Parabolic Initial-boundary Value Problems Research Articles

Solvability for the non-isothermal Kobayashi–Warren–Carter system

In this paper, a system of parabolic type initial-boundary value problems are considered. The system (S)$_\nu$ is based on the non-isothermal model of grain boundary motion by [ 38 ] , which was derived as an extending version of the ``Kobayashi--Warren--Carter model'' of grain boundary motion by [ 23 ] . Under suitable assumptions, the existence theorem of $ L^2 $-based solutions is concluded, as a versatile mathematical theory to analyze various Kobayashi--Warren--Carter type models.

A priori bounds for superlinear elliptic equations with semidefinite nonlinearity

We derive a priori bounds for positive solutions of the superlinear elliptic problems −Δu=a(x)up on a bounded domain Ω in RN, where a(x) is Hölder continuous in Ω. Our main motivation is to study the case where a(x)≥0, a(x)≢0 and a(x) has some zero sets in Ω. We show that, in this case, the scaling arguments reduce the problem of a priori bounds to the Liouville-type results for the equation −Δu=A(x′)up in RN, where A is the continuous function defined on the subspace Rk with 1≤k≤N and x′∈Rk. We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.

Efficient Spectral Collocation Algorithm for Solving Parabolic Inverse Problems

This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to solve nonclassical parabolic initial-boundary value problems. Accordingly, the inverse problem is reduced into a system of ordinary differential equations (ODEs) and afterwards, such system can be solved numerically using implicit Runge–Kutta (IRK) method of order four. Four examples are introduced to demonstrate the applicability, validity, effectiveness and stable approximations of the present method. Numerical results show the exponential convergence property and error characteristics of presented method.

An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

Averaging of Parabolic Initial-Boundary Value Problems with Rapidly Oscillating Principal Part

We justify the Krylov–Bogolyubov averaging method for linear and quasilinear parabolic second order initial-boundary value problems with rapidly time-oscillating principal terms.

Weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation with nonstandard growth

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

Primal hybrid method for parabolic problems

In this article, a class of second order parabolic initial-boundary value problems in the framework of primal hybrid principle is discussed. The interelement continuity requirement for standard finite element method has been alleviated by using primal hybrid method. Finite elements are constructed and used in spatial direction, and backward Euler scheme is used in temporal direction for solving fully discrete scheme. Optimal order estimates for both the semidiscrete and fully discrete method are derived with the help of modified projection operator. Numerical results are obtained in order to verify the theoretical analysis.

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part II: Convergence

In this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial boundary value problems is pursued. Compared to other types of exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility of constructing schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time step sizes is incorporated. The convergence analysis is based upon two fundamental ingredients. The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time step sizes, have been deduced in the recent work [C. Gonzalez and M. Thalhammer, SIAM J. Numer. Anal., 53 (2015), pp. 701--719]. The core of the present work is devoted to the derivation of suitable local and global error representations. In conjunction with the stability bound...

The $L^2$-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations

We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.

Wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems

In this paper, numerical methods based on quasilinearization and Haar and Legendre wavelets to solve a class of semi linear parabolic initial boundary value problem (SPIBVP) have been presented. The Haar and Legendre wavelet methods have been successfully combined with quasilinearization to solve SPIBVP efficiently. The presented numerical scheme has been illustrated using appropriate examples including Fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply.

Boundary control of a parabolic system with distributed parameters on a graph in the class of summable functions

This paper considers a differential system whose state is described by a parabolic initial boundary value problem with distributed parameters on a graph. For the above system, the author studies the problem of boundary control in the class of generalized solutions. Control action and observations are simultaneously boundary. And finally, the existence conditions of a unique control action and the expressions characterizing control actions are obtained.

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part I: Stability

Explicit exponential integrators based on general linear methods are studied for the time discretization of quasi-linear parabolic initial-boundary value problems. Compared to other exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility to construct schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. Employing an abstract framework, the considered problems take the form of initial value problems on Banach spaces: $u'(t) = Q(u(t)) u(t), t \in (0,T), u(0)$ given. A fundamental requirement for the stability and error analysis is that the domains of the defining sectorial operators $Q(v): D = D(Q(v)) \to X$ are independent of $v \in V \subset X$. The scope of applications in particular includes quasi-linear parabolic evolution equations subject to Dirichlet boundary conditions. The work is divided into two parts. In Part I, stability bounds in the norms of certain intermediate spaces between the domain $D$ and the underlying Banach space $X$ are deduced. In view of practical applications, the stability estimates are stated for variable time stepsizes, under mild restrictions on the ratios of subsequent stepsizes. The stability results provide a basic ingredient for the convergence analysis given in Part II.

A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks

This paper presents a novel constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with a constrained backpropogation training approach to obtain an artificial neural network representation of the PDE solution that approximately satisfies the boundary conditions at every integration step. The advantage of CINT over existing methods is that it is readily applicable to solving PDEs on irregular domains, and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. In this paper the CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems with a known analytical solutions that can be used for performance comparison. The numerical results show that, when compared to the most efficient finite element methods, the CINT method achieves significant improvements both in terms of computational time and accuracy.

Identifying an Unknown Coefficient in the Reaction-Diffusion Equation Using He’s VIM

An inverse heat problem of finding an unknown parameter p(t) in the parabolic initial-boundary value problem is solved with variational iteration method (VIM). For solving the discussed inverse problem, at first we transform it into a nonlinear direct problem and then use the proposed method. Also an error analysis is presented for the method and prior and posterior error bounds of the approximate solution are estimated. The main property of the method is in its flexibility and ability to solve nonlinear equation accurately and conveniently. Some examples are given to illustrate the effectiveness and convenience of the method.

Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform meshes. The nonuniform meshes are obtained by equidistributing a positive monitor function, which depends on the second-order spatial derivative of the singular component of the solution. The truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. Semilinear IBVPs are also solved. Numerical experiments are carried out to support the theoretical results.

Discrete Fourier multipliers and cylindrical boundary-value problems

We consider operator-valued boundary-value problems in (0, 2π)n with periodic or, more generally, ν-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary-value problem. As an application, we study vector-valued parabolic initial boundary-value problems in cylindrical domains (0, 2π)n × V with ν-periodic boundary conditions in the cylindrical directions. We show that, under suitable assumptions on the coefficients, we obtain maximal Lq-regularity for such problems. For symmetric operators such as the Laplacian, related results for mixed Dirichlet-Neumann boundary conditions on (0, 2π)n × V are deduced.

Single Point Blow-Up and Regional Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

Let. This article considers the following degenerate semilinear parabolic initial-boundary value problem, where be constants. We obtained the blow up set and find the conditions of single point blow-up.

Existence and Blow-Up of Solutions for a Degenerate Semilinear Parabolic Equation

This article considers the following degenerate semilinear parabolic initial-boundary value problem,where be constants. We obtained the conditions of global existence and blow-up.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

AbstractThis article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion

In this paper, a coupled system of two parabolic initial-boundary value problems is considered. The system presented is a one-dimensional version of the Kobayashi-Warren-Carter model of grain boundary motion [15,16], that is derived as a gradient system of a governing free energy including a weighted total variation. Due to the weighted total variation, some nonstandard terms appear in the mathematical expressions of this system, and such nonstandard terms have made the mathematical treatments to be quite delicate. Recently, a certain definition of the solution have been provided in [21], together with the solvability result. The main objective in this paper is to verify that the system reproduces the foundational rules as a gradient system of parabolic PDEs, such as ``smoothing effect'' and ``energy-dissipation''. Consequently, the existence of a special solution, called ``energy-dissipative solution'', will be demonstrated in the Main Theorem of this paper.