# My research

I am an applied mathematician working in mathematical modelling based on differential equations. The main topics of my interests are associated with natural phenomena, however, occasionally I also work on medical, industrial, and engineering problems. Usually, the main aim of my work is to extract some useful information about a particular phenomenon with the use of analytical tools. Frequently, they include perturbation methods, inverse problems, estimation, and integral equations. On the other hand, I also develop numerical methods that can be used in solving some technically difficult problems and on which the analytics can be tested upon.

My main mathematical interests can be grouped according to the field and are given below sorted chronologically (those on the top are the ones on which I work recently). My papers are listed on the other page.

• Mathematical geoscience with emphasis on climate dynamics and shallow flow.
• Numerical methods for nonlocal PDEs.
• Biomechanics (in particular – models of running).
• Capillary phenomena with strongly nonlinear dynamics (Lucas-Washburn’s equation).
• Anomalous transport in porous media.
• Modelling sensors of volatile substances.
• Modelling in optometry (in particular – corneal topography).

The above list hopefully will be expanded in due course which is what I find the most rewarding benefit of being a mathematician. You always have to learn something new from across a broad spectrum of seemingly distinct fields. The mathematical language is the glue that binds them all.

## Some of my results

### Model of corneal topography

Cornea is the frontal part of the eye and is responsible for about $2/3$ of the refractive power. Moreover, being the first contact with the outside world, the cornea has to be durable to withstand many dangers and exterior factors that can damage the eye. Therefore, cornea combines both optical (refraction) and material (elasticity) properties. It is very interesting and needed to find a mathematical model for the cornea. We have focused on describing its topography since perturbations in its shape can cause various sight disorders such as myopia or hyperopia among others.

In the nondimensional units, the model has the form

$\nabla \cdot \left(\dfrac{\nabla h}{\sqrt{1+\left|\nabla h\right|^2}}\right) + a(\textbf{x}) h = \dfrac{b(\textbf{x})}{\sqrt{1+\left|\nabla h\right|^2}}, \quad h(\textbf{x}) = 0 \text{ for } \textbf{x}\in\Omega, \quad \textbf{x}\in\mathbb{R}^n$,

where $h=h(\textbf{x})$ is the corneal deflection from the reference domain $\Omega$ (usually an ellipse in two dimensions), $a(\textbf{x})$ is the elasticity coefficient, and $b(\textbf{x})$ denotes the intra-ocular pressure. The above model is stated in the $n$-dimensional version however, a meaningful case for applications occurs when $n=2$.

Our results concerning the above model are the following:

• its derivation from physical principles,
• existence and uniqueness results for constant $a$, $b$ and $n=2$,
• various estimates on the solution,
• algorithms for solving inverse problem: find $a$ and $b$ when $h$ is known.

For applications, inverse problems are essential since its stable solution gives the values of biomechanical properties of the cornea based on its shape. In particular, one can obtain a knowledge of the intraocular pressure by measuring the corneal topography.

The above model has been generalized and analyzed thoroughly for $n\in\mathbb{R}$ and arbitrary $a=a(\textbf{x})$ and $b=(\textbf{x})$ by Chiara Corsato, Pierpaolo Omari et al. A very interesting summary of their results can be found in the paper A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis and references therein.

### Anomalous diffusion: time-fractional porous medium equation

When we consider complex porous media with the so-called memory effects, the governing equation for evolution of moisture $u=u(x,t)$ has the form

$$\partial^\alpha_t u = \left(D(u)u_x\right)_x, \quad 0<\alpha\leq 1,$$

where $D=D(u)$ is the moisture-dependent diffusivity while $\alpha$ is the order of Caputo fractional derivative

$$\partial^\alpha_t u = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha} u_t(x,s)ds.$$

The influence of the history of evolution is encoded in the above nonlocal derivative operator. In the usual porous medium (Brooks-Correy) case we usually take $D(u)\propto u^m$ for some $m>0$ which models moisture percolation through various porous media with decent accuracy. For this important case we have obtained the following results.

• Derivation of the above governing equation: Darcy's Law + Nonlocal conservation of mass.
• Existence and uniqueness for compactly supported self-similar solution on the half-line.
• Solution of an inverse problem of diffusivity identification.
• Various estimates and approximations of the solution. In particular, estimate of the compact support (speed of moisture front).
• Efficient numerical method for solving the governing equation on the half-line. The method is based on an integral formulation of the problem and reduces one degree of freedom on the numerical grid. Henceforth, the computational efficiency is greatly increased.