Parabolic Case Research Articles

Gradient-type methods in inverse parabolic problems

This article is devoted to gradient-based methods for inverse parabolic problems. In the first part, we present a priori convergence theorems based on the conditional stability estimates for linear inverse problems. These theorems are applied to backwards parabolic problem and sideways parabolic problem. The convergence conditions obtained coincide with sourcewise representability in the self-adjoint backwards parabolic case but they differ in the sideways case. In the second part, a variational approach is formulated for a coefficient identification problem. Using adjoint equations, a formal gradient of an objective functional is constructed. A numerical test illustrates the performance of conjugate gradient algorithm with the formal gradient.

The effect of band nonparabolicity on modeling few-electron ground states of charge-tunable InAs/GaAs quantum dot

Based on the current spin density functional theory and the nonparabolic effective mass approximation, a three-dimensional model is presented to study the few-electron system in InAs/GaAs quantum dot for which the electron spectra have been obtained from the capacitance–voltage (CV) measurements by Miller et al. [Phys. Rev. B 56 (1997) 6764]. The model is an extension of the single-electron model proposed by Filikhin et al. [Solid State Comm. 140 (2006) 483]. Our results can quantitatively well interpret the experimental CV data. It is shown that the energy differences between the parabolic and nonparabolic approximations are comparable with the exchange-correlation energies. Moreover, the nonparabolic effect is shown to be increasingly more significant than that of parabolic case in higher magnetic fields. It is also more pronounced for larger number of electrons.

An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes

An analytical approach has been developed to derive aberration effects in parabolic and elliptic multilayer optics with weak interaction between photons and matter. The method is based on geometrical ray tracing including refraction effects up to the first order of the refractive index decrement delta. In the parabolic case, the derivation leads to simple parametric equations for the caustic shape. In the elliptic case, the analytical results more involved, but can be well approximated by the parabolic solution. Both geometries are compared with regard to the fundamental impact on their focusing properties.

Exercise boundary of American-style Asian option

In this paper, we study the smoothness of the optimal exercise boundary (i.e. free boundary) of American-style arithmetic average Asian option with floating strike price in finite horizon case (i.e. parabolic case). Applying stochastic analysis the authors of [G. Peskir, N. Yus, On Asian options of American type, Exotic Option Pricing and Advanced Levy Models, Wiley, Eindhoven, 2004, pp. 217–235; G. Peskir, A. Shiryaev, Optimal stopping and free boundary problems, Lectures in Mathematics ETH Zurich, Birkhauser, 2006, pp. 416–436] proved that the optimal exercise boundary is a continuous and monotonic curve. Based on their results we prove that the optimal exercise boundary is infinitely differentiable using PDE method.

An analytic investigation into the behavior of Kepler’s equation1

Kepler’s Equation is solved over the entire range of elliptic and parabolic motion. Global solution accuracy is maintained by defining four M-e plane solution sub-domains. Three computational methods are concatenated in each sub-domain to solve the problem. Perturbation solutions obtain the basic solutions in each sub-domain. Accelerated series convergence is achieved by transforming the perturbation solutions with Pade approximations. A variable-order Newton extrapolation method is used to further transform the Pade results to yield thirteen digits of precision over the entire range of elliptic and parabolic motion. Only four trigonometric evaluations are required for half of the M-e plane. The remaining half of the M-e plane, including the parabolic special case, only requires a cube root and four trigonometric evaluations. Along the M∼0 and e∼0.7 boundary, a closed-form solution is obtained for a pair of complex-valued roots that possess inflection points that limit the radius of convergence of series approximations for Kepler’s equation. The proposed algorithm effectively provides a closed-form solution for Kepler’s equation, in the sense that the solution accuracy exceeds any projected operational mission need.

Non-autonomous stochastic Cauchy problems in Banach spaces

In this paper we study the non-autonomous stochastic Cauchy problem on a real Banach space E, dU(t) = A(t)U(t) dt+B(t) dWH(t), t ∈ [0, T ], U(0) = u0. Here,WH is a cylindrical Brownian motion on a real separable Hilbert spaceH, (B(t))t∈[0,T ] are closed and densely defined operators from a constant domain D(B) ⊂ H into E, (A(t))t∈[0,T ] denotes the generator of an evolution family on E, and u0 ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then, we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapien, and Zabczyk we obtain space-time regularity results in the context of parabolic evolution families on Banach spaces. Afterwards, we apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.

A note on shock profiles in dissipative hyperbolic and parabolic models

This note presents a comparative study of shock profiles in dissipative systems. Main assumption is that both hyperbolic and parabolic model are reducible to the same underlying equilibrium system when dissipative effects are neglected. It will be shown that the highest characteristic speed of equilibrium system determines the critical value of the shock speed for which downstream equilibrium state bifurcates. It will be also shown that it obeys the same transcritical bifurcation pattern in hyperbolic, as well as in parabolic case.

Boundary constructions of petals at the Wolff point in the parabolic case

In the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff point, for a holomorphic self-map f of the open unit disc Δ ⊂ ℂ of parabolic type. The result is obtained in the framework of two interesting dynamical situations which require different kinds of regularity of f at the Wolff point τ: f of non-automorphism type and $$\Re e(f''(\tau )) > 0$$ or f injective of automorphism type, f∈C 3+ɛ(τ) and $$\Re e(f''(\tau )) = 0$$ .

Dirac concentrations in Lotka-Volterra parabolic PDEs

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass, this can be interpreted as well separated populations. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a BV estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counterexamples showing that jumps on the Dirac locations are indeed possible.

Solving Kepler’s Equation using Bézier curves

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bezier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10−13 degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10−5 degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations.

A hyperbolic reaction–diffusion model for the hantavirus infection

AbstractA hyperbolic reaction–diffusion model for the hantavirus infection, generalizing the parabolic set of equations recently derived by Abramson and Kenkre, is proposed within the context of Extended Thermodynamics. The model, as in the parabolic case, captures some of the realistic features of the dynamics of hantavirus in mice population, while it avoids the unphysical features concerning the instantaneous diffusive effects typical of parabolic equations. Traveling wave solutions, related to the spread of the infection in the landscape, are investigated. Both analytical and numerical results obtained herein are discussed and validated from the behavior of the biological system. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Maximum principle for fully nonlinear equations via the iterated comparison function method

We present various versions of generalized Aleksandrov–Bakelman–Pucci (ABP) maximum principle for L p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the “iterated” comparison function method, which was introduced in our previous paper (Koike and Świech in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Świech in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5–6), 967–983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293–1326, 1998; Comm. Partial Diff. Eq. 25, 1997–2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27–76, 1992) and (Crandall and Świech in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.

On the homogenization of some linear problems in domains weakly connected by a system of traps

AbstractThe aim of the paper is to study the asymptotic behaviour of solutions of second‐order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain Ωε weakly connected by a system of traps 𝒫ε, where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain Ωε ⊂Ω a bounded open set of ℝ3 such that Ωε =Ω1ε ∪Ω2ε ∪𝒫ε ∪ Wε, where Ω1ε, Ω2ε are non‐intersecting subdomains strongly connected with respect to Ω, 𝒫ε is a system of traps and meas Wε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain Ωε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Schwarz methods for discrete elliptic and parabolic problems with an application to nuclear waste repository modelling

The paper is devoted to the numerical solution of both elliptic and parabolic problems by overlapping Schwarz methods. It demonstrates that while the two-level Schwarz method is necessary for the efficient solution of discrete elliptic problems, the one-level Schwarz method can be sufficiently efficient in the parabolic case. The paper includes results from numerical simulations of a large-scale model of the performance of a nuclear waste repository.

Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations

Travelling fronts for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$ exist for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both in the parabolic case when $\varepsilon >0$ and in the hyperbolic case when $\varepsilon =0$. We prove that as $\varepsilon \rightarrow 0$, the minimal velocity ${c_{\varepsilon}^*}$ converges to $c^*$, the minimal value when $\varepsilon =0$, and that ${c_{\varepsilon}^*}\geq c^*$ for all $\varepsilon >0$. The proof uses comparison theorems and the variational characterization of the minimal parabolic front velocity. This convergence also yields a reaction-independent sufficient condition for the minimal velocity of the parabolic problem for small positive $\varepsilon$ to be strictly greater than the value predicted by the problem linearized about the unstable equilibrium, that is, for the minimal-velocity travelling front of the viscous equation to be pushed for sufficiently small $\varepsilon$.

The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$-coefficients

Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of $\mathbb R^n$. We prove solvability of the elliptic problem and maximal $L^q$-$L^p$-estimates for the solution of the parabolic problem provided the coefficients $a_{ij} \in L^\infty$ satisfy a Cordes condition and $p \in (1,2]$ is close to $2$. This implies that in two dimensions, i.e., $n=2$, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and $p \in (1,2]$ is close to $2$, for maximal $L^q$-$L^p$-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.

Parabolic finite element equations in nonconvex polygonal domains

Let Ω be a bounded nonconvex polygonal domain in the plane. Consider the initial boundary value problem for the heat equation with homogeneous Dirichlet boundary conditions and semidiscrete and fully discrete approximations of its solution by piecewise linear finite elements in space. The purpose of this paper is to show that known results for the stationary, elliptic, case may be carried over to the time dependent parabolic case. A special feature in a polygonal domain is the presence of singularities in the solutions generated by the corners even when the forcing term is smooth. These cause a reduction of the convergence rate in the finite element method unless refinements are employed.

Diamagnetic susceptibility of a hydrogenic donor in low lying excited states in a Quantum Well

Abstract We present the χ dia of a hydrogenic donor in low lying excited states like 2s, 2p± and 2p0 in a GaAs/ AlxGa1−xAs Quantum Well for various Al compositions x both in the parabolic and non-parabolic conduction band model using the variational method. To our knowledge there has been no such reports both theoretically and experimentally in the literature. It was observed that χ dia increases with increase of x for both the parabolic and non-parabolic cases. The effect of non-parabolicity on χ dia is larger for the lower well width region for 2s and 2p± states. Interestingly, we observe a “turn over” in the variation of χ dia against well widths in the lower well width region, a trend which is similar to the one observed in the variation of binding energy of donors in low lying excited states.

A numerical simulation of mask shape effect for fabrication of hexagonal microlens array by using excimer laser dragging process

This investigation analysed the characteristics of simulating the fabrication of an aspheric hexagonal microlens array using a laser dragging process. Traditionally, an aspheric microlens array with rectangular coordinates was produced using an excimer laser dragging process. For the hexagonal case, the first step in the proposed dragging process is to generate microchannels with a cross-section shape similar to the mask. Then, the dragging procedure is repeated twice, each time after rotating the work piece by 60° relative to the previous channels. This microlens array can be applied in image processing. Variously shaped masks were computer simulated. The results demonstrate that the parabolic-shaped mask can generate excellent axial symmetry for both rectangular and hexagonal microlens array. The analytical results of this study indicate that a hexagonal microlens array has a better axial symmetry than a rectangular microlens array. But for the parabolic case, a hexagonal microlens array has a smaller fill factor than a rectangular microlens array.

The Energy Spectrum of Carriers between Two Concentric Spheres of Kane‐Type Semiconductors

The electronic states of carriers between two concentric spheres of Kane‐type semiconductor are theoretically investigated and compared with the results of the parabolic band approximation. Calculations are performed for a hard‐wall confinement potential and the eigenstates and the eigenvalues of the Kane Hamiltonian are obtained. Taking into account the real band structure (strong spin‐orbital interaction, narrow band gap), the size dependence of the energy of electrons, light holes, and spin‐orbital splitting holes in InSb semiconductor concentric spheres are calculated. According to the obtained results both in parabolic and nonparabolic (Kane model) cases, the electron energy levels come close to each other with the increasing of the radius.