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SPECIAL ISSUE PAPER Fractional ℎ ‐differences with exponential kernels and their monotonicity properties Iyad Suwan Shahd Owies Thabet Abdeljawad Mathematical Methods in the Applied Sciences, Early View First published: 25 January 2020

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.

This paper presents analytical-approximate solutions of the time-fractional Cahn-Hilliard (TFCH) equations of fourth and sixth-order using the new iterative method (NIM) and q-homotopy analysis method (q-HAM). We obtained convergent series solutions using these iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated.

This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leibniz algebras, studying their isomorphism classes. We also compare our results with the current classifications of 2- and 3-dimensional Leibniz algebras. Finally, we introduce and implement the algorithmic procedure used for our goals and devoted to decide

In this paper we consider the propagation of waves in a closed full- or half-waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition which assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, e.g., considered in [6] and allows an extension to open wave guides (see [10]). After application of the Floquet-Bloch transform we consider the Floquet-Bloch variable α as a parameter in the resulting quasi-periodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result which we have found in [4].

The purpose of this paper is to describe the oscillatory properties of second-order Euler-type half-linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non-oscillation of the considered equations, including the so-called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.

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.

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].

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

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit-point case at a(b) and limit-circle case at b(a)) acting in the Hilbert space L2((a,b);C2). In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self-adjoint extensions of the symmetric operator are given.Two classes of dissipative operators are studied. They are called dissipative at a and dissipative at b. For 2 cases, a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl-Titchmarsh function of the corresponding self-adjoint operator. Finally, we prove theorems on completeness of the system ofroot vectors of the dissipative operators.

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 investigate the well-posedness and stability of mild solutions for a class of neutral impulsive stochastic integro-differential equations in a real separable Hilbert space. By the inequality technique combined with theory of resolvent operator, some sufficient conditions are established for the concerned problems. The obtained conclusions are completely new, which generalize and improve some existing results. An example is given to illustrate the effectiveness of our results.

Fragmentation--coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper we consider the discrete decay--fragmentation equation and prove the existence and uniqueness of physically meaningful solutions to this equation using the theory of semigroups of operators. In particular, we find conditions under which the solution semigroup is analytic, compact and has the asynchronous exponential growth property. The theoretical analysis is illustrated by a number of numerical simulations., 24 pages and 16 figures

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 are concerned with the existence and asymptotic behavior of least energy solutions for following nonlinear Choquard equation driven by fractional Laplacian $$(-\Delta)^{s} u+\lambda V(x)u=(I_{\alpha}\ast F(u))f(u) \ \ in \ \ R^{N},$$ where $N> 2s$, $ (N-4s)^{+}, Comment: 19 pages, text overlap with arXiv:1703.08028

This paper is concerned with laminated beams modeled from the well-established Timoshenko system with time delays and boundary feedbacks. By using semigroup method, we prove the global well-posedness of solutions. Assuming the weights of the delay are small, we establish the exponential decay of energy to the system by using an appropriate Lyapunov functional.

A new defect-correction method based on the pressure projection for the stationary Navier-Stokes equations is proposed in this paper. A local stabilized technique based on the pressure projection is used in both defect step and correction step. The stability and convergence of this new method is analyzed detailedly. Finally, numerical examples confirm our theory analysis and validate high efficiency and good stability of this new method.

The Loewner partial differential equation provides a one-parametric family of conformal maps on the unit disk. The images describe a flow of an expanding simply-connected domain, called the Loewner flow, on the complex plane. In this paper, we present a numerical algorithm for solving the radial Loewner partial differential equation. The algorithm is applied to visualization of Loewner flows with the performance and precision. From the theoretical point of view, our algorithm is based on a recursive formula for determining coefficients of polynomial approximations. We prove that each coefficient converges to true values with reasonable regularity.

In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram-type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results.

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.

The present article deals with existence and uniqueness results for a nonlinear evolution initial-boundary value problem, which originates in an age-structured cell population model introduced by Lebowitz and Rubinow (1974) describing the growth of a cell population. Cells of this population are distinguished by age a and cycle length l. In our framework, daughter and mother cells are related by a general reproduction rule that covers all known biological ones. In this paper, the cycle length l is allowed to be infinite. This hypothesis introduces some mathematical difficulties. We consider both local and nonlocal boundary conditions.

The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in $L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with $p$ in a neighborhood of $2$. This system is obtained by adding the semilinear term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide some results about} equivalence between the Gagliardo and non-tangential traces, as well as between the weak canonical conormal derivatives and the non-tangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well posedness results for the Dirichlet and Neumann problems in $L_p$-based Besov spaces on bounded Lipschitz domains in ${\mathbb R}^n$ ($n\geq 3$) are also presented. Then, employing integral potential operators, we show the well-posedness in $L_2$-based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in ${\mathbb R}^n$ $(n\geq 3)$. Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some $L_p$-based Sobolev spaces. Next we use the well-posedness result in the linear case combined with a fixed point theorem in order to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in $L_p$-based Besov spaces, with $p\in (2-\varepsilon ,2+\varepsilon)$ and some parameter $\varepsilon >0$.

In this paper, we prove a Liouville-type theorem for the steady compressible Hall-magnetohydrodynamics system in Π, where Π is whole space R3 or half space R+3. We show that a smooth solution (ρ,u,B,P) satisfying 1/C0 0 is indeed trivial. This generalizes and improves 2 results of Chae.

Recently, Sanhan and Mongkolkeha introduced the concept of Berinde's cyclic contraction, and they established some results. Unfortunately, these results seem to be incorrect. In this paper, some counterexamples are given.

In this paper, we consider a viscoelastic equation with minimal conditions on the L1(0,∞) relaxation function g, namely, g′(t)≤−ξ(t)H(g(t)), where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=sp and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.

International audience; The aim of this paper was to study the junction between a periodic family of beams and two thin plates. This structure depends on 3 small parameters. We use the decompositions of the displacement fields in every beam and plate to obtain a priori estimates. Then in the case for which the displacements of both plates match, we derive the asymptotic behavior of this structure.

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 time-periodic solution to a simplified version of Ericksen-Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time-periodic external force in a periodic domain in RN. By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time-periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.

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, the 2D Navier-Stokes-Voight equations with 3 delays in R2 is considered. By using the Faedo-Galerkin method, Lions-Aubin lemma, and Arzela-Ascoli theorem, we establish the global well-posedness of solutions and the existence of pullback attractors in H1.

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.

In this paper we consider a periodic 2-dimensional quasi-geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution θ∈C([0,T],Ya,σ1−2α(T2)) for small initial data in the Lei-Lin-Gevrey spaces Ya,σ1−2α(T2). Moreover, we establish an exponential type explosion in finite time of this solution.

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.

The goal in the paper is to advertise Dunkl extension of Szasz beta-type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second-order modulus of continuity, the Lipschitz class functions, Peetre's K-functional, and modulus of weighted continuity by Dunkl generalization of Szasz beta-type operators.

In this paper, we introduce and investigate the functions of (μ,ν)-pseudo S-asymptotically ω-periodic of class r(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for (μ,ν)-pseudo S-asymptotic ω-periodicity of (nonautonomous) semilinear integro-differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings.

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 deals with the attraction-repulsion chemotaxis system with nonlinear diffusion ut=∇·(D(u)∇u)−∇·(uχ(v)∇v)+∇·(uγξ(w)∇w), τ1vt=Δv−α1v+β1u, τ2wt=Δw−α2w+β2u, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain Ω⊂Rn(n⩾2), where the coefficients αi, βi, and τi∈{0,1}(i=1,2) are positive. The function D fulfills D(u)⩾CDum−1 for all u>0 with certain CD>0 and m>1. For the parabolic-elliptic-elliptic case in the sense that τ1=τ2=0 and γ=1, we obtain that for any m>2−2n and all sufficiently smooth initial data u0, the model possesses at least one global weak solution under suitable conditions on the functions χ and ξ. Under the assumption m>γ−2n, it is also proved that for the parabolic-parabolic-elliptic case in the sense that τ1=1, τ2=0, and γ⩾2, the system possesses at least one global weak solution under different assumptions on the functions χ and ξ.

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed-point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

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 prove the large-time behavior, as time tends to infinity, of solutions in Hi×H0i×Hi+1(i=1,2) and H4×H04×H4 for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier-Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.

In this paper, we consider a class of asymptotically linear second-order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.

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

We consider, in this paper, the following nonlinear equation with variable exponents: utt−div(|∇u|r(·)−2∇u)+a|ut|m(·)−2ut=b|u|p(·)−2u, where a,b>0 are constants and the exponents of nonlinearity m,p, and r are given functions. We prove a finite-time blow-up result for the solutions with negative initial energy and for certain solutions with positive energy.

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, the notions of annulets and normal filters are introduced in Stone lattices and their properties are studied. A set of equivalent conditions is obtained to characterize normal filters of a Stone lattice. The extensions of the Glivenko-type congruences on a Stone lattice are investigated via annulets and normal filters. A description of the lattice of all extensions of the Glivenko-type congruences on a Stone lattice is given. A one-to-one correspondence between the class of all extensions and the class of all normal filters of a Stone lattice is obtained. Finally, we observe that every 2 extensions of the Glivenko-type congruences are permutable.

In this paper, we apply wavelets to study the Triebel-Lizorkin type oscillation spaces F˙p,qγ1,γ2(Rn) and identify them with the well-known Triebel-Lizorkin-Morrey spaces. Further, we prove that Calderon-Zygmund operators are bounded on F˙p,qγ1,γ2(Rn).

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 the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω+) is embedded in an unbounded fluid domain (Ω−). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain Ω+, we have a 4×4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain Ω−, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Holder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we obtain optimal decay estimates for the solutions to an evolution equation with critical, structural, dissipation, and absorbing power nonlinearity: utt+Δ2θu+2μ(−Δ)θut+|ut|p−1ut=0,t≥0,x∈Rn, with μ>0, θ is a positive integer, and p>1+4θ/n, in space dimension n∈(2θ,4θ). We use these estimates to find the self-similar asymptotic profile of the solutions, when μ≥1.