Wojciech Okrasiński

Nontrivial solutions to nonlinear Volterra integral equations

The Volterra integral equation \[ u(x) = \int_0^x {k(x - s)g(u(s))ds\quad (x > 0)} \] is considered, where

Bessel function model of corneal topography

x \geqq 0

Mathematical modeling of the VLE curve of Lennard-Jones fluids. Application to calculating the vapour pressure

is a monotonic integrable function and g is an increasing, continuous function such that

On approximate solutions to some nonlinear diffusion problems

g(0) = 0

Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity

. Some necessary and sufficient conditions for the existence of nontrivial solutions to the above equation are presented. Physical problems that motivate these results are also discussed.

A note on fractional Bessel equation and its asymptotics

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.

Modeling evaporation using a nonlinear diffusion equation

Abstract In this Letter, we present a straight forward analytical way to obtain the vapour–liquid equilibrium (VLE) curve for Lennard-Jones fluids, based on the symmetrical properties of the derivatives of the densities in the vapour and liquid phases, ρ V and ρ L .

A new numerical procedure to determine the VLE curve

In the paper an approximate solution to a nonlinear diffusion problem in semiconductor production is considered. Some integral operator methods are used to estimate an error between this approximate solution and the exact one.

On approximate mathematical modeling of the vapor–liquid coexistence curves by the Van der Waals equation of state and non-classical value of the critical exponent

In this work we consider nonlinear Volterra equations of a special type. We find necessary and sufficient conditions for blow-up of solutions to this class of equations.

Nonlinear Volterra Integral Equations with Convolution Kernel

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.