• Population Genetics and Coalescent Theory
Theoretical population genetics bridges mathematics and evolutionary biology. A key innovation in the field was the development of the n-coalescent by the probabilist JFC Kingman in 1982. The n-coalescent introduced a retrospective view of a population of individuals allowing probabilistic statements of the past and allowing us to infer parameters of complex population models using genomic data of many individuals. Coalescente theory is based on a Markov process using exponential distribution.
 I have expanded the theory by generalizing the underlying Poisson process of the coalescent process using a recent advancement in the description of the fractional Poisson process that is a semi-Markov process. I call my extension the fractional coalescent, or f-coalescent. The simulated data has been used to validate the model, and the real data has been used to show the advantages of fractional coalescent.
When data is simulated using models with alpha<1 (a key parameter of the fractional Poisson process), or for real datasets (H1N1 influenza, Malaria parasites, Humpback whales), Bayes factor comparisons show an improved model fit of the f-coalescent over the n-coalescent. In this model, the distribution of the number of offspring depends on a parameter alpha which is a potential measure of the environmental heterogeneity that is commonly ignored in current inferences.
  • Fractional Viscoelasticity in Fractal Media
I predicted complex phenomena in viscoelastic materials. Such materials exhibit unusual interactions between elastic and viscous forces. The unique properties of such materials make them ideal for the study of fractional derivatives but they also have practical importance for many smart damping applications (i.e., robotics, automotive, and aerospace structures). While the Scott-Blair fractional model of viscoelasticity has been introduced by using the idealized model of spring and dashpot, I introduced theoretical justification for this model where the results have been experimentally validated using Bayesian uncertainty analysis.
This analysis shows superior results compared to the integer Maxwell model. Also, for the first time I reported using a nonlinear fractional model of viscoelasticity. Importantly, the use of fractional order calculus operations to describe the rate-dependent viscoelastic response yielded a model with self-consistent model parameters. Therefore, the model performs better at predictions across a broad range of experimental rates. This led me to discover a physical explanation for the fractional time derivative applied to viscoelasticity derived by using thermal diffusion and fractal dimensions in fractal media.
The challenge of deriving this explanation has been long-standing (dating back to the work of Bagley and Torvik and Mandelbrot) and this shows the fractional model of viscoelasticity provides more information about the structure of a material. This area opens up an application of fractional calculus which may describe the multi-scale thermomechanical material behavior of many polymers.
  • Numerical Methods for Solving the Fractional System
I introduced a new method for solving fractional differential systems. The method is based on hybrid-function approximation in which the hybrid functions consist of block-pulse functions and Bernoulli polynomials. The method uses the Riemann-Liouville fractional integral operator to reduce the solution of the system to a set of algebraic equations.
 This method can be applied to linear and nonlinear distributed fractional order differential equations, nonlinear fractional integro-differential equations, fractional Bagley-Torvik equations as well as fractional optimal control problems.
  • Hybrid Functions of Block-pulse and Bernoulli Polynomials
I introduced hybrid functions of block-pulse and Bernoulli polynomials as a new base for solving differential systems. My work demonstrated that using Bernoulli polynomials has several advantages over the most commonly used method, Legendre polynomials.
 First, I showed that Bernoulli polynomials have fewer errors in their operational matrix of integrals than in those of shifted Legendre polynomials. Second, Bernoulli polynomials have fewer terms than Legendre polynomials, and those individual terms have lower coefficients. Finally, I have shown that the operational matrix for the hybrid functions of block-pulse and Bernoulli polynomials is more sparse than those that can be achieved through a block-pulse hybrid with Legendre, Chebychev, or Taylor polynomials.
 This makes my method computationally more efficient than other methods, even the widely used Legendre polynomials. This method can be applied to nonlinear, constrained optimal control problems, optimal control of systems described by integro-differential equations, Duffing equation, multi-delay, and piecewise constant delay systems.
  • Least Squares Fit of Highly Oscillatory Functions
Many physical phenomena exhibit a pronounced oscillatory character. Behavior of pendulum-like systems, vibrations, resonances or wave propagation are all phenomena of this type in classical mechanics, while the same is true for the typical behavior of quantum particles. Finding a suitable tool for a good approximation of these kinds of functions is therefore of acute interest from both a mathematical and an application perspective.
 I developed a new base for finding the least squares fit of oscillatory functions. The results obtained with the new base are compared to the ones obtained by means of the Legendre polynomials and with theoretical predictions. The new base is attractive to use in many other mathematical contexts where highly oscillatory functions are involved.