Order Parabolic Equations Research Articles

Discontinuous galerkin method of the first order parabolic differential equation

The objective of this paper is to propose a modest numerical error analysis by applying DG finite element method for the parabolic differential equation. The DG method is an imperative numerical method with much mass compensation and more flexible meshing than other numerical methods. The DG method starts by discretizing the domain into a set of non-overlapping elements. This study gives a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

Investigating invariance and conservation laws for the classes of nonlinear parabolic and wave systems

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.

Null controllability for stochastic coupled systems of fourth order parabolic equations

This paper aims to establish null controllability for systems coupled by two backward fourth order stochastic parabolic equations. The main goal is to control both equations with only one control act on the drift term. To achieve this, we develop a new global Carleman estimate for fourth order stochastic parabolic equations, which allows us to deduce a suitable observability inequality for the adjoint systems.

Discontinuous Galerkin Method of the First Order Parabolic Differential Equation

This paper is concerned with the mathematical study of the discontinuous Galerkin (DG) finite element method for the parabolic differential equation. The DG method is a vital numerical method with much mass compensation and more flexible meshing than other methods. In this study, we give a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The error analysis of this problem is also established. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain

On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domain

Approximation of solutions to integro-differential time fractional order parabolic equations in L^{p}-spaces

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

NONLINEAR INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER PARABOLIC EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

The paper studies the classical solvability of an inverse boundary value problem for a second order parabolic equation with nonlocal boundary conditions. For this purpose, first, the considered problem is reduced to an auxiliary equivalent problem in a certain sense. Then, using the Fourier method the auxiliary problem is presented as a system of integral equations. Further, by means of the contraction mappings principle the unique existence of the solution of the obtained system of integral equations is shown. At the end of investigation the existence and uniqueness theorem for the classical solution of the original inverse boundary value problem is proved based on the equivalence of these problems

On linear higher-order parabolic equations in Morrey spaces

In this paper, linear higher-order parabolic equation is shown to define a semigroup in Morrey spaces. For higher-order diffusion equation, the associated semigroup is proved analytic and sharp semigroup estimates in Morrey scale are derived. The analysis also includes semigroups in Morrey scale related to solving linear higher-order fractional diffusion equations.

Artificial neural network solver for time-dependent Fokker–Planck equations

Stochastic differential equations (SDEs) play a crucial role in various applications for modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a stochastic differential equation is described by the Fokker–Planck equation, which is a second order parabolic partial differential equation (PDE). Previous work combined artificial neural networks and Monte Carlo data to solve stationary Fokker–Planck equations. This paper extends this approach to time dependent Fokker–Planck equations. The main focus is on the investigation of algorithms for training a neural network that has multi-scale loss functions. Additionally, a new approach for collocation point sampling is proposed. A few 1D and 2D numerical examples are demonstrated.

Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the C1,γ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is C1,Dini in space. The free boundary then must immediately become C1,γ for a universal γ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.

Exponential attractors for modified Swift-Hohenberg equation in $\mathbb R^{N}$

A Cauchy problem for a modification of the Swift-Hohenberg equation in $\mathbb R^{N}$ with a mildly integrable potential is considered. Existence of exponential attractors containing a finite-dimensional global attractor in $H^2(\mathbb R^{N})$ is shown under the dissipative mechanism of fourth order parabolic equations in unbounded domains.

Closed-form solutions of higher order parabolic equations in multiple dimensions: A reliable computational algorithm

Parabolic equations play an important role in chemical engineering, vibration theory, particle diffusion and heat conduction. Solutions of such equations are required to analyze and predict changes in physical systems. Solutions of such equations require efficient and effective techniques to get reasonable accuracy in lesser time. For this purpose, current article proposes residual power series algorithm for higher order parabolic equations with variable coefficients in multiple dimensions. The proposed algorithm provides closed-form solutions without linearization, discretization or perturbation. For efficiency testing of the proposed methodology, initially it is implemented to homogeneous multidimensional parabolic models, and exact solutions are computed. In next stage of testing, proposed algorithm is enforced to three-dimensional non–homogeneous fourth order parabolic equation, and closed form solutions are recovered. The obtained results indicate the validity and effectiveness of proposed methodology, hence proposed algorithm can be extended to more complex scenarios in engineering and sciences.

Numerical solutions to two-dimensional fourth order parabolic thin film equations using the Parabolic Monge-Ampere method

<abstract><p>This article presents the Parabolic-Monge-Ampere (PMA) method for numerical solutions of two-dimensional fourth-order parabolic thin film equations with constant flux boundary conditions. We track the PMA technique, which employs special functions to acclimate and force the mesh moving associated with the physical PDE representing the thin liquid film equation. The accuracy and convergence of the PMA approach are investigated numerically using a one two-dimensional problem. Comparing the results of this method to the uniform mesh finite difference scheme, the computing effort is reduced.</p></abstract>

Analyticity and observability for fractional order parabolic equations in the whole space

In this paper, we study the quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in ℝn. We first obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally Hölder-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set in ℝn, by utilizing a telescoping series method.

Well-posedness and dynamic properties of solutions to a class of fourth order parabolic equation with mean curvature nonlinearity

The well-posedness and dynamic properties of solutions to a class of fourth order parabolic equations with mean curvature nonlinearity are studied. More specifically, we first prove the existence and uniqueness of strong solutions by semigroup method. Then, we establish conditions for the global existence and finite time blow-up of solutions with subcritical, critical and supercritical initial energies. At the same time, we study the exponential decay and $ \omega $-limit set of the global solutions, and the upper bound of the blow-up time of the blow-up solutions. In addition, we also study the existence and regularity of ground state solutions for the steady state problems.

Decay estimate and blow-up for a fourth order parabolic equation modeling epitaxial thin film growth

<abstract><p>In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.</p></abstract>

Estimates for fundamental solutions of parabolic equations in non-divergence form

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

NTIM solution of the fractional order parabolic partial differential equations

Abstract In this article, natural transform iterative method has been used to find the approximate solution of fractional order parabolic partial differential equations of multi-dimensions together with initial and boundary conditions. The method is applicable without any discretization or linearization. Three problems have been taken as test examples and the results are summarized through plots and tables to show the efficiency and reliability of the method. By practice of a few iterations, we observe that the approximate solution of the parabolic equations converges to the exact solution. The fractional derivatives are considered in the Caputo’s sense.

Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations

In this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems are eventually globally positive if initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition.