An Exponential Time Differencing Scheme with a Real Distinct Poles Rational Function for Advection-Diffusion Reaction Equations

A second order Exponential Time Differencing (ETD) scheme for advection-diffusion reaction systems is developed by using a real distinct poles rational function for approximating the underlying matrix exponential. The scheme is proved to be second order convergent. It is demonstrated to be robust for reaction-diffusion systems with non-smooth initial and boundary conditions, sharp solution gradients, and stiff reaction terms. In order to apply the scheme efficiently to higher dimensional problems, a dimensional splitting technique is also developed. This technique can be applied to all ETD schemes and has been found, in the test problems considered, to reduce computational time by up to 80%

Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives. The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation

Market Capitalization and Firm Value: The Size Factor

Current multifactor valuation pricing models use size (measured by market capitalization) of a firm as one factor to determine the value of a security. The problem with current standard models was that none of them could explain the value of a security consistently and accurately based on current factors and in particular the size factor. The purpose of this quantitative study using existing time-series data over a 10-year period from 2006 to 2015 was to examine the impact of size factor on the realized rate of return of financial securities, while controlling for the impact of market rate of return. There are currently many valuation models but there is no 2-factor model or a model that uses a size factor that includes mid-cap sized securities. The research questions examined mid-cap sized securities for the size factor in a 2-factor model to determine the accuracy of predicting financial returns compared to the current standard Fama-French 3-factor model. The main theoretical framework that guided the study was the efficient market hypothesis that postulates that the price of a stock reflects all relevant available information. Data were collected for historical returns of 15 individual firms and portfolios of securities based on size. Multiple regression analysis methodology was used to examine the impact of size factor on the realized rate of return of financial securities, while controlling for the impact of market rate of return in the modified 2-factor model that included mid-caps. The results of the study indicate that size is a statistically significant factor in a 2-factor model that included mid-caps. The positive social impact of this study is that it could provide greater confidence in financial markets by providing a fair and equitable means of investment and flow of capital for a robust economy