Second-order Parabolic Equations Research Articles

On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds

The paper is concerned with problems of existence, uniqueness, and stabilization of weak solutions of one class of semilinear second-order parabolic differential equations on closed manifolds. These equations are inhomogeneous analogues of the Kolmogorov-Petrovskii-Piskunov-Fisher equation, and have significant applied and mathematical value.

О корректности начальных задач для уравнения дробной диффузии

The paper studies a second-order parabolic partial differential equation with fractional differentiation with respect to a time variable. The fractional differentiation operator is a linear combination of the Riemann-Liouville and Gerasimov-Caputo fractional derivatives. It is shown that the distribution of orders of fractional derivatives, included in the equation affects the correctness of the initial problems for the equation under consideration.

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

Parabolic differential equations with bounded delay

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

A Solution to Linear Black-Scholes Second-order Parabolic equation in sobolev space

In this paper, a set of functions were constructed that transforms Black-Scholes partial differential equation into weak formulations. The analytical solutions: existence, uniqueness and other estimates were also obtained in weak form with the use of boundary conditions to establish the effects of its financial implications in Sobolev spaces. The regularity conditions of the problem were considered which the coefficients, the boundary of the domain are all smooth functions. To this end, the definitions, assumptions which paved way to useful assertions are illustrated in this paper.

Subdomain solution decomposition method for nonstationary problems

The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Efendiev and Vabishchevich (2021) [22]), a new class of methods of approximate solution of nonstationary problems has been introduced based on decomposition not of operators but of the solution itself. This new approach with subdomain solution selection is used in this paper to construct domain decomposition schemes. The boundary value problem for a second-order parabolic equation in a rectangle with a difference approximation in space is typical. Two and three-level schemes for decomposition of the domain with and without overlapping subdomains are investigated. Our numerical experiments complement the theoretical results.

Global existence of weak solution to a phase‐field model on martensitic phase transformations

We introduce a mesoscale model capable of describing transformations between martensitic variants as well as twinning. The model consists of two nonlinear parabolic equations of second order, and one of equations is degenerate. We prove the existence of global‐in‐time solutions to an initial‐boundary value problem in one space dimension of this model.

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.

Digital Simulations for Three-dimensional Nonlinear Advection-diffusion Equations Using Quasi-variable Meshes High-resolution Implicit Compact Scheme

A two-level implicit compact formulation with quasi-variable meshes is reported for solving three-dimensions second-order nonlinear parabolic partial differential equations. The new nineteen-point compact scheme exhibit fourth and second-order accuracy in space and time on a variable mesh steps and uniformly spaced mesh points. We have also developed an operator-splitting technique to implement the alternating direction implicit (ADI) scheme for computing the 3D advection-diffusion equation. Thomas algorithm computes each tri-diagonal matrix that arises from ADI steps in minimal computing time. The operator-splitting form is unconditionally stable. The improved accuracy is achieved at a lower cost of computation and storage because the spatial mesh parameters tune the mesh location according to solution values' behavior. The new method is successfully applied to the Navier-Stokes equation, advection-diffusion equation, and Burger's equation for the computational illustrations that corroborate the order, accuracies, and robustness of the new high-order implicit compact scheme. The main highlight of the present work lies in obtaining a fourth-order scheme on a quasi-variable mesh network, and its superiority over the comparable uniform meshes high-order compact scheme.

Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity

We deal with the initial-boundary value problems with some restrictions at infinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions.

Integral Analogue of the First Initial-Boundary Value Problem for Second-Order Hyperbolic and Parabolic Equations

Integral Analogue of the First Initial-Boundary Value Problem for Second-Order Hyperbolic and Parabolic Equations

Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

In this paper, we study some controllability and observability problems for stochastic systems coupling fourth- and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on the one hand, we study the controllability of a linear backward system where the couplings are made only through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a simplified nonlinear coupled model of forward equations. This case, which is well known to be harder to solve, follows a methodology that has been introduced recently and relies on an adaptation of the well-known source term method in the stochastic setting together with a truncation procedure. This approach gives a new concept of controllability for stochastic systems.

A game-theoretic perspective to study a nonlinear stochastic parabolic model of population competition

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players.

Well-posedness of solutions to a phase-field model for the martensitic phase transformations

We study a phase-field model, which describes the transformations for the austenite-martensite and the multiple twinning in Martensite. The model consists of two nonlinear parabolic equations of second order. We first show the existence of local solutions to an initial-boundary value problem by utilizing the Banach fixed-point theorem. Then we verify the solutions is global. Finally we investigate the regularity and uniqueness of the solution.

О ЕДИНСТВЕННОСТИ В ЗАДАЧАХ ОПРЕДЕЛЕНИЯ ТОЧЕЧНЫХ ИСТОЧНИКОВ В МАТЕМАТИЧЕСКИХ МОДЕЛЯХ ТЕПЛОМАССОПЕРЕНОСА

We consider the problem of determining point sources for mathematical models of heat and mass transfer. The values of a solution (concentrations) at some points lying inside the domain are taken as overdetermination conditions. A second-order parabolic equation is considered, on the right side of which there is a linear combination of the Dirac delta functions δ(x–xi) with coefficients that depend on time and characterize the intensities of sources. Several different problems are considered, including the problem of determining the intensities of sources if their locations are given. In this case, we present the theorem of uniqueness of solutions, the proof of which is based on the Phragmén–Lindelöf theorem. Next, in the model case, we consider the problem of simultaneous determining the intensities of sources and their locations. The conditions on the number of measurements (the ovedetermination conditions) are described which ensure that a solution is uniquely determined. Examples are given to show the accuracy of the results. This problem arises when solving environmental problems, first of all, the problems of determining the sources of pollution in a water basin or atmosphere. The results are important when developing numerical algorithms for solving the problem. In the literature, such problems are solved numerically by reducing the problem to an optimal control problem and minimizing the corresponding objective functional. The examples show that this method is not always correct since the objective functional can have a significant number of minima.

ОБ ИДЕНТИФИКАЦИИ КОЭФФИЦИЕНТА ТЕПЛООБМЕНА В СЛОИСТОЙ СРЕДЕ

In the article we consider the well-posedness in Sobolev spaces of the inverse problems of determining the heat exchange coefficient at the interface which is included in the transmission condition of the imperfect contact type. In a cylindrical spatial domain, a second-order parabolic equation is considered. The domain is divided into two sub-domains, on the common part of the boundary of which the transmission condition is set. The heat exchange coefficient included in the transmission condition is sought as end segment in series with time-dependent unknown Fourier coefficients. The equation is supplemented with general boundary conditions and initial conditions, as well as overdetermination conditions. The ovedetermination conditions are the values of a solution at some points lying in the spatial domain. Under natural smoothness conditions for the data and the location of the measurement points, the existence and uniqueness theorem local in time is demonstrated. The obtained solution to the problem is regular, i. e., all generalized derivatives included in the equation are summable to some power, and the equation holds almost everywhere. The method is constructive, and the approach allows to develop numerical methods for solving the problem. The proof is based on a priori estimates obtained and the fixed-point theorem.

A Numerical Solution of Heat Equation for Several Thermal Diffusivity Using Finite Difference Scheme with Stability Conditions

The heat equation is a second-order parabolic partial differential equation, which can be solved in many ways using numerical methods. This paper provides a numerical solution that uses the finite difference method like the explicit center difference method. The forward time and centered space (FTCS) is used to a problem containing the one-dimensional heat equation and the stability condition of the scheme is reported with different thermal conductivity of different materials. In this study, results obtained for different thermal conductivity of distinct materials are compared. Also, the results reveal the well-behavior properties of the materials in good agreement.

Contact problem for a second-order parabolic equation with Dini-continuous coefficients

Contact problem for a second-order parabolic equation with Dini-continuous coefficients

Approximate solutions to second-order parabolic equations: Evolution systems and discretization

<p style='text-indent:20px;'>We study the discretization of a linear evolution partial differential equation when its Green's function is known or well approximated. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method that we developed in <i>J. Math. Phys.</i> <b>51</b> (2010), n. 10, 103502 (reference [<xref ref-type="bibr" rid="b15">15</xref>]). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.</p>

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

A numerical scheme is constructed for the second-order parabolic partial differential equation with piecewise smooth initial data. The scheme comprises an orthogonal spline collocation strategy with the Rannacher time-marching. The irregular behavior of the underlying initial conditions of such differential equations results in inaccurate approximations due to the quantization error. For such problems, even the A-stable Crank-Nicolson scheme yields only first-order convergence in the temporal direction, with oscillations near the discontinuity. Applying mathematical perspective to dampen these oscillations, we present a highly accurate orthogonal spline collocation method with a smooth but straightforward time-marching scheme that significantly improves the convergence order. Through rigorous analysis, the present conjunctive scheme's convergence in the spatial direction is shown fourth-order (in L2 and L∞-norms) and third-order (in H1-norm), and it is shown second-order in the temporal direction. The performance and robustness of the contributed scheme are conclusively demonstrated with two test examples.