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Abstract In this paper, we propose one new alternative formula for moment generating function of random vectors via the inverse survival function. A recursive formula for moment generating function of random vector is obtained and as application, we derive the corresponding alternative formula for mixed moment. Several examples are also presented along with the theory. [ABSTRACT FROM AUTHOR]

Abstract This paper mainly focuses on the adaptive synchronization problem of multi-agent systems via distributed impulsive control method. Different from the existing investigations of impulsive synchronization with fixed time impulsive inputs, the proposed distributed variable impulsive protocol allows that the impulsive inputs are chosen within a time period (namely impulsive time window) which can be described by the distances of the left (right) endpoints or the centers between two adjacent impulsive time windows. Obviously, this kind of flexible control scheme is more effective in practical systems (especially for the complex environment with physical restrictions). Moreover, the proposed adaptive control technique is helpful to solve the problem with uncertain system parameters. By means of Lyapunov stability theory, impulsive differential equations and adaptive control technique, three sufficient impulsive consensus conditions are given to realize the synchronization of a class of multi-agent nonlinear systems. Finally, two numerical simulations are provided to illustrate the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

The problem of constructing a canonical representation for an arbitrary continuous piecewise-linear (PWL) function in any dimension is considered in this paper. We solve the problem based on a general lattice PWL representation, which can be determined for a given continuous PWL function using existing methods. We first transform the lattice PWL representation into the difference of two convex functions, then propose a constructive procedure to rewrite the latter as a canonical representation that consists of at most η-level nestings of absolute-value functions in n dimensions, hence give a thorough solution to the problem mentioned above. In addition, we point out that there exist notable differences between a lattice representation and the two novel general constructive representations proposed in this paper, and explain that these differences make all the three representations be of their particular interests. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non-linear diffusion equation. Design/methodology/approach – In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution. Findings – In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems. Originality/value – The basic idea described in this paper is expected to be further employed to solve other functional equations. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to implement the Ritz-Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non-local boundary conditions. Design/methodology/approach – The properties of Bernstein polynomial and Ritz-Galerkin method are first presented, then the Ritz-Galerkin method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. Findings – The authors applied the method presented in this paper and solved three test problems. Originality/value – This is the first time that the Ritz-Galerkin method in Bernstein polynomial basis is employed to solve the model investigated in the current paper. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to consider the time-fractional diffusion-wave equation. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α € (0, 2]. The fractional derivatives are described in the Caputo sense. Design/methodology/approach – The two methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. Findings – Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. Originality/value – In this paper, the variational iteration and homotopy perturbation methods are used to obtain a solution of a fractional diffusion equation. [ABSTRACT FROM AUTHOR]

The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson (Lord Kelvin), it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin's theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader's intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'-type nonlinear partial differential equations. Design/methodology/approach – The PDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by Runge-Kutta fourth order method. Findings – Numerical results for the nonlinear evolution equations such as 1D Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained by applying PDQM. The numerical results are found to be in good agreement with the exact solutions. Originality/value – A comparison is made with those which are already available in the literature and the present numerical schemes are found give better solutions. The strong point of these schemes is that they are easy to apply, even in two-dimensional nonlinear problems. [ABSTRACT FROM AUTHOR]

Purpose – This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense. Design/methodology/approach – By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM). Findings – This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution. Originality/value – The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→;1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a method for the numerical solution of Love''s equation in a particular physical system. The resolution protocol is based on the Boubaker Polynomials Expansion Scheme (BPES), followed by array analyses. The obtained results are compared with some recently published ones. The accuracy and the asymptotic behaviors of the solutions are discussed. [Copyright &y& Elsevier]

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of Rn, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least n-times continuously differentiable. The proof is based on a generalization of the Weier- strass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm W1∞ to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper is to introduce and investigate new classes of generalizations of non-continuous functions, obtain some of their properties and to hold decompositions of strong α-irresolute in topological spaces. [Copyright &y& Elsevier]

Abstract: The general solution, the local and global asymptotic stability of equilibrium points and period three cycles of the third order rational difference equationare studied in this paper. [Copyright &y& Elsevier]

Purpose - Initial value problems for the one-dimensional third-order dispersion equations are investigated using the reliable Adomian decomposition method (ADM). Design/methodology/approach - The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms. Findings - It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non-homogeneous equations. Research limitations/implications - The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data. Practical implications - The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory. Originality/value - The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively-based exact solutions are found. [ABSTRACT FROM AUTHOR]

Abstract: This paper demonstrates that there is an exponentially stable unique equilibrium state in a Hopfield-type neural network that is subject to quite large impulses that are not too frequent. The activation functions are assumed to be globally Lipschitz continuous and unbounded. The analysis exploits an homeomorphic mapping and an appropriate Lyapunov function, and also either a geometric–arithmetic mean inequality or a Young inequality, to derive a family of easily verifiable sufficient conditions for convergence to the unique globally stable equilibrium state. These sufficiency conditions, in the norm ∥·∥ p where p ⩾1, include those governing the network parameters and the impulse magnitude and frequency. [Copyright &y& Elsevier]

Abstract: Some recent contributions to Economic Dynamics have shown a new interest for delay differential equations. In line with these approaches, we re-proposed the problem of the existence of a finite lag between the accrual and the payment of taxes in a framework where never this type of lag has been considered: the well known IS-LM model. The qualitative study of the system of functional (delay) differential equations shows that the finite lag may give rise to a wide variety of dynamic behaviours. Specifically, varying the length of the lag and applying the “stability switch criteria”, we prove that the equilibrium point may lose or gain its local stability, so that a sequence of alternated stability/instability regions can be observed if some conditions hold. An important scenario arising from the analysis is the existence of limit cycles generated by sub-critical and supercritical Hopf bifurcations. As numerical simulations confirm, if multiple cycles exist, the so called “crater bifurcation” can also be detected. Economic considerations about a stylized policy analysis stand by qualitative and numerical results in the paper. [Copyright &y& Elsevier]

In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics. [Copyright &y& Elsevier]

In this paper, balanced truncation of linear time-varying systems is studied in discrete and continuous time. Based on relatively basic calculations with time-varying Lyapunov equations/inequalities we are able to derive both upper and lower error bounds for the truncated models. These results generalize well-known time-invariant formulas. The case of time-varying state dimension is considered. Input-output stability of all truncated balanced realizations is also proven. The method is finally successfully applied to a high-order model. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the asymptotic behaviour of a second order nonlinear differential equation arising in a constrained optimal stopping problem. Two examples are considered to illustrate our main result. The present result extends and supplements a recent result of the author . [Copyright &y& Elsevier]

The article presents information on the experimental results presented by researchers J.J. Kohfeld and G.T. Thompson in their paper on a modification of Reinhard Nordsieck's method for the numerical solution of ordinary differential equations using a multiple precision arithmetic package available on the IBM 7094 at the Ohio State University Computer Center in Columbus, Ohio. The experimental results agree with those presented by Kohfeld and Thompson at h = 0.10 and 0.15 for the Nordsieck and the GSN methods, which is to be expected since round-off error is not critical at those interval lengths.

For a general class of nonlinear differential-algebraic equations of index one, we develop and unify a convergence theory on waveform relaxation (WR). Convergence conditions are achieved for the cases of continuous-time and discrete-time WR approximations. Most of known convergence results in this field can be easily derived from the new theory established here. [ABSTRACT FROM AUTHOR]