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Summary: In the paper, a nonlinear analysis of RC phase‐shift oscillator with operational amplifier is done. Using Kirchhoff's laws, a nonlinear system of three differential equations that describes the behavior of the oscillator is obtained. This system is analyzed using the Poincaré–Andronov–Hopf bifurcation theory. The basic principles of the Poincaré–Andronov–Hopf bifurcation theory are presented in advance, including the Center Manifold Theory and the Theory of Normal Forms. It was found that, under certain conditions, the oscillator system has a limit cycle, which means that the oscillator under consideration generates periodic oscillations. The period, amplitude, and stability of the generated periodic oscillations are determined theoretically. The obtained analytical results are confirmed by a numerical analysis of the nonlinear system of differential equations performed by MATLAB. [ABSTRACT FROM AUTHOR]

Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner-Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. [ABSTRACT FROM AUTHOR]

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation [ABSTRACT FROM AUTHOR]

Dynamic two-phase interaction of soil can be modelled by a displacement-based, two-phase formulation. The finite element method together with a semi-implicit Euler-Cromer time-stepping scheme renders a discrete equation that can be solved by recursion. By experience, it is found that the CFL stability condition for undrained wave propagation is not sufficient for the considered two-phase formulation to be numerically stable at low values of permeability. Because the stability analysis of the two-phase formulation is onerous, an analysis is performed on a simplified two-phase formulation that is derived by assuming an incompressible pore fluid. The deformation of saturated porous media is now captured in a single, second-order partial differential equation, where the energy dissipation associated with the flow of the fluid relative to the soil skeleton is represented by a damping term. The paper focuses on the different options to discretize the damping term and its effect on the stability criterion. Based on the eigenvalue analyses of a single element, it is observed that in addition to the CFL stability condition, the influence of the permeability must be included. This paper introduces a permeability-dependent stability criterion. The findings are illustrated and validated with an example for the dynamic response of a sand deposit. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: In this paper, optimal control for a novel nonlinear mathematical model of tumor under immune suppression is presented. The proposed model is governed by a system of fractional differential equations, where the fractional derivative is defined in the Caputo sense. Modified parameters are introduced to account for the fractional order. Necessary and sufficient conditions that guarantee the existence and the uniqueness of the solution of the control problem are given. Two control variables are proposed to minimize the cost of interventions. Two simple numerical methods are used to study the nonlinear fractional optimal control problem. The methods are the generalized Euler method and the nonstandard generalized Euler method. The stability analysis and the truncation error of the nonstandard generalized Euler method are given. Comparative studies are implemented. [ABSTRACT FROM AUTHOR]

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin-Bona-Mahony equation, Burger's equation, Fornberg-Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka-Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays on time scales, a permanence result for the model is obtained. Furthermore, based on the permanence result, by studying the Lyapunov stability theory of impulsive dynamic equations on time scales, we establish the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system. Finally, we give an example to show the feasibility of our main results, and our example also shows that the continuous time system and its corresponding discrete time system have the same dynamics. Our results of this paper are completely new even if for both the case of the time scale [ABSTRACT FROM AUTHOR]

In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. [ABSTRACT FROM AUTHOR]

In this paper, we study a one-dimensional morphogenesis model considered by C. Stinner et al. (Math. Meth. Appl. Sci. 2012; 35: 445-465). Under homogeneous boundary conditions, we prove the existence of nonconstant positive steady states through local bifurcation theories. Then we rigorously study the stability of these nonconstant solutions when the sensitivity functions are chosen to be linear and logarithmic, respectively. Finally, we present numerical solutions to illustrate the formation of stable inhomogeneous spatial patterns. Our numerical simulations show that this model can develop very complicated and interesting structures even over one-dimensional finite domains. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate the use of differential equations by means of an example from network analysis and show that differential/algebraic equations (DAE), rather than explicit ordinary differential equations (ODE), are more suitable for the description of electrical systems and networks. The main ideas of numerical integration of ODEs are presented. We consider this material from the point of view of replacing the ODE by a difference equation (DE). In particular, the relationship between the ODE and the associated DE is discussed. In the last section the application of integration methods for ODEs upon DAEs and its difficulties are discussed. The paper is intended as a review; but a few new results are also included. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved method for analyzing the stability of the nonlinear hybrid dynamical systems (NHDSs) using a monodromy matrix. We define an n‐dimensional NHDS and focus on an orbit whose initial values exist near the periodic orbit. Because the system has nonlinearity, the circuit equation cannot be expressed in the form of a linear ordinary differential equation, meaning that the existing monodromy‐matrix‐based stability analysis method cannot be used because it requires a matrix exponential. Therefore, we propose a general theory for calculating orbital perturbations during Poincaré observation. Perturbation through switching events is expressed by a state‐transition matrix, which we refer to as the saltation matrix. By computing the perturbations, we obtain the monodromy matrix, whose characteristic multipliers denote the stability of the periodic orbit. We apply the proposed algorithm to an interrupted electric circuit with a nonlinear characteristic to confirm its validity. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

In this paper, helicoidal flat surfaces in the 3‐dimensional sphere S3 are considered. A complete classification of such surfaces, that generalizes a classification of rotational flat surfaces, is given in terms of the first and second fundamental forms for asymptotic parameters. The result consists in a relation between helicoidal flat surfaces and linear solutions of the corresponding homogeneous wave equation for the angle function. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we present the vibration control design for a string with the boundary time‐varying output constraint. The dynamics of the string is a distributed parameter system described by a partial differential equation and two ordinary differential equations. A barrier Lyapunov function with a logarithmic function is adopted to prevent the time‐varying constraint violations. Adaptive control is designed to handle the system parametric uncertainties. Stability analysis and the solvability of the inequality equations are provided. Numerical simulations are provided to illustrate the effectiveness of the proposed control design. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data. [ABSTRACT FROM AUTHOR]

SUMMARY One-dimensional consolidation analysis of layered soils conventionally entails solving a system of differential equations subject to the flow conditions at the bounding upper and lower surfaces, as well as the continuity conditions at the interface of every pair of contiguous layers. Formidable computational efforts are required to solve the ensuing transcendental equations expressing the matching conditions at the interfaces, using this method. In this paper, the jump discontinuities in the flow parameters upon crossing from one layer to the other have been systematically built into a single partial differential equation governing the space-time variation of the excess pore pressure in the entire composite medium, by the use of the Heaviside distribution. Despite the presence of the discontinuities in the coefficients of the differential equation, a closed-form solution in the sense of an infinite generalized Fourier series is obtained, in addition to which is the development of a Green's function for the differential problem. The eigenfunctions of the composite medium are the coordinate functions of the series, obtained computationally through the application of the extended equations of Galerkin. The analysis has been illustrated by solving the consolidation problem of a four-layer composite, and the results obtained agree very well with the results obtained by previous researchers. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: In this paper, finite element method is reformulated using new shape functions to approximate the state variables (ie, displacement field and its derivatives) and inhomogeneous term (ie, inertia term) of Navier's differential equation. These shape functions and corresponding elements are called spherical Hankel hereafter. It is possible for these elements to satisfy the polynomial and the first and second kind of Bessel function fields simultaneously, while the classic Lagrange elements can only satisfy polynomial ones. These shape functions are so robust that with least degrees of freedom, much better results can be achieved in comparison with classic Lagrange ones. It is interesting that no Runge phenomenon exists in the interpolation of proposed shape functions when going to higher degrees of freedom, while it may occur in classic Lagrange ones. Moreover, the spherical Hankel shape functions have a significant robustness in the approximation of folded surfaces. Five numerical examples related to the usage of suggested shape functions in finite element method in solving problems are studied, and their results are compared with those obtained from classic Lagrange shape functions and analytical solutions (if available) to show the efficiency and accuracy of the present method. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we study zero‐sum two‐player stochastic differential games in which the state equations are competing Brownian particles and the cost functional is defined by generalized backward stochastic differential equations with more than one increasing process. After we study the regularity of competing Brownian particles, we establish the dynamic programming principle for the upper and lower value functions and show that these are the unique viscosity solution of the associated upper and lower Isaacs' equations, which are fully nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we give some sufficient conditions to guarantee global asymptotic stability of the zero solution of the third-order nonlinear differential equation: x ′ ′ ′ + g( x, x ′ , x ′ ′ ) + f( x, x ′ ) x ′ + h( x) = 0. Two examples are also given to illustrate our results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper demonstrates the capability of our previously published undoped Double-Gate (DG) MOSFET explicit and analytical compact model to also forecast the effect of the volume inversion (VI) on the intrinsic capacitances. For that purpose, we present simulation results for these capacitances. We show now that the model presents an accurate dependence on the silicon layer thickness, consistent with two-dimensional numerical simulations, for both thin and thick silicon films. As opposed to our previous work, here we test the capacitance model for three different film thicknesses and also show that the transition from VI regime to dual gate behaviour is well simulated. We demonstrate in this paper that even if the current drive and transconductance are enhanced in VI regime, our results show that intrinsic capacitances are higher as well, which may limit the high-speed (delay time) behaviour of DG MOSFETs under VI regime. The good agreement between the numerical simulations and our model shows the high potential of our complete DG MOSFET model. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion. Moreover, the reconstruction of the body's motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order, where a k-th order reconstruction requires the first k - 1 time derivatives of the velocity. Such reconstruction formulas could be used whenever the velocity field is accessible. As an example the formulas are applied to the rotation update in a momentum preserving time stepping scheme for the dynamic Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we derive asymptotic formulas for the signless noncentral q-Stirling numbers of the first kind and for the corresponding series. The signless noncentral q-Stirling numbers of the first kind appear as coefficients of a polynomial of q-number [ t] q, expressing the noncentral ascending q-factorial of t of order m and noncentrality parameter k. In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method. The second main purpose is to derive an asymptotic expression for the signless noncentral q-Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the nth success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q-Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q-Stirling numbers of the first kind and of the corresponding series even for moderate values of m. [ABSTRACT FROM AUTHOR]

The IDO method, known as a numerical method suitable for partial differential equations containing an advection term, is improved in terms of its accuracy. In the method proposed in this paper, the order of Hermite interpolation is increased with a parameter h in order to deal with the higher order differential coefficients as independent variables. (In the conventional IDO method, h = 3 for calculation of the advection term and h = 5 for calculation of the diffusion term.) When this method is applied to a one-dimensional advection-diffusion equation, it is found that the numerical error is reduced to about 1/5 of that in the conventional IDO method if h = 7 is used. By using two mutually overlapping grid systems, the scheme becomes stable. Further stability is achieved by applying the time-splitting method. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(8): 11–18, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10124 [ABSTRACT FROM AUTHOR]

One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous iterative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.

A method is presented in this paper to determine the FM modulation/demodulation limit by numerical calculation. This is the synchronization limit when a phase-locked loop is used as an FM modulator or demodulator. It is shown that the same differential equation can be used both for modulation and demodulation to describe the behavior of the phase-locked loop for the FM signal. An algorithm is shown which solves the derived equation, which is nonlinear In general, by the harmonic analysis, and derives the periodic solution with the same period as that of the modulating signal by a numerical calculation. The limit of existence for the stable periodic solution is the limit of synchronization. A method is presented which can deter- mine the Unit. A phase-locked loop with a sinusoidal phase comparator arid an adequate loop filter is considered as an example. The periodic solution is actually calculated and the FM modulation/demodulation limit is determined from the limit of stability. The obtained result can be accounted for qualitatively and agrees with the experimentally obtained result. [ABSTRACT FROM AUTHOR]

This paper investigates a method for finding roots of simultaneous nonlinear algeraic equations containing complex coefficients. In this method, all roots can be fond for both real and complex roots. For setting of initial points, Moore's solution is used which is a method for a single-variable algebraic equation. Then a sufficient number of initial points is prepared. Subsequently, for each initial point, a perturbation parameter e is introduced and the root is traced for ϵ = 0 to 1. For convergence of solutions , a unique performance function F is defined and the tracking is carried out so that the value of F is minimized. In this tracking, the Cauchy-Riemann differential equation and a variational method are used. An imaginary term Ck (1 - ϵ) i is added to the left-hand side of the equation so that complex conjugate roots do not appear in the tracking process. Numerical calculations have been carried out for three examples for which all solutions have been obtained. [ABSTRACT FROM AUTHOR]

Symbolic computational systems introduce some unique features in computational engineering. There have been several papers published on the solution of differential equations under given boundary conditions by symbolic systems. The finite element formalism has received prime attention in the course of development of symbolic computation in engineering. The main idea has been to develop a symbolic FEM package to reduce the burden of manual algebra, eliminate errors introduced by numerical quadrature, and improve the efficiency of element generation. This work discusses a symbolic solution to electromagnetic linear antenna problems. The solution is a method of moments that transforms Pocklington's integral equation to a matrix equation. The symbolic system is used to produce (1) analytical integration, (2) the parametric expression for the input impedance and (3) computational code for forward and reverse problem of the input impedance. [ABSTRACT FROM AUTHOR]

This paper discusses a least-squares error design of 2-D digital filters under a peak error constraint. Initially we present a filter design procedure in which the peak error constraint is satisfied by iterating the design based on a weighted least-squares error criterion to update the weights. The computational complexity required in the filter design is then reduced by switching from the above procedure to the Lagrange method of unknown constants during the course of iteration. Finally, the trade-off between square error and absolute peak error is verified by using a numerical example. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(9): 75–82, 1998 [ABSTRACT FROM AUTHOR]

This paper presents a mathematically accurate and numerically very fast procedure to compute both expectation and variance (and standard deviation) of the stresses in a visco‐elastic material with stochastically fluctuating elastic properties. The stochastic material properties turn the evolution equation for the viscous strain components into a stochastic ordinary differential equation (ODE). We solve this ODE analytically for arbitrary loading cases which allows for a separation of deterministic and stochastic contributions. As a consequence, we are able to present precise formulas for the expectation and variance of the stresses. Material point simulations for various loading cases and varying viscosities are compared to Monte‐Carlo results which prove the reliability of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal control problem for delayed stochastic differential equations driven by fractional Brownian motions. Some necessary Pontryagin's type conditions are derived by considering the adjoint equations satisfying an anticipated backward stochastic differential equation driven by both fractional Brownian motions and the standard Brownian motions. Some new results on stochastic analysis about the control systems driven by fractional Brownian motions are presented. As an application, a linear quadratic problem is deduced, and a numerical example is shown to prove the effectiveness of our method. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint, which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as a variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem. [ABSTRACT FROM AUTHOR]

In a fairly recent paper (2008 American Control Conference, June 11-13, 1035-1039), the problem of dealing with trading in optimal pairs was treated from the viewpoint of stochastic control. The analysis of the subsequent nonlinear evolution partial differential equation was based upon a succession of Ansätze, which can lead to a solution of the terminal-value problem. Through an application of the Lie Theory of Continuous Groups to this equation, we show that the Ansätze are based upon the underlying symmetries of the equation (their (14)). We solve the problem in a more general context by allowing the parameters to be explicitly time dependent. The extension means that more realistic problems are amenable to the same mode of solution. [ABSTRACT FROM AUTHOR]

In this paper, the Volterra differential equations governing on the predator‐prey dynamics is numerically solved through the combination of evolutionary algorithms with Moving Least Squares as well as the Finite Difference Method. Among the numerous evolutionary algorithms, the Genetic Algorithms and Particle Swarm Optimization, as two powerful and effective approaches, are selected. The penalty method is chosen to impose the initial conditions on the objective function, while the Moving Least Squares is implemented to interpolate the derivatives in the differential equations. In order to illustrate the effectiveness of the Moving Least Squares, the results are compared with those obtained via the Finite Difference Method based evolutionary algorithms. Moreover, the results are associated with those of a numerical method and an analytical scheme to evaluate the obtained populations of the prey and the predator. These comparisons clearly illustrate the reliability and efficiency of the proposed methodology to solve nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse source problem of identification of F( t) function in the linear parabolic equation u t = u xx + F( t) and u0( x) function as the initial condition from the measured final data and local boundary data. Based on the optimal control framework by Green's function, we construct Fréchet derivative of Tikhonov functional. The stability of the minimizer is established from the necessary condition. The CG algorithm based on the Fréchet derivative is applied to the inverse problem, and results are presented for a test example. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of solving a class of periodic discrete-time Riccati equation with an indefinite sign of its quadratic term. Such an equation is closely related to the so-called full-information H ∞ control of discrete-time periodic systems. A globally convergent iterative algorithm with a local quadratic convergence rate is proposed for this purpose. An application to the problem of H ∞ filtering of discrete-time periodic systems is also developed and illustrated via a numerical example. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper presents lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) model and incompressible LBGK model-based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK-based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 105. However, for the pressure-driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK-based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we consider two methods for solving differential and algebraic Lyapunov equations in the time and frequency domains. Solutions of these equations are finite and infinite Gramians. In the first approach, we use the Laplace transform to solve the equations, and we apply the expansion of the matrix resolvent of the dynamical system. The expansions are bilinear and quadratic forms of the Faddeev matrices generated by resolvents of the original matrices. The second method allows computation of an infinite Gramian of a stable system as a sum of sub-Gramians, which characterize the contribution of eigenmodes to the asymptotic variation of the total system energy over an infinite time interval. Because each sub-Gramian is associated with a particular eigenvector, the potential sources of instability can easily be localized and tracked in real time. When solutions of Lyapunov equations have low-rank structure typical of large-scale applications, sub-Gramians can be represented in low-rank factored form, which makes them convenient in the stability analysis of large systems. Our numerical tests for Kundur's four-machine two-area system confirm the suitability of using Gramians and sub-Gramians for small-signal stability analyses of electric power systems. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Numerical analysis of linear visco-elastic materials requires robust and stable methods to integrate partial differential equations in both space and time. In this paper, symmetric space-time finite element operators are derived for the first time for elementary linear elastic spring and linear viscous dashpot. These can thereafter be assembled in parallel and in series to simulate an arbitrarily complex linear visco-elastic behaviour. The flexibility of the proposed method allows the formulation of the behaviour, which closely reflects physical processes. An efficient algorithm is proposed to use the generated elementary matrices in a way that is comparable with finite difference schemes, in terms of both processor and memory costs. This unconditionally stable and convergent procedure is equally valid for space domains in which geometry or material properties evolve with time. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The coherent robust H∞ control problem for a class of linear quantum passive systems with model uncertainties is considered in this study, where both the plant and the controller are described by quantum stochastic differential equations (QSDEs). In the framework of (S,L,H) language, the model uncertainties are translated into the uncertainties in the Hamiltonian, the coupling operators, and the scattering matrices. Based on the quantum bounded real lemma, the robust H∞ controller design of the original uncertain plant is transformed into the controller design of a new scaled system without model uncertainties. Further, the latter is transformed into the solving problem of a couple of Riccati equations and therefore the numerical solutions of the coefficient matrices of QSDEs associated with the controller dynamics are obtained. Finally, numerical simulations on an optical system coupled to three optical channels are performed to verify the effectiveness of the method proposed in this study. [ABSTRACT FROM AUTHOR]

Some draining or coating fluid-flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third-order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a two-dimensional numerical procedure based on the boundary integral equations to model acoustic waves of finite-amplitude. The analysis is performed in the frequency domain. By applying the perturbation technique up to the second-order term, the governing differential equations are written as a system of two Helmholtz equations, one is homogeneous and the other one is inhomogeneous. Both are transformed into integral equations, which can be numerically solved without domain discretization by the use of the dual reciprocity boundary element method (DRBEM). To the authors' knowledge, this is the first application of DRBEM to nonlinear acoustics. The numerical procedure can be applied to predict the propagation of finite but of moderate amplitude acoustic waves in domains of any geometry. The final formulation is validated by comparison with an analytical solution derived by the authors. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the characteristics of layered structures (photonic or electromagnetic bandgaps), including chiral media, are studied by means of two different numerical methods, one in the time domain (finite differences in the time domain, FDTD) and the other in the frequency domain (coupled-mode method, CMM). The results (reflection and transmission coefficients for a plane wave normally incident over a layered structure) obtained by means of both well different techniques are practically identical. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, He's variational iteration method (VIM) is employed successfully for solving modified Camassa-Holm and Degasperis-Procesi equations. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions or the perturbation theory. The results show applicability, accuracy and efficiency of VIM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that VIM can be widely applied in science and engineering problems. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This note is concerned with the stability problem for solutions of a system which arises in the mathematical modelling of a temperature regulation device. It is well-known that, for autonomous systems, a periodic regulation process can be asymptotically stable only if it is stationary. In this paper we consider the case where the “heating” equation is time-dependent. We show that in this case, under suitable hypotheses, the solution is asymptotically stable, and we also give estimates for approximations of solutions. [ABSTRACT FROM AUTHOR]

To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x. [ABSTRACT FROM AUTHOR]

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here. [ABSTRACT FROM AUTHOR]

We present a numerical method that optimizes the open-loop stability of solutions of periodic optimal control problems. We consider general periodic processes that may have several phases, each characterized by its own set of differential equations, and discontinuities of the state variables and the right hand side between phases. Stability is measured in terms of the spectral radius of the monodromy matrix which results in a nonsmooth optimization criterion. We have applied this method to design walking robots that can perform stable periodic gaits without any sensors or feedback; three such examples are presented in this paper. [ABSTRACT FROM AUTHOR]