Second-order Parabolic Equations Research Articles

Existence and Uniqueness of the Solution of a Mixed Problem for a Parabolic Equation Under Nonconventional Boundary Conditions

Our research focuses on examining a mixed problem associated with a second-order parabolic equation that features temporal mixing and variable coefficients, subject to non-local and non-self-adjoint boundary conditions. We establish the problem’s unique solvability by imposing specific conditions on the provided data, utilizing a combination of residue and contour integral methods. Additionally, our study produces an explicit analytical solution for addressing the problem at hand.

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.

Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space

Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.

Computational decomposition and composition technique for approximate solution of nonstationary problems

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit–implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition–composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Schauder and Calderón–Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ∂tu−F(x,t,D2u)=f(x,t)inQ1=B1×(−1,0],provided that the source f and the coefficients of F are Hólder continuous functions, and F enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are C1,Log-Lip-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Doubly reflected generalized BSDEs with jumps and an obstacle problem of parabolic IPDEs with nonlinear Neumann boundary conditions

Abstract A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Insensitizing control problems for the stabilized Kuramoto–Sivashinsky system

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto–Sivashinsky model. The main idea is to look for controls such that some functional of the states (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying the inverse mapping theorem.

An explicit fourth-order accurate compact method for the Allen-Cahn equation

<abstract><p>In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.</p></abstract>

High-order estimates for fully nonlinear equations under weak concavity assumptions

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems.

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

Numerical Simulation of Hydrogen Diffusion in Cement Sheath of Wells Used for Underground Hydrogen Storage

The negative environmental impact of carbon emissions from fossil fuels has promoted hydrogen utilization and storage in underground structures. Hydrogen leakage from storage structures through wells is a major concern due to the small hydrogen molecules that diffuse fast in the porous well cement sheath. The second-order parabolic partial differential equation describing the hydrogen diffusion in well cement was solved numerically using the finite difference method (FDM). The numerical model was verified with an analytical solution for an ideal case where the matrix and fluid have invariant properties. Sensitivity analyses with the model revealed several possibilities. Based on simulation studies and underlying assumptions such as non-dissolvable hydrogen gas in water present in the cement pore spaces, constant hydrogen diffusion coefficient, cement properties such as porosity and saturation, etc., hydrogen should take about 7.5 days to fully penetrate a 35 cm cement sheath under expected well conditions. The relatively short duration for hydrogen breakthrough in the cement sheath is mainly due to the small molecule size and high hydrogen diffusivity. If the hydrogen reaches a vertical channel behind the casing, a hydrogen leak from the well is soon expected. Also, the simulation result reveals that hydrogen migration along the axial direction of the cement column from a storage reservoir to the top of a 50 m caprock is likely to occur in 500 years. Hydrogen diffusion into cement sheaths increases with increased cement porosity and diffusion coefficient and decreases with water saturation (and increases with hydrogen saturation). Hence, cement with a low water-to-cement ratio to reduce water content and low cement porosity is desirable for completing hydrogen storage wells.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Internal and Startup Controls of the Solutions of Boundary-Value Problem for Parabolic Equations with Degenerations

For a second-order parabolic equation with degenerations, we construct the solution of the problem of optimal control for systems described by the first boundary-value problem with internal and startup controls. The coefficients of parabolic equation have power singularities of any order in time and space variables on a certain set of points.

L2-estimates of error in homogenization of parabolic equations with correctors taken into account

We consider second-order parabolic equations with bounded measurable \(\varepsilon\)-periodic coefficients. To solve the Cauchy problem in the layer \( R^d \times(0,T) \) with the nonhomogeneous equation, we obtain approximations in the norm \(\|\cdot\|_{L^2(R^d\times(0,T))}\) with remainder of order \(\varepsilon^2\) as \(\varepsilon \to 0.\)

Applications of a space-time FOSLS formulation for parabolic PDEs

Abstract In this work, we show that the space-time first-order system least-squares formulation (Führer, T. & Karkulik, M. (2021) Space–time least-squares finite elements for parabolic equations. Comput. Math. Appl.92, 27–36) for the heat equation and its recent generalization (Gantner, G. & Stevenson, R. (2021) Further results on a space-time FOSLS formulation of parabolic PDEs. ESAIM Math. Model. Numer. Anal.55, 283–299) to arbitrary second-order parabolic partial differential equations can be used to efficiently solve parameter-dependent problems, optimal control problems and problems on time-dependent spatial domains.

Linear-quadratic mean field games of controls with non-monotone data

In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding N N -player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from the LQ mean field game of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the N N -player game directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the N N -player game to the mean field game and the propagation of chaos property for the related optimal trajectories.

Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

<p style='text-indent:20px;'>In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a <inline-formula><tex-math id="M1">\begin{document}$ \phi(\triangle t) $\end{document}</tex-math></inline-formula>-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter <inline-formula><tex-math id="M2">\begin{document}$ \triangle t $\end{document}</tex-math></inline-formula> goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.</p>