Dynamical analysis of the transmission of seasonal diseases using the differential transformation method

The aim of this paper is to apply the differential transformation method (DTM) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models where the solutions exhibit periodic behavior due to the seasonal transmission rate. These models describe the dynamics of the different classes of the populations. Here the concept of DTM is introduced and then it is employed to derive a set of difference equations for this kind of epidemic models. The DTM is used here as an algorithm for approximating the solutions of the epidemic models in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the DTM is accurate, easy to apply and the calculated solutions preserve the properties of the continuous models, such as the periodic behavior. Furthermore, it is showed that the DTM avoids large computational work and symbolic computation.     

A semi-discretization method based on quartic splines for solving one-space-dimensional hyperbolic equations

In this paper, based on C³ quartic splines, a semi-discretization method containing two schemes is constructed to solve one-space-dimensional linear hyperbolic equations. It is shown that both schemes are unconditionally stable and their approximation orders are of O(k⁵+h⁴) and of O(k⁷+h⁴) with k and h being step sizes in time and space, respectively, which are much higher than those of other published schemes. A numerical example is presented and the results are compared with other published numerical results.     

Solutions to operator equations on Hilbert -modules

Let be a C^*-algebra, E,F and G be Hilbert -modules, , and . We generalize the Douglas theorem about the operator equation TX=T^′ from Hilbert space to Hilbert C^*-module. To the equation TX=T^′ and to the system of two equations TX=T^′ and XS=S^′, we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G=E). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G=E) to the equation TX=T^′ are given too.     

Nodal and multiple constant sign solutions for resonant -Laplacian equations with a nonsmooth potential

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Using a variational approach combined with suitable truncation techniques and the method of upper–lower solutions, we prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal. Our hypotheses on the nonsmooth potential allow resonance at infinity with respect to the principal eigenvalue λ₁>0 of .     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

On the modulus of -convexity

In this paper, we prove that the moduli of W^*-convexity, introduced by Ji Gao [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386], of a Banach space X and of the ultrapower of X itself coincide whenever X is super-reflexive. Moreover, we improve a sufficient condition for uniform normal structure of the space and its dual. This generalizes and strengthens the main results of [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386].     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).

Under various assumptions, the existence of periodic solutions of the problem is obtained by applying Mawhin’s continuation theorem.     

Application of continuous-time programming problems to a class of variational-type inequalities

In this paper, we establish sufficiency criteria under generalized ρ−(η,θ)-invexity conditions for general continuous-time programming problems with nonlinear equality/inequality constraints. Using this we establish some existence criteria for solutions of a class of variational-type inequalities.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Bessel capacities on compact manifolds and their relation to Poisson capacities

A motivation for this paper comes from the role of Choquet capacities in the study of semilinear elliptic partial differential equations. In particular, the recent progress in the classification of all positive solutions of Lu=uα in a bounded smooth domain E⊂Rd was achieved by using, as a tool, capacities on a smooth manifold ∂E. Either the Poisson capacities (associated with the Poisson kernel in E) or the Bessel capacities (related to the Bessel kernel) have been used. In this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities. (Two capacities are called equivalent if their ratio is bounded away from 0 and ∞.) In the literature Bessel capacities are considered mostly in the space Rd. We introduce two versions of Bessel capacities on a compact N-dimensional manifold. A class Cap?,p of equivalent capacities is defined, for ?p?N, on every compact Lipschitz manifold. Another class CB?,p is defined (for all ?>0, p>1) in terms of a diffusion process on a C²-manifold. These classes coincide when both are defined. If the manifold is the boundary of a bounded C²-domain E⊂Rd, then both versions of the Bessel capacities are equivalent to the Poisson capacities.     

The fixed point property in -triples and preduals of -triples

We prove that for every member X in the class of real or complex JB^*-triples or preduals of JBW^*-triples, the following assertions are equivalent:

(1) X has the fixed point property.

(2) X has the super fixed point property.

(3) X has normal structure.

(4) X has uniform normal structure.

(5) The Banach space of X is reflexive.

As a consequence, a real or complex C^*-algebra or the predual of a real or complex W^*-algebra having the fixed point property must be finite-dimensional.

Trajectory analysis of nonlinear kinetic models of population dynamics of several species

The existence and uniqueness of global solutions are proved for kinetic models of coupled population dynamics. Moreover, under consideration, the uniqueness and the stability analysis of this steady-state solution are discussed. This analysis uses, essentially, the dissipativity, a subtangential condition and the positivity of the related C₀-semigroup.     

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov () system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D(m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D(m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D(m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).