arXiv subjects

Łukasz Płociniczak

Publications and source records attributed to Łukasz Płociniczak.

At least 37 records · Page 2Linked to original sources

arXiv2022

A linear Galerkin numerical method for a quasilinear subdiffusion equation

We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with their remainders for discretizations of Caputo derivative and fractional integrals. These constants are the smallest possible which improves the previously established results from the literature.

math.NA↗

arXiv2022

Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.

math.NA↗

arXiv2022

The Bushell-Okrasiński inequality

We present an expository account of the Bushell-Okrasiński inequality, the motivation behind it, its history, and several generalizations. This inequality originally appeared in studies of nonlinear Volterra equations but very soon gained interest of its own. The basic result has quickly been generalized and extended in different directions strengthening the assertion, generalizing the kernel and nonlinearity, providing the optimal prefactor, finding conditions under which it becomes an equality, and formulating variations valid for other than Lebesgue integrals. We review all of these aspects.

math.CA↗

arXiv2022

Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of a nonsmooth initial data. The diffusion coefficient is allowed to depend on time. It is well-known that in such parabolic problems the spatial error increases when time decreases. The rate of this time-dependency is a function of the fractional parameter and the regularity of the initial condition. We use the energy method to find optimal bounds on the error under weak and natural assumptions on the diffusivity. First, we prove the result for the linear problem and then use the "frozen nonlinearity" technique coupled with various generalizations of Gronwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results.

math.NA↗

arXiv2022

Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line

Many physical, biological, and economical systems exhibit various memory effects due to which their present state depends on the history of the whole evolution. Combined with the nonlinearity of the process these phenomena pose serious difficulties in both analytical and numerical treatment. We investigate a time-fractional porous medium equation that has proved to be important in many applications, notably in hydrology and material sciences. We show that solutions of the free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfy a Volterra integral equation with a non-Lipschitz nonlinearity. Based on this result we prove existence, uniqueness, and construct a family of numerical methods that solve these equations outperforming the usual finite difference approach. Moreover, we prove the convergence of these methods and support the theory with several numerical examples.

math.NA↗

arXiv2022

Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

math.NA↗

arXiv2021

Stability of fixed points in an approximate solution of the spring-mass running model

We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.

math.DS↗

arXiv2020

Off-axis vortex beam propagation through classical optical system in terms of Kummer confluent hypergeometric function

The analytical solution for the propagation of the laser beam with optical vortex through the system of lenses is presented. The optical vortex is introduced into the laser beam (described as Gaussian beam) by spiral phase plate. The solution is general as it holds for the optical vortex of any integer topological charge, the off-axis position of the spiral phase plate and any number of lenses. Some intriguing conclusions are discussed. The higher order vortices are unstable and split under small phase or amplitude disturbance. Nevertheless, we have shown that off-axis higher order vortices are stable during the propagation through the set of lenses described in paraxial approximation, which is untypical behavior. The vortex trajectory registered at image plane due to spiral phase plate shift behaves like a rigid body. We have introduced a new factor which in our beam plays the same role as Gouy phase in pure Gaussian beam.

physics.optics↗

arXiv2020

A weighted finite difference method for subdiffusive Black Scholes Model

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has $2-α$ order of accuracy with respect to time where $α\in(0,1)$ is the subdiffusion parameter, and $2$ with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results.

cs.CE↗

arXiv2019

Asymptotic behaviour of a solution to a nonlinear equation modelling capillary rise

We are concerned with the asymptotics and perturbation analysis of a singular second-order nonlinear ODE that models capillary rise of a fluid inside a narrow vertical tube. We prove the convergence of the exact solution to a unperturbed solution when a nondimensional parameter decrease to zero. Furthermore, we provide an accurate method of approximating the asymptotic solution for large times. Due to the a fact that the nonlinear component in the main equation does not satisfy the Lipschitz continuity condition the methods used to prove the main theorems are nonstandard, require careful analysis, and can be useful in dealing with similar nonlinear ODEs.

math.CA↗

arXiv2019

Solution and asymptotic analysis of a boundary value problem in the spring-mass model of running

We consider the classic spring-mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data we conduct a rigorous asymptotic analysis in terms of the Poicaré-Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

math.CA↗

arXiv2019

Hopf bifurcation in a conceptual climate model with ice-albedo and precipitation-temperature feedbacks

In this paper we analyse a dynamical system based on the so-called KCG (Källén, Crafoord, Ghil) conceptual climate model. This model describes an evolution of the globally averaged temperature and the average extent of the ice sheets. In the nondimensional form the model is prone to several simplifications facilitating the subsequent analysis. We consider the limiting case of the stationary snow line for which the phase plane can be completely analysed and the type of each stationary point can be determined. One of them can exhibit the Hopf bifurcation for which occurrence we find the sufficient conditions. Those, in turn, have a straightforward physical meaning and indicate that the model predicts internal oscillations of the climate. Using the typical values of model parameters we conclude that the obtained results are in the same ballpark as the conditions on our planet during the quaternary ice ages. Our analysis is a rigorous justification of a generalization of some previous results by KCG and other authors.

math.DS↗

arXiv2019

Asymptotic analysis of internal relaxation-oscillations in a conceptual climate model

We construct a dynamical system based on the KCG (Källén, Crafoord, Ghil) conceptual climate model which includes the ice-albedo and precipitation-temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations which are a characteristic feature of the Pleistocene ice-ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation-oscillations without any external forcing and for a wide range of parameters.

math.DS↗

arXiv2018

Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.

physics.flu-dyn↗

arXiv2018

Numerical method for the time-fractional porous medium equation

This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the self-similar solution and then to apply the discretization. The main difficulty arises due to the non-Lipschitzian behaviour of the equation's nonlinearity. By the analysis of the recurrence relation for the error we are able to prove that there exists a family of finite difference methods that is convergent for a large subset of the parameter space. We illustrate our results with a concrete example of a method based on the midpoint quadrature.

math.NA↗

arXiv2018

Compactly supported solution of the time-fractional porous medium equation on the half-line

In this work we prove that the time-fractional porous medium equation on the half-line with Dirichlet boundary condition has a unique compactly supported solution. The approach we make is based on a transformation of the fractional integro-differential equation into a nonlinear Volterra integral equation. Then, the shooting method is applied in order to facilitate the analysis of the free-boundary problem. We further show that there exists an exactly one choice of initial conditions for which the solution has a zero which guarantees the no-flux condition. Then, our previous considerations imply the unique solution of the original problem.

math.AP↗

arXiv2017

Quickest drift change detection in Lévy-type force of mortality model

In this paper we give solution to the quickest drift change detection problem for a Lévy process consisting of both a continuous Gaussian part and a jump component. We consider here Bayesian framework with an exponential a priori distribution of the change point using an optimality criterion based on a probability of false alarm and an expected delay of the detection. Our approach is based on the optimal stopping theory and solving some boundary value problem. Paper is supplemented by an extensive numerical analysis related with the construction of the Generalized Shiryaev-Roberts statistics. In particular, we apply this method (after appropriate calibration) to analyse Polish life tables and to model the force of mortality in this population with a drift changing in time.

math.OC↗

arXiv2017

Finite difference method for a Volterra equation with a power-type nonlinearity

In this work we prove that a family of explicit numerical finite-difference methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity. In that case the kernel is not of Lipschitz type, therefore the classical analysis cannot be applied. We indicate several difficulties that arise in the proofs and show how they can be remedied. The tools that we use consist of variations on discreet Gronwall's lemmas and comparison theorems. Additionally, we give an upper bound on the convergence order. We conclude the paper with a construction of a convergent method and apply it for solving some examples.

math.NA↗