Stability estimate for the time-dependent source function component of parabolic equations with coefficients dependent on space and time variables

Determination of an Unknown Heat Source from Overspecified Boundary Data

Space–time localized radial basis function collocation method for solving parabolic and hyperbolic equations

Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon

On recovering space or time-dependent source functions for a parabolic equation with nonlocal conditions

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces $$H^s (\mathbb{T})$$ when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.

The problem of structural design of polymeric and composite viscoelastic materials is currently of great interest. The development of new methods of calculation of the stress–strain state of viscoelastic solids is also a current mathematical problem, because when solving boundary value problems one needs to consider the full history of exposure to loads and temperature on the structure. The article seeks to build an iterative algorithm for calculating the stress–strain state of viscoelastic structures, enabling a complete separation of time and space variables, thereby making it possible to determine the stresses and displacements at any time without regard to the loading history. It presents a modified theoretical basis of the iterative algorithm and provides analytical solutions of variational problems based on which the measure of the rate of convergence of the iterative process is determined. It also presents the conditions for the separation of space and time variables. The formulation of the iterative algorithm, convergence rate estimates, numerical computation results, and comparisons with exact solutions are provided in the tension plate problem example.

Identification of an unknown source depending on both time and space variables by a variational method

Exact Controllability for the Time Dependent Transport Equation

Convergence properties in C inside a region for finite difference schemes with a splitting operator as applied to parabolic systems

In this paper, we consider one inverse control-type problem of determining the leading coefficient of a one-dimensional parabolic equation. The problem under consideration is a variational statement of a coefficient inverse problem for a parabolic equation. The sought for coefficient of the parabolic equation depends on the spatial variable. An integral boundary condition is set for the parabolic equation. The desired highest coefficient of the parabolic equation plays role of the control function, which is an element of the Sobolev space. The set of admissible control functions belong to the Sobolev space. The objective functional for the control problem is compiled based on the integral overdetermination condition set in the inverse problem. This condition may be interpreted as tasks of a weighted mean of the solution of the equation under consideration as per time variable. The solution of the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The existence of a solution of the considered inverse control-type problem is proved. An adjoint boundary value problem for the control problem under study is introduced. The Frechet differentiability of the objective functional in a set of admissible control functions is proved. In addition, an auxiliary boundary value problem is introduced; and using the solution of this problem, an expression for the gradient of the objective functional is found. The necessary optimality condition for the admissible control function is obtained.

Solving the inverse problem of identifying an unknown source term in a parabolic equation

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold.

In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ψ ′ ⁢ ( t ) + A ⁢ ψ = f ⁢ ( t ) {\psi^{\prime}(t)+A\psi=f(t)} with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

For a second-order parabolic equation with degenerations, we construct the solution of the problem of optimal control for systems described by the first boundary-value problem with internal and startup controls. The coefficients of parabolic equation have power singularities of any order in time and space variables on a certain set of points.

We study the problem for a second-order linear parabolic equation with nonlocal integral condition in the time variable and power singularities in the coefficients of any order with respect to the time and space variables. By using the maximum principle and a priori estimates, we establish the existence and uniqueness of the solution of this problem in Holder spaces with power weights.