Fourth-order Parabolic Equation Research Articles

Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative

In this article, we study initial-boundary problems for fourth-order nonlinear parabolic equations modeling thin film growth with Caputo-type time fractional derivative. By means of the theory of abstract fractional calculus and \(L^p-L^q\) estimates, we establish the existence and uniqueness of local mild solutions in the spaces \(C([0,T]; L^{\frac{\beta N}{2-\beta}}(\Omega))\) with \(1<\beta<2\). Moreover, the local solutions can be extended globally if the initial data is sufficiently small. For more information see https://ejde.math.txstate.edu/Volumes/2024/58/abstr.html

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

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

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.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

Reconstruction of the time-dependent source in thermal grooving by surface diffusion

In hot polycrystalline materials, when a vertical flat grain boundary meets a horizontal surface the grain boundary forms a groove in the surface. Mathematically modelling features of such thermal grooving mechanism is therefore very important in characterizing polycrystalline materials composed of tiny grains intersecting an external free surface. With this aim in mind, we formulate and investigate a novel inverse problem of reconstructing the unknown time-dependent source term entering the fourth-order parabolic equation of thermal grooving by surface diffusion from a given integral observation. We formulate and prove in Theorems 2.3–2.7 that this linear inverse problem is well-posed. However, in practice, the ideal regularity of data under which the inverse source problem is stable is never satisfied due to the inherent non-smoothness of the measurement. Consequently, this leads to the inverse problem with raw data becoming ill-posed. In order to obtain accurate and stable solutions, we develop and compare two numerical methods, namely, a time-discrete method and an optimization method. We obtain error estimates and convergence rates for the time-discrete method. For the optimization method, an objective functional, which is proved to be Fréchet differentiable, is introduced and the conjugate gradient method (CGM), regularized by the discrepancy principle, is developed to compute the minimizer yielding the source term. The results of two numerical tests illustrate the performance of the two methods for both exact and noisy measured data.

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

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

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.</p></abstract>

Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation

Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation

The Null Boundary Controllability for a Fourth-Order Parabolic Equation with Samarskii–Ionkin-Type Boundary Conditions

The Null Boundary Controllability for a Fourth-Order Parabolic Equation with Samarskii–Ionkin-Type Boundary Conditions

Energy-stable finite element method for a class of nonlinear fourth-order parabolic equations

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L∞(L2)- and L∞(H1)-norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results.

Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.

Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by L1 interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme.

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.

Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise

In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height h h driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant { h > 0 } \{h>0\} (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant measure and provide evidence for the fact that the finite difference discretization touches down almost surely while our discretization stays away from the ∂ { h > 0 } \partial \{h > 0\} .

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>

Asymptotic estimate of weak solutions in a fourth-order parabolic equation with logarithm

This paper deals with an initial-boundary problem of the fourth-order parabolic equation involving two logarithm terms. First, we give some results for blow-up or global solutions through classifying the initial energy and the Nehari energy. Second, we show asymptotic estimates about blow-up time and a large time estimate of solutions, respectively.