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In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations u(4)+wu′′+a(x)u=f(x,u), where w is a constant, a∈C(R,R) and f∈C(R×R,R). By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well-known (AR)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a generalized Fisher equation with exponential diffusion from the point of view of the theory of symmetry reductions in partial differential equations. The generalized Fisher-type equation arises in the theory of population dynamics. These types of equations have appeared in many fields of study such as in the reaction-diffusion equations, in heat transfer problems, in biology, and in chemical kinetics. By using the symmetry classification, simplified by equivalence transformations, for a special family of Fisher equations, all the reductions are derived fromthe optimal systemof subalgebras and symmetry reductions are used to obtain exact solutions.

We propose a new mathematical model for the spread of Zika virus. Special attention is paid to the transmission of microcephaly. Numerical simulations show the accuracy of the model with respect to the Zika outbreak occurred in Brazil., Comment: This is a preprint of a paper whose final and definite form is with 'Mathematical Methods in the Applied Sciences', ISSN 0170-4214. Submitted Aug 10, 2017; Revised Nov 13, 2017; accepted for publication Nov 14, 2017

To protect fishery populations on the verge of extinction and sustain the biodiversity of the marine ecosystem, marine protected areas (MPA) are established to provide a refuge for fishery resource. However, the influence of current harvesting policies on the MPA is still unclear, and precise information of the biological parameters has yet to be conducted. In this paper, we consider a bioeconomic Gompertz population model with interval-value biological parameters in a 2-patch environment: a free fishing zone (open-access) and a protected zone (MPA) where fishing is strictly prohibited. First, the existence of the equilibrium is proved, and by virtue of Bendixson-dulac Theorem, the global stability of the nontrivial steady state is obtained. Then, the optimal harvesting policy is established by using Pontryagin's maximum principle. Finally, the results are illustrated with the help of some numerical examples. Our results show that the current harvesting policy is advantageous to the protection efficiency of an MPA on local fish populations.

In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.

In this paper, we consider a nonautonomous impulsive plankton model with mutual help of preys. Sufficient conditions ensuring permanence and global attractivity of the model are established by the relation between solutions of impulsive system and corresponding nonimpulsive system. Also, we propose the conditions for which the species of system are driven to extinction. Numerical simulations are given to verify the main results.

In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples.

In this paper, we investigate the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator. This predator-prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang.[26] We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrodinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n,R) and the space of translations Rn, where n equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

In this paper, we construct a more general Besov spaces B˙r1,r2,r3σ,q and consider the global well-posedness of incompressible Navier-Stokes equations with small data in B˙r1,r2,r3σ,∞ for 1r1+1r2+1r3−σ=1,1⩽ri

In this paper, we derive an asymptotic expansion for the semi-infinite sum of Dirac-δ functions centered at discrete equidistant points defined by the set Na={x∈R:∃n∈N∧x=na,∀a>0}. The method relies on the Laplace transform of the semi-infinite sum of Dirac-δ functions. The derived series distribution takes the form of the Euler-Maclaurin summation when the distributions are defined for complex or real-valued continuous functions over the interval [0,∞). For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0+. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite-size effects.

In this paper, we investigate a Vector-Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number R0. The disease-free equilibrium is globally asymptotically stable if R0≤1; when R0>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.

In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

This paper studies the time-averaged energy dissipation rate $\langle \varepsilon_{SMD} (u)\rangle$ for the combination of the Smagorinsky model and damping function. The Smagorinsky model is well known to over-damp. One common correction is to include damping functions that reduce the effects of model viscosity near walls. Mathematical analysis is given here that allows evaluation of $\langle \varepsilon_{SMD} (u)\rangle $ for any damping function. Moreover, the analysis motivates a modified van Driest damping. It is proven that the combination of the Smagorinsky with this modified damping function does not over dissipate and is also consistent with Kolmogorov phenomenology., Comment: 14 pages, 5 figures, Mathematical Methods in the Applied Sciences, ISSN 1099-1476 2017

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin-type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

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.

This paper presents an operatorial model based on fermionic operators for the description of the dynamics of political parties affected by turncoat-like behaviors. By observing the political landscape in place in Italy over the last years, appropriate macro-groups have been identified on the basis of the behavior of politicians in terms of disloyal attitude as well as openness towards accepting chameleons from other parties. Once introduced, a time-dependent number-like operator for each physical observable relevant for the description of the political environment, the analysis of the party system dynamics is carried out by combining the action of a quadratic Hamiltonian operator with certain rules acting periodically on the system in such a way that the parameters entering the model are repeatedly changed so as to express a sort of dependence of them upon the variations of the mean values of the observables. Copyright © 2017 John Wiley & Sons, Ltd.

Synchronization of complex networks with time-varying coupling matrices is studied in this paper. Two kinds of time-varying coupling are taken into account. One is the time-varying inner coupling in the node state space and the other is the time-varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time-varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive-definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness for the global solutions of Cauchy problem for coupled nonlinear Schrodinger equations and obtain the continuous dependence result on the initial data and the stronger decay estimate of global solutions. In particular, we show the existence and uniqueness of self-similar solutions. Also, we build some asymptotically self-similar solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we derive the explicit formulas for computing the zeros of certain cubic quaternionic polynomial. From these, we obtain a necessary and sufficient condition to quaternionic cubic polynomial have a spherical zero, and some examples are also provided. Moreover, we will discuss some applications of the cubic quaternionic formulas. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we introduce a generalized form of Cole-Hopf transformation and apply it to find new closed-form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we study existence of invariant curves of an iterative equation h[2](x)−(1+b)h(x)+bx−ϵf(h(x))+(b−1)α=0, which is from dissipative standard map. By constructing an invertible analytic solution g(x) of an auxiliary equation of the form g(λ2x)−(1+b)g(λx)+bg(x)+(b−1)α=ϵf(g(λx)), invertible analytic solutions of the form g(λg − 1(x)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero-zero-Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

We present a new Lyapunov function for laminar flow, in the x-direction, between two parallel planes in the presence of a coplanar magnetic field for three-dimensional perturbations with stress-free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(Re) and magnetic Reynolds numbers(Rm) below π2/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.

Evolution of human language and learning processes have their foundation built on grammar that sets rules for construction of sentences and words. These forms of replicator–mutator (game dynamical with learning) dynamics remain however complex and sometime unpredictable because they involve children with some predispositions. In this paper, a system modeling evolutionary language and learning dynamics is investigated using the Crank–Nicholson numerical method together with the new differentiation with non-singular kernel. Stability and convergence are comprehensively proven for the system. In order to seize the effects of the non-singular kernel, an application to game dynamical with learning dynamics for a population with five languages is given together with numerical simulations. It happens that such dynamics, as functions of the learning accuracy μ, can exhibit unusual bifurcations and limit cycles followed by chaotic behaviors. This points out the existence of fickle and unpredictable variations of languages as time goes on, certainly due to the presence of learning errors. More interestingly, this chaos is shown to be dependent on the order of the non-singular kernel derivative and speeds up as this derivative order decreases. Hence, can people use that order to control chaotic behaviors observed in game dynamical systems with learning! Copyright © 2016 John Wiley & Sons, Ltd.

Graph theory is a fundamental tool in the study of economic issues, and input–output tables are one of the main examples. We use the interpretation of the labour market through networks to obtain a better understanding on its overall functioning. One benefit of the network perspective is that a large body of mathematics exists to help analyze many forms of networks models. If an economic system has obtained a suitable model, then it becomes possible to utilize relevant mathematical tools, such as graph theory, to better understand the way the labour market works. This interpretation allows us to employ the concepts of coverage, invariance, orbit and the structural functions supply–demand and competition and interpret them from the point of view of circular flow. In this paper, we aim to interpret the labour market through networks that are represented by graphs and where characteristic concepts of chaos theory such as cover, invariance and orbits interact with the concept circular flow. Finally, an example of this approach to labour markets is described, and some conclusions are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents all possible exact explicit peakon, pseudo-peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr-like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo-peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr-like media. Copyright © 2016 John Wiley & Sons, Ltd.

This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.

The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge–Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0–1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a novel view on the scattering of a flexural wave in a Kirchhoff plate by a semi-infinite discrete system. Blocking and channelling of flexural waves are of special interest. A quasi-periodic two-source Green's function is used in the analysis of the waveguide modes. An additional "effective waveguide" approximation has been constructed. Comparisons are presented for these two methods in addition to an analytical solution for a finite truncated system.

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grunwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the truncated Painleve analysis and the consistent tanh expansion method are developed for the modified Boussinesq system, and new exact solutions such as the single-soliton, the two-soliton, the rational solutions, and the explicit interaction solutions among a soliton and the cnoidal periodic waves are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Cauchy problem for the third-order nonlinear Schrodinger equation where ℋ=F−1iξξ−1F and F is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition ∫u0xdx≠0. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid-saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdelyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time-fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time-fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate the fourth-order nonlinear Schrodinger equation with parameterized nonlinearity that is generalized from regular cubic-quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth-order generalized cubic-quintic nonlinear Schrodinger equation through modified F-expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright (c) 2016 John Wiley & Sons, Ltd.

In this paper we consider the $b$-family of equations on the torus $u_t- u_{txx}+ (b+1) u u_x=b u_x u_{xx} + u u_{xxx}$, which for appropriate values of $b$ reduces to well-known models, such as the Camassa-Holm equation or the Degasperis-Procesi equation. We establish a local-in-space blow-up criterion

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient-type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study a multiphasic incompressible fluid model, called the Kazhikhov–Smagulov model, with a particular viscous stress tensor, introduced by Bresch and co-authors, and a specific diffusive interface term introduced for the first time by Korteweg in 1901. We prove that this model is globally well posed in a 3D bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.

We point out some mistakes in a known paper. Some existence results for solutions of two classes of boundary value problems for nonlinear impulsive fractional differential equations are established. Our analysis relies on the well-known Schauder fixed point theorem. Examples are given to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.

The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive-definite quaternion sequence and the positive semi-definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two-sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the nonlinear model of the mechanism with two degrees of freedom will be studied. An approximate analytical solution of the differential equation of motion in the series showed the presence of features in the aspiration of the mass of one of the bodies to zero. It also gives an algorithm for finding the points of degeneracy of communication between small perturbations of the function of the problem and the derivatives of these functions at a time. Copyright © 2016 John Wiley & Sons, Ltd.

This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two-dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.