scholarly journals Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions

2015 ◽  
Vol 26 (01) ◽  
pp. 59-110 ◽  
Author(s):  
Claude Bardos ◽  
Denis Grebenkov ◽  
Anna Rozanova-Pierrat
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Order Term ◽  
Heat Content ◽  
Resistance Coefficient ◽  
Small Time ◽  
Heat Diffusion ◽  
Heat Problem ◽  
Short Time ◽  
Mathematical Justification

We consider a heat problem with discontinuous diffusion coefficients and discontinuous transmission boundary conditions with a resistance coefficient. For all bounded (ϵ, δ)-domains Ω ⊂ ℝn with a d-set boundary (for instance, a self-similar fractal), we find the first term of the small-time asymptotic expansion of the heat content in the complement of Ω, and also the second-order term in the case of a regular boundary. The asymptotic expansion is different for the cases of finite and infinite resistance of the boundary. The derived formulas relate the heat content to the volume of the interior Minkowski sausage and present a mathematical justification to the de Gennes' approach. The accuracy of the analytical results is illustrated by solving the heat problem on prefractal domains by a finite elements method.

A Systematic Asymptotic Expansion Procedure for Slender Ships

1964 ◽  
Vol 8 (03) ◽  
pp. 15-23 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Arbitrary Shape ◽  
Careful Consideration ◽  
Wave Resistance ◽  
Expansion Procedure ◽  
Systematic Solution ◽  
The Individual ◽  
Correct Boundary Conditions

Inner and outer expansions are used to formulate a systematic solution to the problem of the steady translation of a slender ship of arbitrary shape. Careful consideration is givien to finding the correct boundary conditions to be satisfied by successive terms in the expansions, and certain of the individual terms are determined partly or completely as functions of hull shape. Some results are given concerning the second approximations to the potential and wave resistance.

scholarly journals THE VALUATION OF SELF-FUNDING INSTALMENT WARRANTS

2017 ◽  
Vol 20 (04) ◽  
pp. 1750025
Author(s):  
J. N. DEWYNNE ◽  
N. EL-HASSAN
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Multiple Scales ◽  
Correction Term ◽  
Approximate Solutions ◽  
Fair Value ◽  
Small Time ◽  
Point Of View ◽  
First Order Correction ◽  
Short Time

We present two models for the fair value of a self-funding instalment warrant. In both models we assume the underlying stock process follows a geometric Brownian motion. In the first model, we assume that the underlying stock pays a continuous dividend yield and in the second we assume that it pays a series of discrete dividend yields. We show that both models admit similarity reductions and use these to obtain simple finite-difference and Monte Carlo solutions. We use the method of multiple scales to connect these two models and establish the first-order correction term to be applied to the first model in order to obtain the second, thereby establishing that the former model is justified when many dividends are paid during the life of the warrant. Further, we show that the functional form of this correction may be expressed in terms of the hedging parameters for the first model and is, from this point of view, independent of the particular payoff in the first model. In two appendices we present approximate solutions for the first model which are valid in the small volatility and the short time-to-expiry limits, respectively, by using singular perturbation techniques. The small volatility solutions are used to check our finite-difference solutions and the small time-to-expiry solutions are used as a means of systematically smoothing the payoffs so we may use pathwise sensitivities for our Monte Carlo methods.

scholarly journals SURVIVAL PROBABILITY (HEAT CONTENT) AND THE LOWEST EIGENVALUE OF DIRICHLET LAPLACIAN

2011 ◽  
Vol 25 (15) ◽  
pp. 1993-2007
Author(s):  
PAVOL KALINAY ◽  
LADISLAV ŠAMAJ ◽  
IGOR TRAVĚNEC
Keyword(s):  
Survival Probability ◽  
Heat Content ◽  
Simple Algorithm ◽  
Time Behavior ◽  
Dirichlet Laplacian ◽  
Geometric Characteristics ◽  
Long Time ◽  
Time Expansion ◽  
Short Time ◽  
Lowest Eigenvalue

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Padé interpolation between the short-time and the long-time behavior of the survival probability, i.e., between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

scholarly journals Approximation of SDE solutions using local asymptotic expansions

2021 ◽  
Author(s):  
A. M. Davie
Keyword(s):  
Asymptotic Expansion ◽  
Strong Approximation ◽  
Optimal Transport ◽  
Small Time ◽  
Degenerate System ◽  
Weak Approximation ◽  
Forward Equation ◽  
Smooth Coefficients ◽  
One Step ◽  
Edgeworth Type Expansion

AbstractWe develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

A Method of Calculating Radiative Heat Diffusion in Particle Simulations

1985 ◽  
Vol 6 (2) ◽  
pp. 207-210 ◽  
Author(s):  
L. Brookshaw
Keyword(s):  
Boundary Conditions ◽  
Radiative Heat ◽  
Three Dimensional ◽  
Heat Diffusion ◽  
New Method ◽  
Particle Simulations

AbstractA new method for solving heat diffusion in three dimensional particle simulations is described. The difficulties encounted by other authors is discussed, in particular the difficulty of including boundary conditions in particle simulations. One and three dimensional tests of the method are described.

scholarly journals SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

2017 ◽  
Vol 5 ◽  
Author(s):  
YUZURU INAHAMA ◽  
SETSUO TANIGUCHI
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Compact Manifold ◽  
Cut Locus ◽  
Finite Set ◽  
Rough Path ◽  
Hypoelliptic Heat Kernel ◽  
Short Time

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the ‘cut locus’ case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ rough path theory.

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