Multi-periodic Solutions Research Articles

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

Optimal sourcing policies for single and multiple period scenarios

Determining the optimum number of suppliers and the optimum quantities to order from each of them is a critical problem for any supply chain. The objective of this paper is to identify the optimal sourcing policy of a retailer for the single and multi-period context when the firm can source its order to multiple suppliers along with a back-up supplier for the emergency situations. The expected total profit is mathematically modelled for single and multi-period scenarios. The optimal sourcing policy is obtained by maximising the expected total profit with respect to the order quantities. Closed form solution is obtained for uniformly distributed demand for both single and multi-period scenarios. It is observed that the multi-period solution is less sensitive compared to the single-period solution. Also it is found that it is optimal for the firm to lessen the amount of supplier diversification in case of planning for multiple periods.

Optimal sourcing policies for single and multiple period scenarios

Determining the optimum number of suppliers and the optimum quantities to order from each of them is a critical problem for any supply chain. The objective of this paper is to identify the optimal sourcing policy of a retailer for the single and multi-period context when the firm can source its order to multiple suppliers along with a back-up supplier for the emergency situations. The expected total profit is mathematically modelled for single and multi-period scenarios. The optimal sourcing policy is obtained by maximising the expected total profit with respect to the order quantities. Closed form solution is obtained for uniformly distributed demand for both single and multi-period scenarios. It is observed that the multi-period solution is less sensitive compared to the single-period solution. Also it is found that it is optimal for the firm to lessen the amount of supplier diversification in case of planning for multiple periods.

MULTIPERIODIC SOLUTIONS OF LINEAR SYSTEMS INTEGRO-DIFFERENTIAL EQUATIONS WITH Dc -OPERATOR AND  -PERIOD OF HEREDITARY

The article explores the questions of the initial problem and the problem of multiperiodicity solutions of linear systems integro-differential equations with an operator of the form mmctctcD∂∂++∂∂+∂∂=11τ, ()constcccm−=,,1 and with finite hereditary period 0>=conste by variable τ that describe hereditary phenomena. Along with the equation of zeros of the special differentiation operator cD are considered linear systems of homogeneous and inhomogeneous integro-differential equations, sufficient conditions are established for the unique solvability of the initial problems for them, both necessary and sufficient conditions of multiperiodic existence are obtained by ()t,τ with periods ()ωθ, of the solutions. The integral representations of multiperiodic solutions of linear inhomogeneous systems with the uniqueness property are determined 1) in the particular case when the corresponding homogeneous systems have exponential dichotomy and 2) in the general case when the homogeneous systems do not have multiperiodic solutions, except for the trivial one. The article proposes a research technique for solving problems that satisfy initial conditions and have the property of multiperiodicity with a given e heredity period for linear systems of integro-differential equations with a special partial differential operator cD. Multiperiodic solutions obtained along characteristics 00ττcctt−+= with fixed ()00,tτ are used as an application in the theory of quasiperiodic solutions of systems of integro-differential equations.

Bounded and multiperiodic solutions of the system of partial integro-differential equations

Bounded and multiperiodic solutions of the system of partial integro-differential equations

On multi-periodic solutions of quasilinear autonomous systems with an operator of differentiation on the Lyapunov’s vector field

On multi-periodic solutions of quasilinear autonomous systems with an operator of differentiation on the Lyapunov’s vector field

Integration of a Linear Equation with Differential Operator, Corresponding to the Main Diagonal in the Space of Independent Variables, and Coefficients, Constant on the Diagonal

We consider an n-th order linear equation with differential operator corresponding to the direction of the main diagonal in the space of independent variables; its coefficients. in general, are variable but constant on the diagonal. We establish conditions on variable eigenvalues which give a possibility to apply some known methods of the theory of for ODEs, when integrating the equation. On this basis, the structures of solutions to the homogeneous equation are determined. We give conditions for existence of multiperiodic solutions of the equations related to variable eigenvalues and initial functions. An integral representation of multiperiodic solution to the inhomogeneous equation is given. We also suggest the concepts of variable frequency and variable period.

Using deficit functions for aircraft fleet routing

We consider the problem of minimizing the number of airplanes needed to fly a fixed daily repeating schedule of flights. We use deficit functions (DF) to decompose an aviation schedule of aircraft flights into aircraft chains (routes) called a chain decomposition. Each chain visits periodically a set of airports and is served by several cockpit crews circulating along the airports of this set. The initial step in our approach is to find the minimal number of aircraft needed to carry out the flight schedule. This is achieved by using the fleet size theorem based on a DF representation of an aircraft flight schedule. A DF is a step function associated with an aircraft terminal which changes by + 1 and −1 at flight departure and arrival times, respectively. DF theory was developed in the 1960–70s by Linis and Maksim (1967) and Gertsbakh and Gurevich (1977). Although the initial application of DFs was to the Russian AEROFLOT fleet it has subsequently attracted more attention on bus scheduling than aircraft scheduling. Here we discuss the revival of this method and its crucial use to construct the so-called chain decomposition of the schedule for a single period. We provide a justification for maximizing the number of balanced chains (flight sequences with the same start and end terminals). To do this we propose The Maximal Balanced Chain Problem. These are then converted into a set of infinite periodic flight sequences, each of which can be carried out by a single aircraft. The conversion is carried out by mapping the single period set of chains into an Euler graph. To construct the set of mutiperiod chains that are “balanced” (return to the same terminal at the start) we find all edge disjoint cycle covers of the Euler graph using a modified version of Hierholzer's algorithm. These cycles are converted back into a balanced multiperiod chain solution and modified to conform to any maintenance constraints. To insure maintenance check constraints are satisfied for multiperiod chains, it may be necessary to add deadhead flights. Minimizing the cost of deadhead trips and overnight stays provide the basis for selecting an optimal routing solution.

General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonal

General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonal

The reliable design of a hierarchical multi-modes transportation hub location problems (HMMTHLP) under dynamic network disruption (DND)

Abstract In ground and air transportation systems, services are operated as hub-and-spoke networks. For a given discrete planning horizon, disrupted hub nodes and edges can change the location of hub nodes and routes of transferring (allocation) at each level of hierarchy, so that the system can be converted to a flexible network capable of expanding or contracting in the long term. Because of the probability of hub nodes’, hub edge’ disruption and the periodic changes demands, A novel mixed-integer mathematical programming formulation that is effective for a hierarchical multi-modes transportation hub location problem (HMMTHLP) is presented. The main issue is how to tackle the hub nodes’ and edges’ disruption in a dynamic system. Various instances from the two different sets of data such as CAB dataset and Airport-Railway Network real-life case study (ARWN) are provided for computational analysis. The value of these tests in a multi-period solution with managerial insights of the two different cases are presented. A Monte Carlo simulation is used in a two-stage heuristic algorithm; this methodology provides a strategy with which to compare and discuss the locations-allocations of a dynamic network with or without any disruption.

Analyzing the combined multi-waves polynomial solutions in a two-layer-liquid medium

In this work, we investigate the behavior of mixed wave solutions in two-layer-liquid with dispersive waveguide. These combined multi-wave solutions were obtained in polynomial types for the (3 + 1)-dimensional Yu–Toda–Sasa–Fukuyama (YTSF) equation with variable coefficients. By virtue of the generalized unified method and symbolic computations, we formally derive polynomial solutions for this equation. The obtained solutions include multi-soliton solutions, multi-periodic solutions, and multi-elliptic solutions. Furthermore, the physical insight and the movement role of the waves in the two-layers are discussed and analyzed graphically for different selections of the arbitrary functions of the obtained solutions.

Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory.

The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multi-periodic, quasi-periodic and multi-pulse localized solutions. For illustration the theory is applied to a complex nonlinear 2+1 Klein-Gordon equation which has two Lighthill surfaces in the manifold of periodic travelling waves.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Effect of gravity-induced asymmetry on the nonlinear vibration of an overhung rotor

In this study a mechanical model of an overhung rotor is explored to determine the effect of gravity on the nonlinear dynamics of an aero-engine. The model is an overhung disc with rotor-stator contact. The model has two degrees of freedom with lumped parameters; friction is neglected in the contact and the equations of motion are non-dimensionalised. A parametric study of the non-dimensional gravity parameter is conducted. The bifurcation plots show that gravity plays a crucial role in the nonlinear dynamics of such systems. With zero gravity, as explored in earlier studies, the model has synchronous whirling solutions, and asynchronous partially contacting solutions that are periodic only when viewed in a rotating coordinate system. If the gravity parameter is non-zero, then the dynamics observed are much richer and show additional multi-periodic and chaotic solutions in the stationary frame and continuous contact (full annular rub).

Nonlinear Fourier Methods for Ocean Waves

Multiperiodic Fourier series solutions of integrable nonlinear wave equations are applied to the study of ocean waves for scientific and engineering purposes. These series can be used to compute analytical formulae for the stochastic properties of nonlinear equations, in analogy to the standard approach for linear equations. Here I emphasize analytically computable results for the correlation functions, power spectra and coherence functions of a nonlinear random process associated with an integrable nonlinear wave equation. The multiperiodic Fourier series have the advantage that the coherent structures of soliton physics are encoded in the formulation, so that solitons, breathers, vortices, etc. are contained in the temporal evolution of the nonlinear power spectrum and phases. I illustrate the method for the Korteweg-deVries and nonlinear SchrÖdinger equations. Applications of the method to the analysis of data are discussed.

Comparative study on vertical deformation based on GPS and leveling data

The development of GPS (Global Positioning System) technology has led to increasingly widely and successful applications of GPS surveys for monitoring crustal movements. However, multi-period GPS survey solutions have not been applied in monitoring vertical crustal movements with normal backgrounds. In this paper, we carried out a comparative study on the vertical deformation of the comprehensive profile of the cross-fault zone in Shanyin, Shanxi province, China, based on GPS and precise leveling observation data for multiple time periods. The vertical deformation rates observed with repeating GPS survey are obviously different (over 20 mm/y at some sites) from those with repeating leveling survey within a relatively short period. However, the deviations in the vertical displacement between GPS and leveling in a long-term survey (over three years) showed good consistency at 3–4 mm/y at most sites, on GPS forced offset surveying and fixed survey instruments in a long-term survey (over three years). Therefore, GPS vertical displacement results can be applied to the study of vertical crustal movements.

A new approach to steady state analysis of nonlinear electrical circuits

PurposeThe purpose of this study is to identify a novel methodology for direct calculation of steady-state periodic solutions for electrical circuits described by nonlinear differential equations, in the time domain.Design/methodology/approachAn iterative algorithm was created to determine periodic steady-state solutions for circuits with nonlinear elements in a chosen set of time instants.FindingsThis study found a novel differential operator for periodic functions and its application in the steady-state analysis.Research limitations/implicationsThis approach can be extended to the determination of two- or multi-periodic solutions of nonlinear dynamic systems.Practical implicationsThe complexity of the steady-state analysis can be reduced in comparison with the frequency-domain approach.Originality/valueThis study identified novel difference equations for direct steady-state analysis of nonlinear electrical circuits.

Construction of rational solutions of the real modified Korteweg-de Vries equation from its periodic solutions

In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit λj → λ1 of the Lax pair eigenvalues used in the n-fold Darboux transformation that generates the order-n periodic solution from a constant seed solution. Further, this special kind of breather solution of order n can be used to generate the order-n rational solution by taking the limit λ1 → λ0, where λ0 is a special eigenvalue associated with the eigenfunction ϕ of the Lax pair of the mKdV equation. This eigenvalue λ0, for which ϕ(λ0)=0, corresponds to the limit of infinite period of the periodic solution. Our analytical and numerical results show the effective mechanism of generation of higher-order rational solutions of the mKdV equation from the double eigenvalue degeneration process of multi-periodic solutions.

Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations

In this paper, we present a generalized unified method for finding multiwave solutions of the time-fractional (2+1)-dimensional Nizhnik–Novikov–Veselov equations. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation. Multiauxiliary equations have been introduced in this method to obtain not only multisoliton solutions but also multiperiodic or multielliptic solutions. It is shown that the considered method is very effective and convenient for solving wide classes of nonlinear partial differential equations of fractional order.

Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (3 + 1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation

Under investigation in this paper is a generalized (3 + 1)-dimensional variable-coefficient BKP equation, which can be used to describe the propagation of nonlinear waves in fluid mechanics and other fields. With the aid of binary Bell’s polynomials, an effective and straightforward method is presented to explicitly construct its bilinear representation with an auxiliary variable. Based on the bilinear formalism, the soliton solutions and multi-periodic wave solutions are well constructed. Furthermore, the tanh method and the tan method are employed to construct more traveling wave solutions of the equation. Finally, the asymptotic properties of the multi-periodic wave solutions are systematically analyzed to reveal the connection between periodic wave solutions and soliton solutions. It is interesting that the periodic waves tend to solitary waves under a limiting procedure. Our results can be used to enrich the dynamical behavior of higher-dimensional nonlinear wave fields.

On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic, and breakdown at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behaviour in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein-Gordon equation which has two singular curves in the manifold of periodic travelling waves.