Evolutionary Differential Equations Research Articles

On the semigroup approach to the interval-valued differential evolution equations

In this paper, we introduce the semigroups of semilinear mappings on the space of nonempty compact intervals of $${\mathbb {R}}$$ . Some important properties of strongly continuous semigroups of interval-valued mappings are investigated. Depending on the different types of generalized Hukuhara differences, the interval-valued infinitesimal generators and resolvent operators are defined and the Hille–Yosida like representation of resolvent operators is given. As an application, the unique existence of mild solutions for Cauchy problems of interval-valued evolution equations is given.

On the dynamics of the zeros of solutions of the Airy equation

We study the dependence of the zeros of solutions of the Airy equation on two parameters introduced in the equation. The parameters characterize the general solution of the equation. A system of infinitely many nonlinear evolution differential equations is obtained, displaying interesting properties.

Mathematical modelling of cell migration: stiffness dependent jump rates result in durotaxis

Durotaxis, the phenomena where cells migrate up a gradient in substrate stiffness, remains poorly understood. It has been proposed that durotaxis results from the reinforcement of focal adhesions on stiff substrates. In this paper we formulate a mathematical model of single cell migration on elastic substrates with spatially varying stiffness. We develop a stochastic model where the cell moves by updating the position of its adhesion sites at random times, and the rate of updates is determined by the local stiffness of the substrate. We investigate two physiologically motivated mechanisms of stiffness sensing. From the stochastic model of single cell migration we derive a population level description in the form of a partial differential equation for the time evolution of the density of cells. The equation is an advection–diffusion equation, where the advective velocity is proportional to the stiffness gradient. The model shows quantitative agreement with experimental results in which cells tend to cluster when seeded on a matrix with periodically varying stiffness.

On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative

Abstract Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.

On Stability of a Class of Linear Systems with Distributed and Lumped Parameters

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

Elliptical Gaussian beam propagation in nonlinear fibres with focusing and defocusing refractive profiles

The paper shows systematic analysis of elliptical Gaussian beam (GB) propagating in self-focusing/self-defocusing media of Kerr type with focusing/defocusing linear refractive profiles. Instead of solving commonly accepted Nonlinear Schrödinger Equation (NLS) we propose equations of geometrical optics: complex eikonal equation and complex transport one. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process and virial theory used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the elliptical GB cross-section. For combined effects of diffraction, nonlinear refraction and focusing/defocusing graded-index factor we observe three types of the solutions for the elliptical radius of principle beam widths: self-focusing or self-defocusing together with diffraction widening, stationary and oscillatory ones under and without collapse effect. We prove here that principle curvatures of the wave front can introduce into our description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discuss the evolution of elliptical Gaussian beam in nonlinear self-defocusing fibre with focusing refractive profile presenting conditions for existence of elliptical spatial solitons. We discuss here also the influence of initial wave front curvatures on elliptical Gaussian beam evolution in nonlinear inhomogeneous fibres of Kerr-type.

Sequential evolution conformable differential equations of second order with nonlocal condition

In this paper, we investigate second order evolution differential equation in the frame of sequential conformable derivatives with nonlocal condition. First, we establish Duhamel’s formula in terms of a standard cosine family of linear operators. Then, we prove some results concerning the existence, uniqueness, stability, and regularity of mild solution concept. Moreover, we present a concrete application of the main results.

Second order evolution differential functional equations with infinite delay

Second order evolution differential functional equations with infinite delay

Bounds of the entropy production of the irreversible thermodynamics and its relation with the non-equilibrium temperature

Based on the entropy balance equation for a multidimensional linear Fokker–Planck equation we have developed connections between the several thermodynamically inspired quantities such as the total entropy production, the entropy production of the irreversible thermodynamics and the non-equilibrium temperature. Then we have calculated both upper and lower bounds of these properties with the help of the entropy balance equation and the Schwartz inequality which has been applied to the differential equation for the time evolution of the entropy. With these we have shown that there is a lower bound of time derivative of the entropy in addition to its upper limit. The present formalism has been applied for the coupling of a Brownian particle with the Markovian and the non-Markovian thermal baths, respectively.

On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients

In this paper we introduce a randomized version of the backward Euler method, that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carath\'eodory type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of $0.5$ in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the numerical solution of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation

Abstract We study the thermoconvective instability in a rotating ferromagnetic fluid confined between two parallel infinite plates with temperature modulation at the boundaries. We use weakly nonlinear stability theory to analyze the stationary convection in terms of critical Rayleigh numbers. The influence of parameters such as the Taylor number, the ratio of the magnetic force to the buoyancy force and the magnetization on the flow behaviour and structure are investigated. The heat transfer coefficient is analyzed for both the in-phase and the out-of-phase modulations. A truncated Fourier series is used to obtain a set of ordinary differential equations for the time evolution of the amplitude of convection for the ferromagnetic fluid flow. The system of differential equations is solved using a recent multi-domain spectral collocation method that has not been fully tested on such systems before. The solutions sets are presented as sets of trajectories in the phase plane. For some supercritical values of the Rayleigh number, spiralling trajectories that transition to chaotic solutions are obtained. Additional results are presented in terms of streamlines and isotherms for various Rayleigh numbers.

Solitary wave solutions of the higher-order evolution equations for two ordering parameters in the shallow water waves

By introducing the relation between the orders of the two independent small expansion parameters, the amplitude parameter α and the long wavelength parameter β, which are taken in the form β=0αn and α=0βm, with n and m specified to two particular cases, we derive some new higher order partial differential nonlinear evolution equations for unidirectional shallow water waves under the influence of gravity as well as surface tension. We then show that it is possible to construct one-solution which satisfy these estimates using the Hirota’s bilinear method.

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state.

Optimal Estimates in Problems of Extrapolation, Filtration, and Interpolation of Functionals of Random Processes with Values in a Hilbert Space

General formulas are obtained for efficient calculation of optimal estimates for functionals of random processes with values in a Hilbert space. In a special case where the process under study is a solution of a nonlinear evolutionary differential equation with a small nonlinearity, the optimal estimates are expanded in powers of a small parameter and the expansion coefficients are given in the form of algorithms and are calculated explicitly in terms of known quantities of the differential equation.

An analytic approach for the study of pulsar spindown

In this work we develop an analytic approach to study pulsar spindown. We use the monopolar spindown model by Alvarez and Carramiñana (2004 Astron. Astrophys. 414 651–8), which assumes an inverse linear law of magnetic field decay of the pulsar, to extract an all-order formula for the spindown parameters using the Taylor series representation of Jaranowski et al (1998 Phys. Rev. D 58 6300). We further extend the analytic model to incorporate the quadrupole term that accounts for the emission of gravitational radiation, and obtain expressions for the period P and frequency f in terms of transcendental equations. We derive the analytic solution for pulsar frequency spindown in the absence of glitches. We examine the different cases that arise in the analysis of the roots in the solution of the non-linear differential equation for pulsar period evolution. We provide expressions for the spin-down parameters and find that the spindown values are in reasonable agreement with observations. A detection of gravitational waves from pulsars will be the next landmark in the field of multi-messenger gravitational wave astronomy.

English

Advanced building design is a rather new interdisciplinary research branch, combining knowledge from physics, engineering, art and social science; its support from both theoretical and computational mathematics is needed. This paper shows an example of such collaboration, introducing a model problem of optimal heating in a low-energy house. Since all particular function values, needed for optimization are obtained as numerical solutions of an initial and boundary value problem for a sparse system of parabolic partial differential equations of evolution with at least two types of physically motivated nonlinearities, the usual gradient-based methods must be replaced by the downhill simplex Nelder-Mead approach or its quasi-gradient modifications. One example of the real low-energy house in Moravian Karst is demonstrated with references to other practical applications.

Existence results for Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Generation of localized patterns in anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion via a variational approach.

We use the time-dependent variational method to examine the formation of localized patterns in dynamically unstable anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion. The governing equation is an extended nonlinear Schrödinger equation known for modified Frankel-Kontorova models of atomic lattices and here derived from an extended Bose-Hubbard model of bosonic lattices with local three-body interactions. In presence of modulated waves, we derive and investigate the ordinary differential equations for the time evolution of the amplitude and phase of dynamical perturbation. Through an effective potential, we find the modulationally unstable domains of the lattice and discuss the effect of the fourth-order dispersion in the dynamics. Direct numerical simulations are performed to support our analytical results, and a good agreement is found. Various types of localized patterns, including breathers and solitonic chirped-like pulses, form in the system as a result of interplay between the cubic-quintic nonlinearities and the second- and fourth-order dispersions.

Existence of solutions for quasilinear random impulsive neutral differential evolution equation

This paper deals with the existence of solutions for quasilinear random impulsive neutral functional differential evolution equation in Banach spaces and the results are derived by using the analytic semigroup theory, fractional powers of operators and the Schauder fixed point approach. An application is provided to illustrate the theory.