Fourth-order Parabolic Equation Research Articles

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines

A functional inequality and its applications to a class of nonlinear fourth-order parabolic equations

A functional inequality and its applications to a class of nonlinear fourth-order parabolic equations

On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator

We study a nonlocal problem for a fractional partial differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator. By separation of variables, we prove a theorem on the existence and uniqueness of a regular solution of this problem.

On a Fourth Order Parabolic Equation with Mixed Type Boundary Conditions in a Nonrectangular Domain

Abstract This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x 4u = f in the non-necessarily rectangular domain Q ={(t,x) ∈ℝ2 : 0 < t < T, ϕ1(t) < x < ϕ2(t)}. The equation is subject to mixed type conditions ∂xu = ∂x 3u + βu = 0, on the lateral boundary of Q. The right-hand side term f of the equation lies in Lω 2 (Q) the space of square-integrable functions on Q with the measure ωdtdx. Our aim is to find sufficient conditions on the coefficient β and on the functions ϕi,i = 1, 2 and on the weight ω such that the solution of this equation belongs to the anisotropic Sobolev space Hω 1,4 (Q) = {u ∈ Lω 2 (Q) : ∂tu, ∂x ju ∈ Lω 2 (Q) , j = 1, 2, 3, 4}. The analysis is performed by using the domain decomposition method.

Solutions for a fourth-order nonlinear parabolic equation describing Marangoni convection

In this paper, we consider the global existence and uniqueness of solutions for a fourth-order nonlinear parabolic equation describing Marangoni convection. Moreover, the existence of global attractor in and space is also considered.

Solutions of fourth-order parabolic equation modeling thin film growth

In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method [1–3] we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namely C0([0,T];Lp(Ω)) where p=nα2−α with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are Lp–Lq-estimates, regularization property of the linear part of e−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.

The Bang-bang principle of time optimal controls for the Kuramoto-Sivashinsky-KdV equation with internal control

Summary This paper is concerned with the time optimal control problem governed by the internal controlled Kuramoto–Sivashinsky–Korteweg-de Vries equation, which describes many physical processes in motion of turbulence and other unstable process systems. We prove the existence of optimal controls with the help of the Carleman inequality, which has been widely used to obtain the local controllability or null controllability of parabolic differential systems. More precisely, with the help of the Carleman inequality, we obtain a relationship between the null controllability and time optimal control problem. Moreover, we give the bang-bang principle for an optimal control of our original problem by using the one of approximate problems. This method is new for time optimal control problems. The bang-bang principle established here seems also to be new for fourth-order parabolic differential equations. Copyright © 2015 John Wiley & Sons, Ltd.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L 1 norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

Approximation of Smectic-A liquid crystals

In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables $({\bf u},p)$ and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector ${\bf n}$. We start from the formulation given in \cite{E} by using the so-called layer variable $\phi$ such that ${\bf n}=\nabla \phi$ and the level sets of $\phi$ describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations $({\bf u},p)$ with a fourth order parabolic equation for $\phi$. We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and $C^0$-finite elements in space. Finally, numerical simulations are presented for $2D$-domains, showing the evolution of the system until it reaches an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.

Some properties of solutions for a parabolic equation modeling epitaxial thin film growth

In this paper, we study a fourth-order parabolic equation with Dirichlet boundary which arises in epitaxial growth of nanoscale thin films. We proved the equation with condition possesses a global attractor in , which attracts any bounded subset of in the -norm. In addition to this, we consider the equation under the conditions is blow-up.

Existence and Uniqueness of Weak Solutions to a Thin Film Type Equation

This paper is concerned with a fourth order parabolic equation in multidimensional spacewith boundary condition u=Δu=0 and initial function u0. The minimizer method yields the existence and uniqueness for the elliptic equation. Finally, the existence and uniqueness of solutions of the corresponding parabolic equation are obtained from the semi-discrete problem.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient $L^2$ norm.

Solving Fourth Order Parabolic PDE with Variable Coefficients Using Aboodh Transform Homotopy Perturbation Method

Here, a new method called Aboodh transform homotopy perturbation method (ATHPM) is used to solve one dimensional fourth order parabolic linear partial differential equations with variable coefficients. The proposed method is a combination of the new integral transform “Aboodh transform” and the homotopy perturbation method. Some cases of one dimensional fourth order parabolic linear partial differential equations are solved to illustrate ability and reliability of mixture of Aboodh transform and homotopy perturbation method. We have compared the obtained analytical solution with the available Aboodh decomposition solution and homotopy perturbation method solution which is found to be exactly same. The results obtained reveal that the combination of Aboodh transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems.

Finite difference discretization of a fourth-order parabolic equation describing crystal surface growth

In this paper, by using finite difference method, we consider the approximate solutions for a fourth-order parabolic equation describing crystal surface growth.

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

The purpose of this paper is to obtain the approximate solution of fourth-order parabolic partial differential equations by the reduced differential transform method (RDTM).This method provides the solution in the form of a convergent series with easily calculable terms. Comparing RDTM with some other methods in the literature shows present approach is very simple, effective, powerful and can be easily applied to other linear or nonlinear PDEs in science and engineering.

Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches

This paper is devoted to some aspects of well-posedness of the Cauchy problem (the CP, for short) for a quasilinear degenerate fourth-order parabolic thin film equation (the TFE-4) (0.1)ut=−∇⋅(|u|n∇Δu)in RN×R+,u(x,0)=u0(x)in RN, where n>0 is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, n∈(0,32)) requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces. The main goal of the paper is to detect proper solutions of the CP for the degenerate TFE-4 by uniformly parabolic analytic ε-regularizations at least for values of the parameter n sufficiently close to 0.Firstly, we study an analytic “homotopy” approach based on a priori estimates for solutions of uniformly parabolic analytic ε-regularization problems of the form ut=−∇⋅(ϕε(u)∇Δu)in RN×R+, where ϕε(u) for ε∈(0,1] is an analytic ε-regularization of the problem (0.1), such that ϕ0(u)=|u|n and ϕ1(u)=1, using a more standard classic technique of passing to the limit in integral identities for weak solutions. However, this argument has been demonstrated to be non-conclusive, basically due to the lack of a complete optimal estimate-regularity theory for these types of problems.Secondly, to resolve that issue more successfully, we study a more general similar analytic “homotopy transformation” in both the parameters, as ε→0+ and n→0+, and describe branching of solutions of the TFE-4 from the solutions of the notorious bi-harmonic equationut=−Δ2uin RN×R,u(x,0)=u0(x)in RN, which describes some qualitative oscillatory properties of CP-solutions of (0.1) for small n>0 providing us with the uniqueness of solutions for the problem (0.1) when n is close to 0.Finally, Riemann-like problems occurring in a boundary layer construction, that occur close to nodal sets of the solutions, as ε→0+, are discussed in other to get uniqueness results for the TFE-4 (0.1).

Time-periodic solution of a 2D fourth-order nonlinear parabolic equation

By using the Galerkin method, we study the the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a fourth-order nonlinear parabolic equation in 2D case.

A note on solving the fourth-order parabolic equation by the sinc-Galerkin method

A variable coefficient fourth-order partial differential equation in one space variable that arises in the study of transverse vibrations of a uniform flexible beam is examined. The numerical solution is obtained by using the sinc-Galerkin method. The method is then tested on three problems which have appeared in physical applications. The numerical results demonstrate the reliability and efficiency of the algorithm developed.

Well-posedness for the Navier-slip thin-film equation in the case of complete wetting

We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourth-order parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x,xβ) (where x denotes the distance from the contact line) with β=13−14≈0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second-order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L2-spaces for the linearized evolution, after suitable subtraction of a(t)+b(t)xβ-terms.

Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations

When developing efficient numerical methods for solving parabolictypes of equations, severe temporal stability constraints on thetime step are often required due to the high-order spatialderivatives and/or stiff reactions. The implicit integration factor(IIF) method, which treats spatial derivative terms explicitly andreaction terms implicitly, can provide excellent stabilityproperties in time with nice accuracy. One major challenge for the IIFis the storage and calculation of the dense exponentials of the sparse discretization matrices resulted from thelinear differential operators. The compact representation of the IIF(cIIF) can overcome this shortcoming and greatly savecomputational cost and storage. On the other hand, the cIIF is oftenhard to be directly applied to deal with problems involving cross derivatives. In this paper, bytreating the discretization matrices in diagonalized forms, wedevelop an efficient cIIF method for solving a family of semilinear fourth-orderparabolic equations, in which the bi-Laplace operator is explicitlyhandled and the computational cost and storage remain the same as to theclassic cIIF for second-order problems. In particular, the proposed method can deal with not onlystiff nonlinear reaction terms but also various types ofhomogeneous or inhomogeneous boundary conditions. Numericalexperiments are finally presented to demonstrate effectiveness and accuracyof the proposed method.