Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations

Periodic travelling waves (wavetrains) are an important solution type for many partial differential equations. In this paper I review the use of numerical continuation for studying these solutions. I discuss the calculation of the form and stability of a given periodic travelling wave, and the calculation of boundaries in a two-dimensional parameter plane for wave existence and stability. I also describe the automated implementation of these numerical continuation procedures via the software package wavetrain (http://www.ma.hw.ac.uk/wavetrain). I conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves.     

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an It? stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].     

Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

Dichotomy and conjugate gradients in the stiff initial value problem

We propose, analyze, and experiment with solution techniques which employ the conjugate gradient algorithm coupled with prediction steps for solving the algebraic equations arising at each mesh point in the numerical development of solutions of the model stiff system $\text{x}\text{}\text{= Ax}$. A stability and error analysis based on a dichotomization of the solutions of the system into rapidly and slowly decaying modes is made, to demonstrate the numerical stability of these methods. Stiff problems are characterized by this dichotomy, and we note that the conjugate gradient algorithm improves in effectiveness with the exaggeration of this characterization.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.      

A new analytical modelling for fractional telegraph equation via Laplace transform

The main aim of the present work is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM). The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). The beauty of the paper is error analysis which shows that our solution obtained by proposed method converges very rapidly to the known exact solution. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.     

An inverse backward problem for degenerate parabolic equations

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017     

2‐D solitary waves in thermal media with nonsymmetric boundary conditions

Optical solitary waves and their stability in focusing thermal optical media, such as lead glasses, are studied numerically and theoretically in (2 + 1) dimensions. The optical medium is a square cell and mixed boundary conditions of Newton cooling and fixed temperature on different sides of the cell are used. Nonlinear thermal optical media have a refractive index which depends on temperature, so that heating from the optical beam and heat flow across the boundaries can change the refractive index of the medium. Solitary wave solutions are found numerically using the Newton conjugate‐gradient method, while their stability is studied using a linearized stability analysis and also via numerical simulations. It is found that the position of the solitary wave is dependent on the boundary conditions, with the center of the beam moving toward the warmer boundaries, as the parameters are varied. The stability of the solitary waves depends on the symmetry of the boundary conditions and the amplitude of the solitary waves.     

Numerical treatment of degenerate diffusion equations via Feller's boundary classification,and applications

A numerical method is devised to solve a class of linear boundary‐value problems for one‐dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite‐difference scheme, grid, and treatment of the boundary data. Second‐order accuracy, unconditional stability, and unconditional convergence of solutions of the finite‐difference scheme to a constant as the time‐step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A fast BIE iteration method for an arbitrary body in a flow of incompressible inviscid fluid

The paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics. The stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid. With a discrete numerical treatment, we approximate the exact matrix by a certain Töeplitz one and then apply a fast algorithm for this matrix, on each iteration step. We illustrate the convergence of this iteration scheme by a number of numerical examples, both for hard and soft boundary conditions. It appears that the method is highly efficient for hard boundaries, being much less efficient for soft boundaries.     

Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load

In the present study, a modified nonlocal elasticity theory is used for flutter and divergence analyses of the cantilever carbon nanotubes (CNTs) conveying fluid. The CNT is embedded in viscoelastic foundation and is subjected to an axial compressive load acting at the free end. An extreme high-order governing equation as well as higher-order boundary conditions is developed using Hamilton's principle for vibration and stability analysis of the CNT. The numerical solution for flutter and divergence velocities is computed using the extended Galerkin method. The validity of the present analysis is confirmed by comparing with molecular dynamics simulation (MDS) and numerical solutions available in the literature. In the numerical results, the effects of nonlocal parameter, surface effects, viscoelastic foundation and compressive axial load on the stability boundaries of the system are investigated. The results show that the stability boundaries of the CNT are strongly dependent on the small scale coefficient and surface effects.     

A stability and numerical study of the solutions of a Timoshenko system with distributed delay

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of stability of the polynomial type, we prove that rate obtained is optimal. We conclude the work performing a numerical study of the solutions and their energies, associated to discrete system.     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.     

A nonlinear model for rotor-shaft joints of high speed rotor systems with internal damping

Viscous internal damping in joints of high speed rotor systems causes instabilities above a certain frequency of revolution. In the majority of cases a nonlinearity adjusts the stability margin towards higher frequencies. In this paper an analytical solution of a nonlinear four degrees of freedom rotor model with internal damping is proposed, which enables to clearly analyse the influence of shaft stiffness, connection stiffness, rotor mass and shaft mass. The steady state solution of the unbalance case and the stability boundaries are deduced analytically. The stabilizing effect of the nonlinearity is shown. The analytical solutions are in good agreement with numerical results obtained by FERAN, a rotor dynamic simulation tool. A model, representing the rotor–shaft connection with an o–ring has been analyzed by a hydro pulse rig. Beneath the linear way, two further approaches to describe the measured hysteresis, a cubic and a bilinear force law are shown in the paper. The different analytical and numerical results for the whole rotor system with these three approaches are compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled system of Korteweg-de Vries equation-type in domains with oscillatory moving boundaries

We consider the initial-boundary value problem in a bounded domain with oscillatory moving boundaries and nonhomogeneous boundary conditions for the coupled system of equations of KdV type modelling strong interactions between internal solitary waves. We give a result of global existence and uniqueness for strong solutions for the coupled system of equations of Korteweg – de Vries type as well as the exponential decay of small solutions in asymptotically cylindrical domains. We present a numerical examples based in semi-implicit finite differences showing the numerical effect of the oscillatory moving boundaries for this kind of systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new PEC algorithm for the numerical solution of ordinary differential equations

Significant time reduction in obtaining numerical solutions of ordinary differential equations for which function evaluations are time consuming can be obtained with PEC methods as compared to PECE methods. In this report we present two PEC methods: a fourth-order algorithm for which stability characteristics and numerical examples are presented, and a second-order algorithm which is just mentioned. It is believed that PEC methods represent a useful addition to the library of solution techniques.     

On a new fractional-order Logistic model with feedback control

In this paper, we formulate and analyze a new fractional-order Logistic model with feedback control, which is different from a recognized mathematical model proposed in our very recent work. Asymptotic stability of the proposed model and its numerical solutions are studied rigorously. By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function, we show that a unique positive equilibrium point of the new model is asymptotically stable. As an important consequence of this, we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability. Furthermore, we construct unconditionally positive nonstandard finite difference(NSFD) schemes for the proposed model using the Mickens' methodology. It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model. Finally, we report some numerical examples to support and illustrate the theoretical results. The results indicate that there is a good agreement between the theoretical results and numerical ones.     

Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.