Parabolic Initial-boundary Value Problems Research Articles

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Error Identities for Parabolic Initial Boundary Value Problems

Error Identities for Parabolic Initial Boundary Value Problems

Error Identities for Parabolic Equations with Monotone Spatial Operators

Abstract The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper.

Численное интегрирование одномерных задач параболического типа на адаптивных сетках

A method of numerical integration of one-dimensioned parabolic initial boundary value problems with contrast structures using movable adaptive grids is described. A difference scheme which is resistant to machine rounding errors and time-monotonic is constructed. Algorithms for constructing adaptive grids in space and time are proposed. A series of numerical experiments using the example of a singularly perturbed “reaction-diffusion-advection” problem is provided.

Оптимальное мультипликативное управление полулинейным параболическим уравнением

An analysis of optimal control problems for a nonlinear parabolic initial-boundary value problem that models the dynamics of the collective behavior of a bacterial community is presented. Estimates for the solution of the initial-boundary value problem are obtained, the solvability of control problems is proved, and optimality conditions are derived. The weak "bang-bang" principle is set for the problem with the final observation.

Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem

Abstract In this article, we study a class of variational inequality problems with non-Newtonian polytropic parabolic operators. We introduce a mapping with an adjustable parameter to control the polytropic term, which exactly meets the conditions of Leray-Schauder fixed point theory. At the same time, we construct a penalty function to transform the variational inequality into a regular parabolic initial boundary value problem. Thus, the existence is treated with a Leray-Schauder fixed point theory as well as a suitable version of Aubin-Lions lemma. Then, the uniqueness and stability of the solution are analyzed.

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called “h-version’’ Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

On Solving Initial Boundary Value Problems for a Parabolic Equation in Dini Classes

The method of potentials is one of the classical methods for solving initial boundary value problems for parabolic equations and systems. The method is based on studying the smoothness of parabolic potentials in various functional spaces and studying the solvability of the corresponding integral equations and systems to which the problems are reduced. This method allows one to find a constructive form of solutions to initial-boundary value problems and directly investigate the smoothness of solutions. This article continues a series of works on constructing the solvability theory of parabolic initial boundary value problems in domains with nonsmooth lateral boundaries in the time variable. The article considers the first and second initial boundary value problems with zero initial condition for a uniformly parabolic equation that is one-dimensional in the spatial variable. It is assumed that the equation coefficients are bounded and uniformly continuous with a modulus of continuity satisfying twice the Dini condition. The specific feature of the study is that the right side of the equation can grow to infinity in a certain way when approaching the initial data assignment line. The solvability (in the classical sense) of these problems is established in a semi-bounded curvilinear domain in a plane with a nonsmooth lateral boundary from the Dini-Hölder class, admitting, in particular, so-called "beaks". The solution is constructed as a sum of the plane parabolic potential and the single layer potential, the kernel of which is the fundamental solution of the equation. The smoothness of the solution of these problems in the Dini-Hölder classes is investigated, and the corresponding correctness estimates are obtained. These estimates characterize the behavior of solutions and their first-order spatial derivative in the closure of the domain. The results obtained can be used to study heat and mass transfer processes. For the proof, the smoothness in the Dini-Hölder space of a plane parabolic potential with an unlimited density of the Dini class in a curvilinear domain with a nonsmooth lateral boundary is investigated. The obtained result can be used in solving other initial boundary value problems for an inhomogeneous parabolic equation.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

Maximal L^1-regularity for parabolic initial-boundary value problems with inhomogeneous data

End-point maximal L^1-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal L^1-regularity for initial-boundary value problems is established in time end-point case upon the homogeneous Besov space {dot{B}}_{p,1}^s({mathbb {R}}^n_+) with 1< p< infty and -1+1/p<sle 0 as well as optimal trace estimates. The main estimates obtained here are sharp in the sense of trace estimates and it is not available by known theory on the class of UMD Banach spaces. We utilize a method of harmonic analysis, in particular, the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity in the Besov and the Lizorkin–Triebel spaces.

Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz–Minkowski plane R12

In this paper, we investigate the evolution of spacelike curves in the Lorentz–Minkowski plane R12 along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike grim reaper curve as time tends to infinity.

An existence result for a strongly nonlinear parabolic equations with variable nonlinearity

We prove the existence of a solution for the strongly nonlinear parabolic initial boundary value problem associated to the equation ut − div a(x, t, ∇u) + g(x, t, u, ∇u) = f, where the vector field a(x, t, ξ) exhibits non-standard growth conditions.

Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary

Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.

A model in one-dimensional thermoelasticity

We study a one-dimensional nonlinear hyperbolic–parabolic initial boundary value problem occurring in the theory of thermoelasticity. We prove existence and uniqueness of the local-in-time strong solution. Also, some global-in-time weak measure valued solutions are proven to exist. To this end we introduce an auxiliary problem with artificial viscosity and prove its global-in-time well-posedness. Next, we show that solutions of the auxiliary problem converge, at some short time interval to the strong solution, and to our measure valued solution for an arbitrary time.

Numerical investigation of time delay parabolic differential equation involving two small parameters

PurposeThe purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP).Design/methodology/approachTo solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization.FindingsIt is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm.Originality/valueThis paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.

Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setti...

Existence of weak solutions for a nonlinear parabolic equations by Topological degree

We study the nonlinear parabolic initial boundary value problem associated to the equation ut − diva(x, t, u, grad u) = f(x, t), where the terme − diva(x, t, u, grad u) is a Leray-Lions operator, The right-hand side f is assumed to belong to L^q(Q). We prove the existence of a weak solution for this problem by using the Topological degree theory for operators of the form L + S, where L is a linear densely defined maximal monotone map and S is a bounded demicontinuous map of class (S+) with respect to the domain of L.

Iterative Methods for Mesh Approximations of Optimal Control Problems Controlled by Linear Equations with Fractional Derivatives

We consider the constrained optimal control problems governed by a parabolic initial-boundary value problem with time-fractional derivative and mixed boundary conditions. Control is carried out on the right side of the equation and on the right side of Neumann boundary condition. Finite element method with the quadratures is used for the approximation of the problem with respect to the spatial variable, and $$L1$$ -approximation for the time-fractional derivative taken in different definitions. Stability estimates in $$L^{2}$$ -norm via $$L^{2}$$ -norms of the control functions are obtained for the discrete state equation. They are used to prove the convergence and convergence rate of the proposed iterative methods for discrete optimal control problems. The main results are generalized to a problem with a state equation with fractional derivatives in time and space. The results of a series of numerical tests and their analysis are presented.

On a nonclassical problem for the heat equation and the Feller semigroup generated by it

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.