Coefficient In Parabolic Equation Research Articles

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.

Internal and Startup Controls of the Solutions of Boundary-Value Problem for Parabolic Equations with Degenerations

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.

Cauchy problem for non-uniformly parabolic equations with power singularities

The Cauchy problem for non-uniformly 2b-parabolic equations with degenerations is investigated. Coefficients of parabolic equations can have power singularities of arbitrary order with respect to any variables on some set of points. Using prior estimates and Arzelа’s and Riesz’s theorems the existence and integral representation of the unique solution to the formulated Cauchy problem are established. Estimates for the solution of the Cauchy problem and its derivatives in Hölder spaces with power weight are found. The order of the power weight is defined in terms of orders of the power singularities and degenerations in coefficients of 2b-parabolic equations. Cite as: І. P. Medynsky, “Fundamental solutіons for degenerate parabolіc equatіons: existence, propertіes and some theіr applіcatіons,” Mat. Met. Fiz.-Mekh. Polya , 64 , No. 2, 31–41 (2021) (in Ukrainian), https://doi.org/10.15407/mmpmf2021.64.2.31-41

On long-time asymptotics of solutions of parabolic equations with increasing leading coefficients

Sharp sufficient conditions on the coefficients of a second-order parabolic equation are examined under which the solution of the corresponding Cauchy problem with a power-law growing initial function stabilizes to zero. An example is presented showing that the found sufficient conditions are sharp. Conditions on the coefficients of a parabolic equation are obtained under which the solution of the Cauchy problem with a bounded initial function stabilizes to zero at a power law rate.

Оптимальное управление коэффициентами параболического уравнения с интегральным граничным условием

Оптимальное управление коэффициентами параболического уравнения с интегральным граничным условием

The optimal control problem for the coefficients of a parabolic equation under phase constraints

For the optimal control problem on the coefficients of parabolic equations, we study correctness problems for the setting in the weak topology on the space of controls. We show that in this problem, the objective functional is bounded from below, the set of optimal controls is nonempty, and every minimizing sequence weakly converges to the set of optimal controls. We establish a necessary optimality condition in the form of a generalized Lagrange multiplier rule.

The inverse problem of the simultaneous determination of the right-hand side and the lowest coefficient in parabolic equations with many space variables

We obtain existence and uniqueness theorems for the solution of the inverse problem of the simultaneous determination of the right-hand side and the lowest coefficient inmultidimensional parabolic equations with integral observation. We give an estimate of the maximum of the modulus of unknown coefficients with constants explicitly expressed in terms of the input data of the problem.

An Optimization Problem with Controls in the Coefficients of Parabolic Equations

1 Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia. Mathematics Department, Faculty of Science, Minia University, Mina, Egypt. CS Department, Faculty of Computer and Information Technology, Taif University, Taif, Saudi Arabia. Mathematics Department, Faculty of Science, Fayoum University, Fayoum, Egypt. 5 Faculty of Science and Arts, King Khalid University, Bbelkrn, Saudi Arabia.

The Inverse Problem of the Simultaneous Determination of the Right-Hand Side and the Lowest Coefficient in Parabolic Equations with Many Space Variables

Получены теоремы существования и единственности решения обратной задачи одновременного определения правой части и младшего коэффициента в многомерном параболическом уравнении при условии интегрального наблюдения. Дана оценка максимума модуля неизвестных коэффициентов с константами, явно выписанными через входные данные задачи. Библиография: 15 названий.

Stabilization of Solutions of Cauchy Problems for Divergence-Free Parabolic Equations with Decreasing Minor Coefficients

Exact sufficient conditions on minor coefficients of parabolic equation are considered in this work. These conditions guarantee the stabilization of solutions of the Cauchy problem to zero in some classes of increasing initial functions.

Approximate Solution of an Unknown Coefficients in Parabolic Equation

In this paper, an inverse parabolic equation is solved by using the homotopy analysis method (HAM) and the homotopy perturbation method (HPM). The approximation solution of this equation is calculated in the form of series which its components are computed easily. HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM. Illustrative examples are presented to exhibit a comparison between the HAM and the HPM.

Stable estimation of two coefficients in a nonlinear Fisher–KPP equation

We consider the inverse problem of determining two non-constant coefficients in a nonlinear parabolic equation of the Fisher–Kolmogorov–Petrovsky–Piskunov type. For the equation ut = DΔu + μ(x) u − γ(x)u2 in (0, T) × Ω, which corresponds to a classical model of population dynamics in a bounded heterogeneous environment, our results give a stability inequality between the couple of coefficients (μ, γ) and some observations of the solution u. These observations consist in measurements of u: in the whole domain Ω at two fixed times, in a subset ω⊂⊂Ω during a finite time interval and on the boundary of Ω at all times t ∈ (0, T). The proof relies on parabolic estimates together with the parabolic maximum principle and Hopf’s lemma which enable us to use a Carleman inequality. This work extends previous studies on the stable determination of non-constant coefficients in parabolic equations, as it deals with two coefficients and with a nonlinear term. A consequence of our results is the uniqueness of the couple of coefficients (μ, γ), given the observation of u. This uniqueness result was obtained in a previous paper but in the one-dimensional case only.

The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation

Existence and uniqueness theorems for the solution to the inverse problem of determining the lower-order coefficient in multidimensional parabolic equations with integral observation are obtained. An estimate of the maximum of the modulus of the unknown coefficient with a constant explicitly expressed via the input data of the problem is given.

Stability of the determination of a time-dependent coefficient in parabolic equations

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.

Stability for determining the principal coefficient of parabolic equation

This work investigates stability of an inverse problem for a backward linear parabolic heat problem by using an initial temperature measurement. This kinds of problem has been widely used in a various field of pure and applied science. The necessary condition of the minimizer for the cost functional are constructed by using the optimal control theory. Stability of the minimizer for the cost functional is proved based on the necessary condition.

Determination of Unknown Coefficients in Parabolic Equations

In this paper, we consider solving for a coefficient inverse problem in the parabolic equation. A new numerical method for the identification of the space-dependent coefficient is developed in a reproducing kernel space. The coefficients can be solved by a lower triangle linear system. Some numerical experiments are presented to show the efficiency of the proposed method.

On the inverse problem of determining the leading coefficient in parabolic equations

We study the unique solvability of the inverse problem of determining the leading coefficient in the parabolic equation on the plane with coefficients depending on both time and spatial variables under the condition of integral overdetermination with respect to time. We obtain sufficient conditions for the unique solvability of the inverse problem. We present nontrivial examples of problems for which such conditions hold. It is shown that the imposed conditions necessarily hold if either the time interval is sufficiently large or the space interval on which the problem is considered is sufficiently small.

Identifying the coefficient of first-order in parabolic equation from final measurement data

This paper studies an inverse problem of recovering the first-order coefficient in parabolic equation when the final observation is given. Such problem has important application in a large field of applied science. The original problem is transformed into an optimal control problem by the optimization theory. The existence, uniqueness and necessary condition of the minimum for the control functional are established. By an elliptic bilateral variational inequality which is deduced from the necessary condition, an algorithm and some numerical experiments are proposed in the paper. The numerical results show that the proposed method is an accurate and stable method to determine the coefficient of first-order in the inverse parabolic problems.

Estimation of piecewise constant coefficients of parabolic equations: applications to the detection of buried objects

A one-dimensional inverse problem arising in infrared thermography for the detection and characterization of buried objects is introduced. Mathematically, the problem is to reconstruct a piecewise constant coefficient of a scalar heat equation in a finite rod from measurements taken at one of its extremities. The problem is posed in the well known least-squares setting and solved by a quasi-Newton method. The contributions of this article include: (i) the parameterization of a piecewise constant function by a small number of unknown parameters which represent its constant values and locations of discontinuities; (ii) the application of the adjoint field technique in the calculation of the gradient of a discretized objective function and (iii) the application of the considered inverse problem in the detection and characterization of buried objects. Numerical results illustrate the good performance of the proposed algorithm.

On The Sobolev Space Theory of Parabolic and Elliptic Equations inC1Domains

Existence and uniqueness results are given for second-order parabolic and elliptic equations with variable coefficients in C1 domains in Sobolev spaces with weights allowing the derivatives of solutions to blow up near the boundary. The number of derivatives can be negative and fractional. The coefficients of parabolic equations are only assumed to be measurable in time.