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Partial differential equations

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Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell Apr 2018

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Brandon Russell

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so-called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H^1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating, periodic, bounded, and measurable coefficients. Finally, we connect these …

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A Regression Model To Investigate The Performance Of Black-Scholes Using Macroeconomic Predictors, Timothy A. Smith, Ersoy Subasi, Aliraza M. Rattansi Dec 2013

A Regression Model To Investigate The Performance Of Black-Scholes Using Macroeconomic Predictors, Timothy A. Smith, Ersoy Subasi, Aliraza M. Rattansi

Timothy Smith

As it is well known an option is defined as the right to buy sell a certain asset, thus, one can look at the purchase of an option as a bet on the financial instrument under consideration. Now while the evaluation of options is a completely different mathematical topic than the prediction of future stock prices, there is some relationship between the two. It is worthy to note that henceforth we will only consider options that have a given fixed expiration time T, i.e., we restrict the discussion to the so called European options. Now, for a simple illustration of …

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Renormalization Group Analysis Of Nonlinear Diffusion Equations With Periodic Coefficients, G. A. Braga, Fred Furtado, J. M. Moreira, L. T. Rolla Jun 2012

Renormalization Group Analysis Of Nonlinear Diffusion Equations With Periodic Coefficients, G. A. Braga, Fred Furtado, J. M. Moreira, L. T. Rolla

Fred Furtado

In this paper we present an efficient numerical approach based on the renormalization group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the veri. cation of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the renormalization group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the …

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Accuracy, Resolution And Stability Properties Of A Modified Chebyshev Method, Jodi Mead, Rosemary A. Renaut Jul 2010

Accuracy, Resolution And Stability Properties Of A Modified Chebyshev Method, Jodi Mead, Rosemary A. Renaut

Jodi Mead

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N−2M) for stable explicit solvers. Theoretically, time steps may be increased to O(N−M) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys., 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N, namely N ≲ 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is …

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