Order Pseudoparabolic Equation Research Articles

Mixed Problem with Integral Condition for a Loaded Third Order Pseudoparabolic Equation

Mixed Problem with Integral Condition for a Loaded Third Order Pseudoparabolic Equation

Solution of a Characteristic Problem for the Third Order Pseudoparabolic Equation with the Bessel Operator by the Method of Transmutation Operators

Solution of a Characteristic Problem for the Third Order Pseudoparabolic Equation with the Bessel Operator by the Method of Transmutation Operators

On one coefficient inverse boundary value problem for a linear pseudoparabolic equation of the fourth order

<abstract><p>In the present work, we consider an inverse boundary value problem for a fourth order pseudo parabolic equation with periodic and integral condition. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. By the contraction mapping principle is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis.</p></abstract>

Inverse problem for fractional order pseudo-parabolic equation with involution

In this paper, we consider an inverse problem on recovering the right-hand side of a fractional pseudo-parabolic equation with an involution operator. The major obstacle for considering the inverse problems is related with the well-posedness of the problem. Inverse problems are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed since the solution is highly sensitive to variations in the final data. The advantage of this paper is two-fold. On the one hand, we investigate the solvability of the direct problem and prove the solvability to this problem. On the other hand, we study the inverse problem based on this direct problem and prove the solvability results in this problem, too. First, we investigate the Cauchy problem for the time-fractional pseudo-parabolic equation with the involution operator, and secondly, we consider the inverse problem on recovering the right-hand side from an overdetermined final condition and prove that it is solvable. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, and functional analysis. In particular, we use the L-Fourier analysis method to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The classical and generalized solutions of the inverse problem are studied.

Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms

In this paper, we consider the initial boundary value problem for a fourth order pseudo-parabolic equation with memory and source terms. Under suitable assumptions on the function $g$, the initial data and the parameters in the equation, we not only prove the existence of global weak solutions by the combination of the Galerkin method and potential well theory, but also establish an explicit decay rate estimate of the energy adopting the ideas of Marcelo M. Cavalcanti et al. (J. Differ. Equations 203 (2004) 119-158) and Patrick Martinez (ESAIN: Control Optim. Calc. Var. 4 (1999) 419-444). Furthermore, the finite time blow up results for the solutions with both positive and negative initial energy are obtained under certain conditions.

On a Nonlocal Boundary Value Problem for a Third-Order Pseudo-Parabolic Equation

In this paper we consider a nonlocal boundary value problem for a third order pseudo-parabolic equation. For the solution of the problem a priori estimates in differential and difference settings are obtained. The obtained a priori estimates imply stability of the solution of the problem with respect to the initial data and with respect to the right-hand side on a layer as well as the convergence of the solution of the difference problem to that of the differential problem.

Determination of an Unknown Coefficient in the Third Order Pseudoparabolic Equation with Non-Self-Adjoint Boundary Conditions

The solvability of the inverse boundary problem with an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved.

Cauchy Problem in the Non-Classical Treatment for One Pseudoparabolic Equation

In the paper, we consider the Cauchy problem for a fifth order pseudoparabolic equation that appears in studying the issues of fluid filtration in fissured media, the moisture transfer in soils and etc. The Cauchy problem with non-classic conditions not requiring the agreement conditions are studied for a discontinuous coefficient pseudoparabolic equation. The equivalence of these conditions with the Cauchy classic condition is substantiated in the case when the solution of the stated problem is sought in S.L.Sobolev anisotropic space ( ) ( ) 3,2 p WG .

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.

Goursat Problem in the Non-Classical Treatment for a Sixth Order Pseudoparabolic Equation

In this paper substantiated for a differential equation of pseudoparabolic type with discontinuous coefficients a Goursat problem with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions boundary condition is substantiated classical, in the case if the solution of the problem in the anisotropic S. L. Sobolev's space is found. The considered equation as a pseudoparabolic equation generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (telegraph equation, Aller's equation , moisture transfer generalized equation, Manjeron equation, Boussinesq - Love equation and etc.). It is grounded that the Goursat boundary conditions in the classic and non-classic treatment are equivalent to each other, and such boundary conditions are demonstrated in geometric form. Even from geometric interpretation can see that the grounded non-classic treatment doesn't require any additional conditions of agreement type. Thus, namely in this paper, the non-classic problem with Goursat conditions is grounded for a pseudoparabolic equation of sixth order. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev anisotropic space ( ) ( )

Final-Boundary Value Problem in the Non-Classical Treatment for a Sixth Order Pseudoparabolic Equation

In this paper substantiated for a differential equation of pseudoparabolic type with discontinuous coefficients a final-boundary problem with non-classical boundary conditions is considered, which requires no matching conditions. The considered equation as a pseudoparabolic equation generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (telegraph equation, Aller's equation , moisture transfer generalized equation, Manjeron equation, Boussinesq-Love equation and etc.). It is grounded that the final-boundary conditions in the classic and non-classic treatment are equivalent to each other, and such boundary conditions are demonstrated in geometric form. Even from geometric interpretation can see that the grounded non-classic treatment doesn't require any additional conditions of agreement type. Thus, namely in this paper, the non-classic problem with final-boundary conditions is grounded for a pseudoparabolic equation of sixth order. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev anisotropic space WP(4,2)(G) .

Задача с условиями на всей границе для одного псевдопараболического уравнения шестого порядка

Задача с условиями на всей границе для одного псевдопараболического уравнения шестого порядка

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.