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[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.

Let e ℓ , for ℓ = 1,2,3, be orthogonal unit vectors in R 3 and let Ω ⊂ R 3 be a bounded open set with smooth boundary ∂Ω. Denoting by x a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T = a(x)∂xe1 + b(x)∂ye2 + c(x)∂ze3, into the conservation of energy law, here a, b, c ∶ Ω → R are given functions. With the S-spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c ∶ Ω → R the fractional powers of T exist in the sense of the S-spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.

This paper is focused on following time-harmonic Maxwell equation: ∇×(μ−1(x)∇×u)−ω2e(x)u=f(x,u),inΩ,ν×u=0,on∂Ω, where Ω⊂R3 is a bounded Lipschitz domain, ν:∂Ω→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as |u|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor e∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

This paper deals with an adaptation of the Poincare-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line x > as follows: ut−lux−uxx=−β(uvx)x,x>0,t>0,λv−vxx=u,x>0,t>0,lu(0,t)+ux(0,t)=vx(0,t)=0,t>0,u(x,0)=u0(x),x>0, where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 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 deal with a new approximation of a diffusive predator--prey model with Leslie--Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator-prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator-prey interaction in the presence of Leslie--Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.

This paper studies the inhomogeneous nonlinear Schrödinger equations, which may model the propagation of laser beams in nonlinear optics. Using the cross-constrained variational method, a sharp condition for global existence is derived. Then, by solving a variational problem, the strong instability of solitary waves of this equation is proved.

In this paper, we study a fractional order time delay for nonlinear financial system. By using Laplace transformation, stability and Hopf bifurcation analysis have been done for the model. Furthermore, numerical simulation has been carried out for better understanding of our results.

In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

This paper attempts to focus on the split common fixed point problem for demicontractive mappings. We give an accelerated hybrid projection algorithm which combines the hybrid projection method and the inertial technique. The strong convergence theorems of this algorithm are obtained under mild conditions by a self-adaptive step-size sequence, which does not need prior knowledge of operator norms. Some numerical experiments in infinite Hilbert space are provided to illustrate the reliability and robustness of the algorithm and also to compare it with existin

In this paper, we establish existence and finite time blow up of weak solutions of a parabolic equation of p(x)-Laplacian type with the Dirichlet boundary condition. Moreover, we obtain upper and lower bounds for the blow up time of solutions, by employing concavity method and differential inequality technique respectively.

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

In the paper, we consider the local-in-time and the global-in-time infinity-ion-mass convergence of bipolar Euler-Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi→∞. We use the method of asymptotic expansions to handle the local-in-time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling, but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data is sufficiently close to the constant equilibrium state, we establish the global-in-time infinity-ion-mass convergence.

[EN] Historically, the Poincare disk model has taken a pioneering role in the development of non-Euclidean geometry. But still today, this model is critical as a playground for simulations and new theories. In this paper, we discuss how to implement and simulate acoustic wave phenomena on the Poincare disk with the help of so-called metamaterials. After formally developing a theory based on the acoustic potential for the curved spacetime of the Poincare model, we also provide practical instructions on how to manufacture the appropriate metadevice in a laboratory environment. Finally, as an example, analytical results for the nontrivial radial contributions in concentric wave propagation are derived, and the corresponding numerical predictions are presented., Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF), Grant/Award Number: TIN2017-89314-P; Programa de Apoyo a la Investigacion y Desarrollo 2018, Grant/Award Number: PAID-06-18

In this paper, we study the Jost solutions of the impulsive Dirac systems (IDS) on entire axis and examine analytic and asymptotic properties of these solutions. Furthermore, we obtain a general form of the scattering matrix of the IDS and its characteristic properties. Finally, we also compare the similar properties for the IDS with the mass on entire axis with an example.

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $10$ is a parameter.

In this paper, we propose a center Lipschitz condition for the second derivative together with the use of restricted domains in order to improve the starting points for Newton's method when compared with previous results. Moreover, we present some numerical examples validating the theoretical results.

In this paper, we investigate the potential for a class of non-Gaussian processes so-called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M-Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes which is previously chosen, diminishing and non-summable. While the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence as well as the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparisons with other algorithms.

In this paper we use the steerable special relativistic (space-time) Fourier transform (SFT), and relate the classical convolution of the algebra for space-time $Cl(3,1)$-valued signals over the space-time vector space $\R^{3,1}$, with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space-time signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of space-time signals in terms of finite linear combinations of classical convolutions.

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact soluti ...

In this paper, we are concerned with the asymptotic behavior of the Neumann-Poincare operator for Helmholtz system. By analyzing the asymptotic behavior of spherical Bessel function near the origin and/or approach higher order, we prove the asymptotic behavior of spectral of Neumann-Poincare operator when frequency is small enough and/or the order is large enough. The results show that spectral of Neumann-Poincare operator is continuous at the origin and converges to zero from the complex plane in general., 12 pages, 11 figures

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system involving competing potential functions \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u, & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $f$ is a function of $C^1$ class, superlinear and subcritical nonlinearity, $2_s^{\ast}=\frac{6}{3-2s}$, $s>\frac{3}{4}$, $t\in(0,1)$, $V(x)$ $K(x)$ and $Q(x)$ are positive continuous function. Under some suitable assumptions on $V$, $K$ and $Q$, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small $\varepsilon>0$, of which it is concentrating on the set of minimal points of $V(x)$ and the sets of maximal points of $K(x)$ and $Q(x)$. The methods are based on the Nehari manifold, arguments of Brezis-Nirenberg and concentration compactness of P. L. Lions.

In this paper, the well-posedness of Cauchy's problem of fractional Schr\"odinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local solution with initial data in $H^{[\frac{n}{2}]+1}$ will blow up in finite time no matter how small the initial data is, which follows from a new weight function and ODE inequalities. Moreover, the upper-bound of the lifespan can be estimated., Comment: 15 pages. Welcome all comments

In this paper, we investigate that the H\"older regularity of solutions to the time fractional Schr\"odinger equation of order $1, Comment: 25 pages

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.

In this paper, we study the stability of a one-dimensional Bresse system with infinite memory type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. Unlike [6], [10], and [22], we consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions and generalizes that of [6], [10], and [22].

In this paper, we study the following Schrodinger-Kirchhoff–type equation with critical or supercritical growth −(a+b∫R3|∇u|2dx)△u+V(x)u=f(x,u)+λ|u|p−2u,x∈R3, where a>0, b>0, λ>0, and p≥6. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ>0 by variational method. Moreover, we regard b as a parameter and obtain a convergence property of the nontrivial solution as b↘0. Our main contribution is related to the fact that we are able to deal with the case p>6.

In this paper, we introduce different kinds of growth orders for the set of entire solutions to the most general framework of higher-dimensional polynomial Cauchy-Riemann equations ∏i=1p(D−λi)kif=0, where D:=∂f∂x0+∑i=1nei∂f∂xi is the hypercomplex Cauchy-Riemann operator, λi are arbitrarily chosen nonzero complex constants, and ki are arbitrarily chosen positive integers. The core ingredient is a projection formula that establishes a relation to the ki-monogenic component functions, which are null-solutions to iterates of the Cauchy-Riemann operator that we studied in earlier works. Furthermore, we briefly outline the analogies of the Lindelof-Pringsheim theorem in this context.

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi-Newton's method, which is based on Frechet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi-Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.

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 study constraint minimizers of the following L2−critical minimization problem: e(N):=inf{E(u),u∈H1(R3)and∫R3|u|2dx=N>0}, where E(u) is the Schrodinger-Poisson-Slater functional E(u):=∫R3|∇u|2dx−12∫R3∫R3u2(y)u2(x)|x−y|dydx−35∫R3m(x)|u|103dx, and N denotes the mass of the particles in the Schrodinger-Poisson-Slater system. We prove that e(N) admits minimizers for N N∗, where Q(x) is the unique positive solution of △u−u+u73=0 in R3. Some results on the existence and nonexistence of minimizers for e(N∗) are also established. Further, when e(N∗) does not admit minimizers, the limit behavior of minimizers as N↗N∗ is also analyzed rigorously.

In this paper, we study the following fractional Schrodinger equations: (−△)αu+λV(x)u=κ|u|q−2u|x|s+β|u|2α∗−2u,u∈Hα(RN),N⩾3,(1) where (−△)α is the fractional Laplacian operator with α∈(0,1),2≤q≤2α,s∗=2(N−s)N−2α≤2α∗=2NN−2α, 0≤s≤2α, λ>0, κ and β are real parameter. 2α∗ is the critical Sobolev exponent. We prove a fractional Sobolev-Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.

This paper mainly focus on the exponential stabilization problem of coupled systems on networks with mixed time-varying delays. Periodically intermittent control is used to control the coupled systems on networks with mixed time-varying delays. Moreover, based on the graph theory and Lyapunov method, two different kinds of stabilization criteria are derived, which are in the form of Lyapunov-type theorem and coefficients-type criterion, respectively. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, a decision theorem is also presented. It is straightforward to show the stability of original system can be determined by that of modified system with added absolute value into the coupling weighted-value matrix. Finally, the feasibility and validity of the obtained results are demonstrated by several numerical simulation figures.

In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second-order quasilinear equation (ϕp(x′))′+g(x)=e(t), where ϕp:R→R,ϕp(u)=|u|p−2u,p>1, g satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase-plane analysis methods and a generalized version of the Poincare-Birkhoff twist theorem.

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrodinger equations: −Δu+V1(x)u=f1(x,u)+λ(x)v,x∈R2,−Δv+V2(x)v=f2(x,v)+λ(x)u,x∈R2, where the nonlinearities f1(x,s) and f2(x,s) are superlinear at infinity and have exponential critical growth of the Trudinger-Moser type. The potentials V1(x) and V2(x) are nonnegative and satisfy a condition involving the coupling term λ(x), namely, λ(x)2

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.

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.

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.

This paper is concerned with the following nonlinear fractional Schrodinger equation e2s(−Δ)su+V(x)u=|u|p−2u,inRN, where e>0 is a small parameter, V(x) is a positive function, 0 2s). Under some suitable conditions, we prove that for any positive integer k, one can construct a nonradial sign-changing (nodal) solutions with exactly k maximum points and k minimum points near the local minimum point of V(x).

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.

In this paper, we study a model of hematopoiesis with time-varying delays and discontinuous harvesting, which is described by a nonsmooth dynamical system. Based on a newly developed method, nonsmooth analysis, and the generalized Lyapunov method, some new delay-dependent criteria are established to ensure the existence and global exponential stability of positive periodic solutions. Moreover, an example with numerical simulations is presented to demonstrate the effectiveness of theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.