Parabolic quasi‐concavity for solutions to parabolic problems in convex rings

We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi‐concavity, which involves time and space jointly and is a stronger property than the spatial quasi‐concavity, and study the convexity of superlevel sets of the solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving Multi‐Scale problems with heterogeneous finite difference methods

Multi‐scale differential equations are problems in which the variables can have different length scales. The direct numerical solution of differential equations with multiple scales is often difficult due to the work for resolving the smallest scale. We present here a strategy which allows the use of finite difference methods for the numerical solution of parabolic multi‐scale problems, based on a coupling of macroscopic and microscopic models for the original equation.     

Time‐dependent operators on some non‐orientable projective orbifolds

G. R. Franssens In this paper, we present an explicit construction for the fundamental solution of the heat operator, the Schrödinger operator, and related first‐order parabolic Dirac operators on a class of some conformally flat non‐orientable orbifolds. More concretely, we treat a class of projective cylinders and tori where we can study parabolic monogenic sections with values in different pin bundles. We present integral representation formulas together with some elementary tools of harmonic analysis that enable us to solve boundary value problems on these orbifolds. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007     

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

In this work, we utilize the existing Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, together with the telescoping series method, to establish an observability inequality from measurable subsets in time‐space variable for the parabolic equation with Grushin operator in some multidimension domains. We can apply this observability inequality to show the bang–bang property for both time optimal and norm optimal control problems for this kind of singular parabolic equation. Copyright © 2016 John Wiley & Sons, Ltd.     

An ETD Crank‐Nicolson method for reaction‐diffusion systems

A novel Exponential Time Differencing Crank‐Nicolson method is developed which is stable, second‐order convergent, and highly efficient. We prove stability and convergence for semilinear parabolic problems with smooth data. In the nonsmooth data case, we employ a positivity‐preserving initial damping scheme to recover the full rate of convergence. Numerical experiments are presented for a wide variety of examples, including chemotaxis and exotic options with transaction cost. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

End effects in thermoelasticity

This paper derives spatial decay bounds in a dynamical problem of thermoelasticity defined on a semi‐infinite cylindrical region. Previous results for isothermal elastodynamics and the parabolic heat equation lead us to suspect that the solution of the thermoelastic problem should tend to zero, faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance. This is reminiscent of the decay rate in the second‐order parabolic problems. Similar arguments are sketched for the case of a non‐prismatic cylinder, and a non‐linear heat conduction problem for a rigid body is considered. Some extensions and generalizations are indicated in the final section. Copyright © 2001 John Wiley & Sons, Ltd.     

Propagation of Singularities of Solutions to Hyperbolic‐Parabolic Coupled Systems

In this paper, the authors study the propagation of singlarities for a semilinear hyperbolic‐parabolic coupled system, which comes from the model of thermoelasticity. Both of the Cauchy problem and the problem inside of a domain are considered. We obtain that the microlocal singularities of solutions to the semilinear hyperbolic‐parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of paradifferential operators. Furthermore, for the Cauchy problem of the semilinear coupled system, if the initial data have singularities at the origin, we prove that the solutions have the same order regularity with respect to spatial variables as in hyperbolic problems in the forward characteristic cone issuing from the origin, which improves the previous results for semilinear systems in thermoelasticity.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Elliptic and parabolic problems in unbounded domains

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi‐explicit representation formulas via operator‐valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

We consider partial differential equations of drift‐diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure‐valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0⁺. The time evolution of these singular masses is also explicitly described and, as a by‐product, uniqueness of this measure solution is obtained.     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data

We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

In this study, we investigate the concept of the complete flux (CF) obtained as a solution to a local boundary value problem (BVP) for a given parabolic singularly perturbed differential‐difference equation (SPDDE) with modified source term to propose an efficient complete flux‐finite volume method (CF‐FVM) for parabolic SPDDE which is μ‐ and ?‐uniform method where μ, ? are shift and perturbation parameters, respectively. The proposed numerical method is shown to be consistent, stable, and convergent and has been successfully implemented on three test problems.     

A functional partial differential equation arising in a cell growth model with dispersion

In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Analytic approach to solve a degenerate parabolic PDE for the Heston model

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second‐order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half‐plane. Copyright © 2017 John Wiley & Sons, Ltd.     

半线性椭圆方程支配系统的最优性条件

本文讨论了可能具有多值解的椭圆型偏微分方程支配系统的最优控制问题,我们通过构造一个抛物方程控制问题的逼近序列,并利用抛物方程控制问题的结果,得到了椭圆系统最优控制的必要条件．