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.
