Equation Of Parabolic-hyperbolic Type Research Articles

Unknown coefficient problem for mixed equation of parabolic-hyperbolic type with non-local boundary conditions on characteristics

For an equation of a mixed parabolic-hyperbolic type with a characteristic line of type change, we study the inverse problem associated with the search for an unknown coefficient at the lowest term of the parabolic equation. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal condition on the characteristics in the hyperbolic part and the Dirichlet condition in the parabolic part of the domain. In order to determine the unknown coefficient by the solution on the parabolic part of the domain, the integral overdetermination condition is proposed. Global results on the unique solvability of the inverse problem in the sense of the classical solution are proved.

Coefficient Identification Problem for the System of Heat and Wave Equations Associated with a Non-Characteristic Type Change Line

The solvability of the inverse problem associated with the search for an unknown coefficient at the lowest term of a mixed parabolic-hyperbolic type equation with a non characteristic line of type change is studied. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal condition on the characteristics in the hyperbolic part and initial-boundary conditions in the parabolic part of the domain. To determine unknown coefficient, with respect to the solution, defined in the parabolic part of the domain, the integral overdetermination condition is specified. The unique solvability of the inverse problem in the sense of the classical solution is proved.

Boundary value problem with the Bitsadze-Samarsky condition for a loaded equation of parabolic-hyperbolic type in a doubly connected region

Boundary value problem with the Bitsadze-Samarsky condition for a loaded equation of parabolic-hyperbolic type in a doubly connected region

Boundary Value Problems for a Mixed Equation of Parabolic-Hyperbolic Type of the Third Order

Boundary Value Problems for a Mixed Equation of Parabolic-Hyperbolic Type of the Third Order

Local Boundary Value Problems for a Degenerate Loaded Equation of Parabolic-Hyperbolic Type of the Second Kind

Local Boundary Value Problems for a Degenerate Loaded Equation of Parabolic-Hyperbolic Type of the Second Kind

On a Boundary Value Problem for a Third-Order Equation of Parabolic-Hyperbolic Type with a Fractional Order Operator

On a Boundary Value Problem for a Third-Order Equation of Parabolic-Hyperbolic Type with a Fractional Order Operator

Dirichlet’s problem for a third-order parabolic hyperbolic type equation of the second kind

Equations of mixed type, the degeneracy line of which is the envelope of a family of characteristics, therefore, is itself also a characteristic, in the literature it is customary to call equations of mixed type of the second kind, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In the present work, a boundary value problem for a homogeneous equation of parabolic-hyperbolic type of the third order of the second kind is investigated. Necessary and sufficient conditions for the existence and uniqueness of a generalized solution of the problem are found. In some special cases, the representation of the solution of the problem under study is written out explicitly.

О ЧИСЛЕННОМ РЕШЕНИИ ОБРАТНОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ СМЕШАННОГО ПАРАБОЛО–ГИПЕРБОЛИЧЕСКОГО ТИПА ПО ОПРЕДЕЛЕНИЮ ПРАВОЙ ЧАСТИ УРАВНЕНИЯ

The paper proposes  algorithms  for  numerical  solving  an  inverse  problem for a mixed parabolic-hyperbolic type equation with a nonlocal condition for finding the right-hand side of the equation. It was assumed that the function included in the nonlocal condition and the function that is additional information for solving the inverse problem may be known with some error, since they are the result of practical measurements. Three algorithms for the numerical solution are proposed.  The numerical  experiments of testing the proposed algorithms are presented.

Local boundary value problems for a loaded equation of parabolic-hyperbolic type degenerating inside the domain

Local boundary value problems for a loaded equation of parabolic-hyperbolic type degenerating inside the domain

Dezin problem analog for a parabolic-hyperbolic type equation with periodicity condition

In this paper we consider an inhomogeneous second-order parabolic-hyperbolic mixed type equation, represented as one-dimensional heat equation in the parabolic part and the one-dimensional wave equation in the hyperbolic part. For the equation, an analog of the Dezin problem is investigated, which means to find a solution to the equation that satisfies inner-boundary condition, relating the value of the desired function on the equation type change line to the value of the normal derivative on the hyperbolicity region boundary, and inhomogeneous periodicity nonlocal boundary conditions. A substitution is given that allows us to reduce the problem to an equivalent one and, without losing generality, restrict ourselves to investigate the problem with homogeneous conditions for an inhomogeneous equation.The solution is constructed as the Fourier series on the orthonormal system of eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the solution uniqueness to the problem is established.In case when the uniqueness criterion is violated, an example of a nontrivial solution to a homogeneous problem is given, and a necessary and sufficient condition for the existence of a solution to an inhomogeneous problem is obtained.In justifying the solution existence, the problem of small denominators in the sum of the series with respect to the ratio of the rectangle sides in the hyperbolic part of the domain. An estimate of the denominator separation from zero under certain conditions with respect to the problem parameters is obtained. This estimate allows us to substantiate the uniform convergence of the series and their derivatives up to the second-order inclusive under certain conditions for given functions.

Solvability of a Boundary Value Problem for a Fourth Order Equation of Parabolic-Hyperbolic Type in a Pentagonal Domain

Solvability of a Boundary Value Problem for a Fourth Order Equation of Parabolic-Hyperbolic Type in a Pentagonal Domain

An Inverse Problem for an Equation of Parabolic-Hyperbolic Type to Find the Factors of the Right-Hand Side Depending on the Spatial Coordinates

An Inverse Problem for an Equation of Parabolic-Hyperbolic Type to Find the Factors of the Right-Hand Side Depending on the Spatial Coordinates

Unique Solvability of a Boundary Value Problem for a Loaded Fractional Parabolic-Hyperbolic Equation with Nonlinear Terms

This work is devoted to study the existence and uniqueness of solution of an analogue of the Gellerstedt problem with nonlocal assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov–Caputo operator of differentiation. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of successive approximations of factorial law for Volterra type nonlinear integral equations.

On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

Solvability of BVPs for the Parabolic-Hyperbolic Equation with Non-linear Loaded Term

This work is devoted to prove the existence and uniqueness of solution of BVP with non-local assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation involving Caputo derivatives. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of integral equations

Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.

Gellerstedt Type Problem for the Loaded Parabolic-Hyperbolic Type Equation with Caputo and Erdelyi–Kober Operators of Fractional Order

The work is devoted to the proof of the uniqueness and existence of solution to local and nonlocal problems with an integral gluing condition for a loaded parabolic-hyperbolic type equation with differential and integral operators of fractional order, in which the trace of the solution appears in the Erdelyi–Kober integral operator. Using the method of energy integrals, the uniqueness of the solution is proved, and the existence of the solution is proved by the method of integral equations.

Tricomi Problem for Second Kind Parabolic Hyperbolic Type Equation

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, $$R_{00}^{\lambda}$$ 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 $$R_{00}^{\lambda}$$ 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.

On a boundary value problem for a third-order parabolic-hyperbolic type equation with a displacement boundary condition in its hyperbolicity domain

Исследована краевая задача со смещением для неоднородного уравнения параболо-гиперболического типа третьего порядка с волновым оператором в области гиперболичности, когда в качестве одного из граничных условий задана линейная комбинация с переменными коэффициентами производных от значений искомой функции на независимых характеристиках, а также на линии изменения типа и порядка. Найдены необходимые и достаточные условия существования и единственности регулярного решения задачи. В некоторых частных случаях представление решения исследуемой задачи выписано в явном виде.

Задачи типа Геллерстедта для нагруженного уравнения параболо-гиперболического типа с операторами Капуто и Эрдейли-Кобера дробного порядка

Задачи типа Геллерстедта для нагруженного уравнения параболо-гиперболического типа с операторами Капуто и Эрдейли-Кобера дробного порядка