Łukasz Płociniczak

Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications

Abstract In this paper we investigate the porous medium equation with a time-fractional derivative. We justify that the resulting equation emerges when we consider a waiting-time (or trapping) phenomenon that can have its place in the medium. Our deterministic derivation is dual to the stochastic CTRW framework and can include nonlinear effects. With the use of the previously developed method we approximate the investigated equation along with a constant flux boundary conditions and obtain a very accurate solution. Moreover, we generalise the approximation method and provide explicit formulas which can be readily used in applications. The subdiffusive anomalies in some porous media such as construction materials have been recently verified by experiment. Our simple approximate solution of the time-fractional porous medium equation fits accurately a sample data which comes from one of these experiments.

Bessel function model of corneal topography

In this paper we propose a new nonlinear mathematical model of corneal topography. This model is stated as a two-point boundary value problem. We derive the governing equation from the first physical principles and provide a mathematical analysis concerning its solution. The existence and uniqueness theorems are proved and various estimates are shown. At the end, we fit the simplified model based on a modified Bessel function of the First Kind with the corneal data. The fitting error is of order of 1%, which is sufficiently accurate for this type of data and biomedical applications.

A note on fractional Bessel equation and its asymptotics

In this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

ODE/PDE analysis of corneal curvature

The starting point for this paper is a nonlinear, two-point boundary value ordinary differential equation (BVODE) that defines corneal curvature according to a static force balance. A numerical solution to the BVODE is computed by first converting the BVODE to a parabolic partial differential equation (PDE) by adding an initial value (t, pseudo-time) derivative to the BVODE. A numerical solution to the PDE is then computed by the method of lines (MOL) with the calculation proceeding to a sufficiently large value of t such that the derivative in t reduces to essentially zero. The PDE solution at this point is also the solution for the BVODE. This procedure is implemented in R (an open source scientific programming system) and the programming is discussed in some detail. A series approximation to the solution is derived from which an estimate for the rate of convergence is obtained. This is compared to a fitted exponential model. Also, two linear approximations are derived, one of which leads to a closed form solution. Both provide solutions very close to that obtained from the full nonlinear model. An estimate for the cornea radius of curvature is also derived. The paper concludes with a discussion of the features of the solution to the ODE/PDE system.

On Asymptotics of Some Fractional Differential Equations

Abstract In this paper we study the large-argument asymptotic behaviour of certain fractional differential equations with Caputo derivatives. We obtain exponential and algebraic asymptotic solutions. The latter, decaying asymptotics differ significantly from the integer-order derivative equations. We verify our theorems numerically and find that our formulas are accurate even for small values of the argument. We analyze the zeros of fractional oscillations and find the approximate formulas for their distribution. Our methods can be used in studying many other fractional equations.

Diffusivity identification in a nonlinear time-fractional diffusion equation

Abstract This paper deals with the estimates of the convergence rates for various problems associated with diffusivity identification in a time-fractional nonlinear diffusion equation. We find the convergence rate of the corresponding Erd´elyi-Kober type operator as the anomalous parameter approaches the classical limit. Further, we use this result in proving an estimate of the difference between the identified diffusivity and its approximate value which was derived by the use of the previously developed framework. The exact formula for the diffusivity is computationally expensive thus having an accurate and easily calculable approximation is very relevant. In the last part of the paper we take up the problem of regularization strategy for solving the inverse problem. Calculation of the diffusivity requires a computation of the derivative which is a unstable operation and can amplify the measurement noise. We discuss how this ill-possedness can be mollified and prove some corresponding estimates on the convergence rates of the regularization

Eigenvalue asymptotics for a fractional boundary-value problem

This letter presents a result concerning eigenvalue approximation of a boundary-value problem with the Caputo fractional derivative. This approximation is derived by the use of the asymptotic (for large x and @l) form of the exact solution. The growth order of the eigenvalues is given and it is shown that their number is finite. Moreover, a simple method of estimating the size of the spectrum is proposed. The issue of a finite number of eigenvalues is a very peculiar and characteristic feature of differential equations with fractional order derivative. The paper is concluded with a numerical verification that our approximations are very accurate. This shows that the devised formulas can be readily used in applications of fractional boundary-value problems.

Vortex microscope: analytical model and experiment

We present the analytical model describing the Gaussian beam propagation through the off axis vortex lens and the set of axially positioned ideal lenses. The model is derived on the base of Fresnel diffraction integral. The model is extended to the case of vortex lens with any topological charge m. We have shown that the Gaussian beam propagation can be represented by function G which depends on four coefficients. When propagating from one lens to another the function holds its form but the coefficient changes.