The Tricomi problem of a quasi-linear Lavrentiev–Bitsadze mixed type equation

In this paper, we consider the Tricomi problem of a quasi-linear Lavrentiev–Bitsadze mixed type equation $$\begin{array}{lll}({\rm sgn}\,u_y) {\frac{\partial ^2 u}{\partial x^2}} + {\frac{\partial ^2 u}{\partial y^2}}-1=0,\end{array}$$ whose 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.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Correctness of the Dirichlet problem for the Lavrent’ev–Bitsadze equation

It is shown that the Dirichlet problem in a multidimensional domain for the Lavrent’ev–Bitsadze equation is uniquely solvable. A criterion of the uniqueness of the solution is obtained.     

Bitsadze–Samarskii Boundary Condition for Elliptic-Parabolic Volume Potential

A spectral decomposition of the Green’s function of the Holmgren problem in a cylindrical domain is used to obtain Bitsadze–Samarskii boundary conditions for a regular elliptic-parabolic volume potential.     

The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type

We study the problem with boundary conditions of the first and second kind on the boundary of a rectangular domain for an equation with two internal perpendicular lines of change of a type. With the use of the spectral method we prove the unique solvability of the mentioned problem. The eigenvalue problem for an ordinary differential equation obtained by separation of variables is not self-adjoint, and the system of root functions is not orthogonal. We construct the corresponding biorthogonal system of functions and prove its completeness. This allows us to establish a criterion for the uniqueness of the solution to the problem under consideration. We construct the solution as the sum of the biorthogonal series.     

A. A. Dezin’s problem for inhomogeneous Lavrent’ev–Bitsadze equation

We establish a uniqueness criterion for solution of nonlocal Dezin’s problem for an equation of mixed elliptic-hyperbolic type. The solution is constructed as a sum of a series in eigenfunctions of the corresponding one-dimensional spectral problem. In the proof of its convergence there arises a problem on small denominators arises. Under certain restrictions on the given parameters and functions we prove the convergence of constructed series in the class of regular solutions.     

Tricomi problem for an advance–delay equation of mixed type with variable deviation of the argument

We study a boundary value problem for an equation of mixed type with the Lavrent’ev–Bitsadze operator in the leading part and with variable deviation of the argument in lower-order terms. The general solution of the equation is constructed. We prove a uniqueness theorem without any conditions on the value of the deviation. The problem is uniquely solvable. We derive integral representations of the solutions in closed form in the elliptic and hyperbolic domains.     

Tricomi problem for a functional-differential advanced-retarded Lavrent’ev–Bitsadze equation

We study a boundary value problem for the Lavrent’ev–Bitsadze equation with functional delay and advance. The general solution is constructed. The problem is uniquely solvable.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

Eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation

We determine eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation.     

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.     

On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.     

The completeness of the eigenfunctions of the Tricomi problem for the Lavrent'ev–Bitsadze equation with the Frankl gluing condition

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.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.     

A Neumann problem of Ambrosetti–Prodi type

The aim of this paper is to establish an Ambrosetti–Proditype result for the problem

$$\left\{ \begin{array}{ll}-\Delta{u} = g(x, u,\nabla{u}) + t\varphi \quad {\rm in}\, \Omega,\\ \frac{\partial{u}}{\partial\eta} = 0 \qquad\qquad\qquad\quad {\rm on}\, \partial\Omega ;\end{array} \right.$$

i.e., under appropriate conditions, we will show that there exists a constant t ₀ such that the problem above has no solution if t > t ₀, at least a solution if t =  t ₀ and at least two solutions if t < t ₀. The proof is based on a combination of upper and lower solutions method and the Leray–Schauder degree.

     

A singular Cauchy problem for the Euler–Poisson–Darboux equation

We consider a singular Cauchy problem for the Euler–Poisson–Darboux equation of Fuchsian type in the time variable with ramified Cauchy data. In this paper we establish an expansion of the solutions in a series of hypergeometric functions and then investigate the nature of the singularities of the solutions.     

On Periodic Solutions of Quasilinear Parabolic Equations with Boundary Conditions of Bitsadze–Samarskii Type

Doklady Mathematics - We consider a quasilinear parabolic boundary value problem with a nonlocal boundary condition of Bitsadze–Samarskii type. A theorem on the existence and uniqueness of a...