractional Riemann – Liouville derivative that generalize the Aller – Lykov equation. The first boundary value problem for the generalized moisture transfer equation is considered. To prove the uniqueness of a solution we employ the energy inequalities method; an a priori estimate is obtained in terms of the fractional Riemann – Liouville derivative. The existence of the solution for the problem is proved by the Fourier method.

Three-dimensional analogue of the problem of Tricomi with non-characteristic parallel planes of change of types of the equation is investigated for a parabolic-hyperbolic equation. The uniqueness of the solution of the problem is proved by the method of a priori estimates, and the existence of the solution of the problem is enlightened to the existence of a solution of the system of the second type Voltaire integral equation.

We consider a nonlocal boundary-value problem for a third-order parabolic-hyperbolic equation with degeneracy of type and order in the domain of hyperbolicity, containing second-order derivatives in the boundary conditions. Sufficient conditions of the unique solvability of the problem are obtained. The Tricomi method is used to prove the uniqueness theorem for a solution. The solution of the problem is expressed in the explicit form. [ABSTRACT FROM AUTHOR]

In the work Tricomi problem was investigated for a parabolic-hyperbolic type equation in a mixed domain. If the parabolic degeneration line is a characteristic of a hyperbolic equation, then Tricomi problem for considered equation will not be uniquely solvable. Therefore, another formulation of Tricomi problem was proposed which the gluing condition is given as in [1-2]. To study Tricomi problem in the hyperbolic part of the domain, class of the regular solutions of the view changed Cauchy problem for the equation of the hyperbolic part are introduced. An explicit form of the solution is found for the Cauchy problem from this class. The solution of the Tricomi problem in the hyperbolic part of the domain is found as a regular solution from the class of the view changed Cauchy problem, and in the parabolic part of the domain as the solution of the first boundary value problem. For proving the existence of the solution of the problem, the theory of second kind Volterra integral equations is used. [ABSTRACT FROM AUTHOR]

In this paper we obtain an a priori estimate for solution of Tricomi problem for the Lavrent’ev-Bitsadze equation, from which the uniqueness of regular solution follows. Presented a numerical finite-difference method for solving the investigated problem. We obtain an a priori estimate for solution of the difference scheme, from which follows the second-order convergence.

The theorem about the a priori estimate for the solution of Tricomi problem for Lavrentiev-Bitsadze equation is proved. From this theorem, in particular, follows the uniqueness of a regular solution of the investigated problem.

In this article, we study a boundary value problem for a parabolic-hyperbolic equation with Caputo fractional derivative. Under certain conditions, we prove its unique solvability using methods of integral equations and Green's functions.

In the present work we investigate the Tricomi problem with an integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional differential operator. Using the method of energy integrals, we prove the uniqueness of the solution for the considered problem. The existence of the solution have been proved applying methods of ordinary differential equations and Fredholm integral equations. The solution is represented in an explicit form.

We obtain conditions on the complex parameter, when there is an unique solution of the Tricomi problem for an equation with two perpendicular lines of degeneracy.

The Tricomi and Tricomi–Neumann problems for the Lavrent'ev-Bitsadze equation. It is proved in this work that when the Frankl gluing condition is fulfilled, the system of eigenfunctions can either be: a Riesz basis; not complete; not minimal; complete and minimal, but not a basis. [ABSTRACT FROM PUBLISHER]

For Tigan system existence conditions of homoclinic trajectories are obtained. Algorithms for computing bifurcation values of parameters corresponding to homoclinic trajectories are formulated. [ABSTRACT FROM AUTHOR]

The existence of a principal eigenvalue is established for the Tricomi problem in normal domains; that is, the existence of a positive eigenvalue of minimum modulus with an associated positive eigenfunction. The argument here uses prior results of the authors on the generalized solvability in weighted Sobolev spaces and associated maximum/minimum principles cite{[LP2]} coupled with known results of Krein-Rutman type.

Some examples that show that the homogeneous Tricomi problem for a multidimensional hyperbolic-parabolic equation has the infinite set of nontrivial solutions are constructed. [ABSTRACT FROM AUTHOR]

Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system, describing three-mode model of rotating fluid convection, is obtained. New applications of the Fishing principle are developed. [ABSTRACT FROM AUTHOR]

We study the homogeneous Tricomi problem for a mixed type equation with two spectral parameters λ and λ and specify the conditions on these parameters which ensue uniqueness of solutions to this problem. Hence, we describe the sets on the complex plane which do not contain the spectrum of the Tricomi problem. [ABSTRACT FROM AUTHOR]

In this paper the unique solvability of the analog of the Tricomi problem for the third order loaded differential and integro-differential equations with parabolic-hyperbolic operators is proved. The existence of a solution is proved by the reduction to the integral equations with a shift of the second kind. We obtain necessary and sufficient conditions for the existence of a unique solution. [ABSTRACT FROM AUTHOR]

In the present work we investigate the Tricomi problem with an integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional differential operator. Using the method of energy integrals, we prove the uniqueness of the solution for the considered problem. The existence of the solution have been proved applying methods of ordinary differential equations and Fredholm integral equations. The solution is represented in an explicit form. [ABSTRACT FROM AUTHOR]

In this article, we study a boundary value problem for a parabolic-hyperbolic equation with Caputo fractional derivative. Under certain conditions, we prove its unique solvability using methods of integral equations and Green's functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Tricomi problem of a quasi-linear Lavrentiev-Bitsadze mixed type equationwhose coefficients depend on the first-order derivative of the unknown function. We prove the existence of solution to this problem by using the hodograph transformation. The method can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics. [ABSTRACT FROM AUTHOR]

For Lorenz-like system, conditions of Shilnikov chaos are obtained. Algorithms for computation of bifurcational values of parameters, which correspond to homoclinic trajectories are formulated. Modified Lorenz, Chen and Lu systems are considered. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Tricomi problem for a genuinely nonlinear Lavrentiev–Bitsadze mixed-type equation whose coefficients depend on the first-order derivative of an unknown function. We prove the existence of a solution to this problem by a method which can be used to study more difficult problems for nonlinear mixed-type equations arising in gas dynamics. [Copyright &y& Elsevier]

We study a mixed type equation, which is analogous, in parts, to Tricomi equation and Keldysh equation. For this equation, the characteristics in the hyperbolic region are transversal to the transition locus in a subset of the locus, but is tangential to it in another subset. We propose a formulation of a closed boundary value problem for such a mixed type equation, and prove that it admits a unique H1 solution with some higher regularity inside the domain. Our result shows the variety and the complexity of the boundary value problems for mixed-type equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The famous Tricomi equation was established in 1923 by F. G. Tricomi who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945 F. I. Frankl established a generalization of these problems for the well-known Chaplygin equation subject to a certain Frankl condition. In 1953 and 1955 M. H. Protter generalized these problems even further by improving the Frankl condition. In 1977 we generalized these results in several n-dimensional simply connected domains. In 1990 we proposed the exterior Tricomi problem in a doubly connected domain. In 2002 we considered uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem. In 2006 G. C. Wen investigated the exterior Tricomi problem for general mixed type equations. In this paper we establish uniqueness of quasi-regular solutions for the exterior Tricomi and Frankl problems for quaterelliptic - quaterhyperbolic mixed type partial differential equations of second order with eight parabolic degenerate lines and propose certain open problems. These mixed type boundary value problems are very important in fluid mechanics. [ABSTRACT FROM AUTHOR]

We consider a mixed-type singular differential equation in a bounded spatial domain in a specific form and prove the unique solvability of the Tricomi problem for the mentioned equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a class of equations of mixed type, we study an analog of the Dirichlet problem satisfying the conjugation conditions on the line of change of the type. We establish a uniqueness and existence criterion for the solution of this problem. [ABSTRACT FROM AUTHOR]

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)). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for absence of global solutions to semilinear evolution differential equations and inequalities of higher order (in the time variable). These results generalize the well-known assertions for parabolic and hyperbolic equations. The proofs base on the test function method by Veron, Mitidieri, Pokhozhaev, and Tesei. [ABSTRACT FROM AUTHOR]

For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray–Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]