Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Soliton solutions of a BBM(m, n) equation with generalized evolution

We consider a BBM(m, n) equation which is a generalization of the celebrated Benjamin-Bona-Mahony equation with generalized evolution term. By using two solitary wave ansatze in terms of sech^p(x) and tanh^p(x) functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. Note that, it is always useful and desirable to construct exact analytical solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.     

Composition operators from F(p, q, s) spaces to the nth weighted-type spaces on the unit disc

In this note we characterize the boundedness and compactness of the composition operator from the general function space F(p, q, s) to the nth weighted-type space on the unit disk, where the nth weighted-type space has been recently introduced by Stevo Stevi?.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian

In this paper we consider the multiplicity of positive solutions for the one-dimensional p-Laplacian differential equation (?p^′(u^′))+q(t)f(t,u,u^′)=0, t∈(0,1), subject to some boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to some multipoint boundary value problems.     

Property (X) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t²k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X⁺), which is very important for the existence of periodic solutions and oscillation theory.     

Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative

This paper presents sufficient conditions for the existence and multiplicity of positive solutions to the one-dimensional p-Laplacian differential equation (?p^′(u^′))+a(t)f(u,u^′)=0, subject to some boundary conditions. We show that it has at least one or two or three positive solutions under some assumptions by applying the fixed point theorem.     

The equivalence of curves in SL(n, R) and its application to ruled surfaces

The generating system of the differential algebra for invariant differential polynomials with two parametric curves is obtained. Conditions for the equivalence of two parametric curves families are given. We are also proved that the generating differential invariants of two parametric curves are independent. Finally, we reduce the SL(n, R)-equivalent problem for ruled surfaces to that of parametric curves.     

Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x=0 and x=-1. We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p, any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε→0) of the form ∑Y_n(x/ε^′)ε^′n, where ε^′p+1=ε, for x=ε^′X with X large enough, but independent of ε. In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.     

Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations

Commonly studied models of the consecutive-k-out-of-n: F repairable systems in the existing literatures were considering the systems which had one repairman without vacation or infinite repairmen without vacations. In addition to those models, multiple repairmen without vacations are studied occasionally. However, technical personnel are very short in some fields. Some failed components cannot be repaired in time. This paper deals with the phenomenon of waiting for repair by supposing R repairmen with multiple vacations in the system. Using the pairs (i, |j|), the factor that the R repairmen taking multiple vacations was embedded into the classical C(k, n: F) system. Reliability indexes are presented. Finally, the Runge–Kutta method was used to a special case, and the experimental results demonstrate the necessity and validity of the new model.     

Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u₀(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P^∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.     

Existence of three positive solutions for boundary value problems with p-Laplacian

By using fixed point theorem, we study the following equation g(u^′′(t))+a(t)f(u)=0 subject to boundary conditions, where g(v)=|v|^p−2v with p>1; the existence of at least three positive solutions is proved.