Infinite Interval Research Articles

A numerical method for linear Volterra integral equations on infinite intervals and its application to the resolution of metastatic tumor growth models

A Nyström method for linear second kind Volterra integral equations on unbounded intervals, with sufficiently smooth kernels, is described. The procedure is based on the use of a truncated Lagrange interpolation process and of a truncated Gaussian quadrature formula. The stability and the convergence of the method in suitable weighted spaces of functions are studied and some numerical examples showing its reliability are presented. In particular, the proposed method has been tested for the numerical resolution of some Volterra integral equations arising from the reformulation of differential models describing metastatic tumor growth whose unknown solutions represent biological observables as the metastatic mass or the number of metastases.

Optimal stopping problems for maxima and minima in models with asymmetric information

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

Multiple solutions for a nonlocal fractional boundary value problem with fractional integral conditions on infinite interval

This paper discusses the existence and multiplicity of positive solutions for fractional differential equations with nonlocal fractional integro-differential boundary conditions set on an unbounded domain. Using properties of Green’s function and the fixed point theory, some existence results were obtained. Three examples illustrate the existence theorems.

Deficiency indices and discreteness property of block Jacobi matrices and Dirac operators with point interactions

The first part of the paper concerns with infinite symmetric block Jacobi matrices J with p×p-matrix entries. We present new conditions for general block Jacobi matrices to be self-adjoint and have discrete spectrum. In the second part of the paper a special classes of block Jacobi matrices JX,α are investigated. Our approach here substantially relies on a close connection between Jacobi matrices JX,α from this class and symmetric 2p×2p Dirac operators DX,α with point interactions in L2((a,b);C2p) established in our previous papers. In particular, their deficiency indices are related by n±(DX,α)=n±(JX,α) and under a simple additional assumption they are discrete only simultaneously. For block Jacobi matrices of this class we present several conditions ensuring equality n±(JX,α)=k with any k≤p. It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality n±(JX,α)=p for JX,α it is first established for Dirac operator DX,α. We also find several conditions for matrix Schrödinger and Dirac operators with point interactions on finite or infinite intervals to have discrete spectrum.

Asymptotic expansion of the integral with two oscillations on an infinite interval

In this paper, we focus on constructing the asymptotic expansion for the highly oscillatory integral including of the product of exponential and Bessel oscillations with the stationary point. Based on the exact integral representation of Bessel function, the integral is transformed into a double oscillatory integral. For the resulting inner semi-infinite integral, we present a new way of a combination of the integration by parts and the Filon-type methods to produce the asymptotic expansion. Furthermore, the original oscillatory integral can be expanded in the sum of Gaussian hypergeometric function. The corresponding asymptotic property is also analysed. With increasing the oscillatory parameter, the error of the proposed asymptotic expansions decreases very fast. Numerical experiments are provided to illustrate the effectiveness of the expansion.

Lyapunov modal analysis and participation factors applied to small-signal stability of power systems

In large dynamical systems subjected to noise or forced oscillations, disturbances constantly arise over time. In this case, the small-signal stability is determined by the energy of perturbations accumulated in the system. To analyze this perturbation energy, this study proposes a novel physically motivated Lyapunov modal analysis (LMA) framework that combines selective modal analysis with the spectral decompositions of specially chosen Lyapunov functions. Based on this approach, we propose new modal indicators that characterize individual modes and modal interactions in connection with specific state variables. Conventional participation factors characterize the relative contributions of the system modes to the evolution of states. In contrast, the proposed Lyapunov participation factors make similar contributions to the Lyapunov functions that estimate the integral energy associated with the states and signals on an infinite or finite time interval. The Lyapunov modal interaction factors characterize the pairwise interaction between modes in terms of their mutual actions produced in states over time. We prove that the proposed modal indicators characterize the stability of individual modes and resonant modal interactions in linear parameter-varying systems and demonstrate the application of these indicators for the small-signal stability analysis of a two-area power system model. New indicators can be calculated independently for a selected part of the system spectrum and used to quickly assess the behavior of critical modes in large-scale dynamical systems.

Tamed-adaptive Euler-Maruyama approximation for SDEs with locally Lipschitz continuous drift and locally Hölder continuous diffusion coefficients

We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals.

Iterative Solutions for the Differential Equation with p -Laplacian on Infinite Interval

This paper systematically investigates a class of fourth-order differential equation with p -Laplacian on infinite interval in Banach space. By means of the monotone iterative technique, we establish not only the existence of positive solutions but also iterative schemes under the suitable conditions. At last, we give an example to demonstrate the application of the main result.

Clocked population protocols

We define an extension to the standard population protocol that provides each agent with a clock signal that indicates when the agent has waited long enough for the protocol to have converged. We represent “long enough” as an infinite time interval, and treat the protocol as operating over transfinite time. Over finite time intervals, the protocol behaves as in the standard model. At nonzero limit ordinals, corresponding to clock ticks, the protocol switches to a limit of previous configurations supplemented by a clock signal appearing as an extra component in some of the agents' states. Fairness generalizes to transfinite executions straightforwardly. We show that a clocked population protocol running in less than ωk time for any fixed k≥2 is equivalent in power to a nondeterministic Turing machine with space complexity logarithmic in the population size, and can be simulated using only finitely many clock ticks.

Uniform asymptotics for the compound risk model with dependence structures and constant force of interest

Consider a nonstandard compound renewal risk model, which is more realistic in catastrophe insurance and can extend classical risk models to the case when perhaps not only one individual claim occurs in a catastrophe such as windstorm, earthquake, epidemic, strike or riot. In the presence of heavy tails and dependence structures among modelling components, we obtain the uniform asymptotics of the tail probability of discounted aggregate claims and the finite-time ruin probability for all time varying in a relevant finite or infinite interval.

Asymptotic Behavior of Solutions to a Cauchy Problem with a Turning Point in the Case of Change of Stability

We prove the existence of solutions of a perturbed Cauchy problem on an infinite interval and obtain asymptotic estimates of the proximity of solutions to the perturbed and nonperturbed problems on an infinite interval containing an unstable interval.

Solving vector interval-valued optimization problems with infinite interval constraints via integral-type penalty function

This paper aims at establishing necessary optimality conditions for a nondifferentiable multiobjective interval-valued optimization problem with infinite interval-valued constraints, and then deriving the equivalence of the weakly LU efficient solution between the primal optimization problem and its integral-type penalized optimization problem, which provides us supporting theories when we solve this kind of optimization problem by means of penalty function method. The equivalence of the weakly LU efficiency is established under some suitable conditions. Some examples are provided to illustrate the advantages of our results in some cases.

Averaging Theory for Fractional Differential Equations

Abstract The theory of averaging is a classical component of applied mathematics and has been applied to solve some engineering problems, such as in the filed of control engineering. In this paper, we develop a theory of averaging on both finite and infinite time intervals for fractional non-autonomous differential equations. The closeness of the solutions of fractional no-autonomous differential equations and the averaged equations has been proved. The main results of the paper are applied to the switched capacitor voltage inverter modeling problem which is described by the fractional differential equations.

Optimal Control for Some Classes of Dynamic Equations on the Infinite Interval of Time Scale

We study a linear (in control) system of dynamic equations on an infinite interval of time scale and establish sufficient conditions for the existence of optimal controls in terms of the right-hand sides of the system and a function contained in the quality criterion. The relationship between the solutions of the initial-value problem on the semiaxis and the corresponding problem on a finite interval of the time scale is analyzed.

Multicomponent stress-strength reliability estimation based on unit generalized Rayleigh distribution

PurposeThe purpose of this article is to derive inference for multicomponent reliability where stress-strength variables follow unit generalized Rayleigh (GR) distributions with common scale parameter.Design/methodology/approachThe authors derive inference for the unknown parametric function using classical and Bayesian approaches. In sequel, (weighted) least square (LS) and maximum product of spacing methods are used to estimate the reliability. Bootstrapping is also considered for this purpose. Bayesian inference is derived under gamma prior distributions. In consequence credible intervals are constructed. For the known common scale, unbiased estimator is obtained and is compared with the corresponding exact Bayes estimate.FindingsDifferent point and interval estimators of the reliability are examined using Monte Carlo simulations for different sample sizes. In summary, the authors observe that Bayes estimators obtained using gamma prior distributions perform well compared to the other studied estimators. The average length (AL) of highest posterior density (HPD) interval remains shorter than other proposed intervals. Further coverage probabilities of all the intervals are reasonably satisfactory. A data analysis is also presented in support of studied estimation methods. It is noted that proposed methods work good for the considered estimation problem.Originality/valueIn the literature various probability distributions which are often analyzed in life test studies are mostly unbounded in nature, that is, their support of positive probabilities lie in infinite interval. This class of distributions includes generalized exponential, Burr family, gamma, lognormal and Weibull models, among others. In many situations the authors need to analyze data which lie in bounded interval like average height of individual, survival time from a disease, income per-capita etc. Thus use of probability models with support on finite intervals becomes inevitable. The authors have investigated stress-strength reliability based on unit GR distribution. Useful comments are obtained based on the numerical study.

New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

This paper is concerned with the existence of multiple positive solutions for a nonlinear Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. We first transform the given problem into an equivalent fixed point problem on cone, and then solve it by using the generalized Avery–Henderson fixed point theorem. Finally, an example is constructed to illustrate the application of the obtained results.

Analysis of Forced Convection Boundary Layer Flow and Heat Transfer Past a Semi-Infinite Static and Moving Flat Plate Using Nanofluids-by Haar Wavelets

The paper presents, the steady state two-dimensional forced convection boundary layer flow of heat transfer past a semi-infinite static flat plate (Blasius problem) and moving flat plate (Sakiadis problem) in the water based nanofluid with various nanoparticles. The self-similar solution exists for the boundary layer equations and using suitable similarity variables, the governing equations have been converted into coupled nonlinear ordinary differential equations (NODEs) with an infinite domain. The governing problems over an infinite interval were solved using semi-numerical technique which makes the use of power of Haar wavelets coupled with collocation method. The solutions obtained using wavelet methods have been confirmed to be more accurate as compared to other previously published results. The several physical interesting results of the problem are concentrated and verified through numerical schemes. Three different types of nonmetallic or metallic nanoparticles such as alumina (Al2O3), copper (Cu) and titania (TiO2) in the base fluid of water with Prandtl number Pr = 6.2, to study the effect of solid volume fraction parameter Φ of the nanofluids. The effect of local skin friction coefficients, Nusselt number, velocity and temperature profiles are plotted for various values of nanoparticle volume fractions and for different nanoparticles are analyzed in detail, the numerical results are presented in terms of Tables. It predicts that, the solid volume fraction affects the fluid flow and heat transfer characteristics.

Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model

In this article, the authors simulate and study dark and bright soliton solutions of 1D and 2D regularized long wave (RLW) models. The RLW model occurred in various fields such as shallow-water waves, plasma drift waves, longitudinal dispersive waves in elastic rods, rotating flow down a tube, and the anharmonic lattice and pressure waves in liquid–gas bubble mixtures. First of all, the tanh–coth method is applied to obtain the soliton solutions of RLW equations, and thereafter, the approximation of finite domain interval is done by truncating the infinite domain interval. For computational modeling of the problems, a meshfree method based on local radial basis functions and differential quadrature technique is developed. The meshfree method converts the RLW model into a system of nonlinear ordinary differential equations (ODEs), then the obtained system of ODEs is simulated by the Runge–Kutta method. Further, the stability of the proposed method is discussed by the matrix technique. Finally, in numerical experiments, some problems are considered to check the competence and chastity of the developed method.

Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies

We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non-resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the frequency vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with “holes”. If we want the set to be connected we have to require some non-resonance condition on the frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.

The geometric origin of perspectivist science in G.W. Leibniz. Analysis based on unpublished manuscripts

The perspectivist research carried out by G.W. Leibniz between 1679 and 1686 in the field of geometry is analysed. This work is reflected in six, as yet unpublished, texts, of which the main three are analysed: Constructio et usus scalae perspectivae, Origo regularum artis perspectivae and Scientia perspectiva. The philosophical perspectivism advocated by the German thinker is widely known, but his geometric research on perspective is much less so. This article seeks to remedy this situation. The first of these writings (Constructio et usus scalae perspectivae) includes Leibniz's experimentation with the perspectivist methodology of scales. Then, in Origo regularum artis perspectivae quales, Leibniz constructs his perspectivist regula generalis. Finally, Leibniz wrote Scientia perspectiva and readdresses the main rule of perspective and experiments with the theoretical limits of the analysis carried out in this discipline. Primarily, he supposed a ‘minimum distance’ between the elements that constitute it, and then theorised an ‘infinite interval’ between these same elements.