Authors:Zhenjie Liu Pages: 219 - 232 Abstract: Abstract This paper is devoted to the construction of solutions for one-dimensional wave equations with Dirichlet or Neumann boundary conditions by means of a Nash-Moser iteration scheme, for a large set of frequencies. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0137-5 Issue No:Vol. 7, No. 3 (2017)

Authors:Maxim Limonov; Roman Nedela; Alexander Mednykh Pages: 233 - 243 Abstract: Abstract In this paper we give a few discrete versions of Robert Accola’s results on Riemann surfaces with automorphism groups admitting partitions. As a consequence, we establish a condition for \(\gamma \) -hyperelliptic involution on a graph to be unique. Also we construct an infinite family of graphs with more than one \(\gamma \) -hyperelliptic involution. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0138-4 Issue No:Vol. 7, No. 3 (2017)

Authors:Ying Wang; Yunxi Guo Pages: 245 - 254 Abstract: Abstract In this paper, we developed, for the first time, the exact expressions of several periodic travelling wave solutions and a solitary wave solution for a shallow water wave model of moderate amplitude. Then, we present the existence theorem of the global weak solutions. Finally, we prove the stability of solution in \(L^{1}(R)\) space for the Cauchy problem of the equation. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0139-3 Issue No:Vol. 7, No. 3 (2017)

Authors:S. A. Bishop; E. O. Ayoola; G. J. Oghonyon Pages: 255 - 265 Abstract: Abstract New results on existence and uniqueness of solution of impulsive quantum stochastic differential equation with nonlocal conditions are established. The nonlocal conditions are completely continuous. The methods applied here are simple extension of the methods applied in the classical case to this noncummutative quantum setting. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0140-x Issue No:Vol. 7, No. 3 (2017)

Authors:Perumal Muthukumar; Saminathan Ponnusamy Pages: 267 - 283 Abstract: Abstract In this article, we define discrete analogue of generalized Hardy spaces and its separable subspace on a homogenous rooted tree and study some of its properties such as completeness, inclusion relations with other spaces, separability, growth estimate for functions in these spaces and their consequences. Equivalent conditions for multiplication operators to be bounded and compact are also obtained. Furthermore, we discuss about point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0141-9 Issue No:Vol. 7, No. 3 (2017)

Authors:Vladimir Ryazanov Pages: 285 - 289 Abstract: Abstract It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for a special case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0142-8 Issue No:Vol. 7, No. 3 (2017)

Authors:K. Sayevand; K. Pichaghchi Pages: 291 - 318 Abstract: Abstract In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called \(\mathsf {WKB}\) method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the \(\mathsf {WKB}\) to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the \(\mathsf {WKB}\) method in the scope of the fractional differential equation. By means of this extension, the \(\mathsf {WKB}\) analysis based on the Borel resummation, for fractional differential operators of \(\mathsf {WKB}\) type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified \(\mathsf {WKB}\) . PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0143-7 Issue No:Vol. 7, No. 3 (2017)

Authors:B. C. Chanyal; S. K. Chanyal Pages: 319 - 334 Abstract: Abstract Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac–Maxwell’s equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0144-6 Issue No:Vol. 7, No. 3 (2017)

Authors:Valentin Lychagin; Valeriy Yumaguzhin Pages: 107 - 115 Abstract: Abstract In this paper (cf. Lychagin and Yumaguzhin, in Anal Math Phys, 2016) a class of totally geodesics solutions for the vacuum Einstein equations is introduced. It consists of Einstein metrics of signature (1,3) such that 2-dimensional distributions, defined by the Weyl tensor, are completely integrable and totally geodesic. The complete and explicit description of metrics from these class is given. It is shown that these metrics depend on two functions in one variable and one harmonic function. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0130-z Issue No:Vol. 7, No. 2 (2017)

Authors:Boris Rubin Pages: 117 - 150 Abstract: Abstract We suggest new modifications of the Helgason’s support theorem and description of the kernel for the hyperplane Radon transform and its dual. The assumptions for functions are formulated in integral terms and close to minimal. The proofs rely on the properties of the Gegenbauer–Chebyshev integrals which generalize Abel type fractional integrals on the positive half-line. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0133-9 Issue No:Vol. 7, No. 2 (2017)

Authors:Min Wang Pages: 151 - 163 Abstract: Abstract This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP \((F,\phi , K)\) ) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP \((F,\phi ,K)\) ) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0134-8 Issue No:Vol. 7, No. 2 (2017)

Authors:Haibo Lin; Suqing Wu; Dachun Yang Pages: 187 - 218 Abstract: Abstract Let \((\mathcal {X},d,\mu )\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu )\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by T and b from the atomic Hardy space \(\widetilde{H}^1(\mu )\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty }(\mu )\) . From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu )\) for all \(p\in (1,\infty )\) . Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu )\) function from \(\widetilde{H}^1(\mu )\) into \(L^{1/{(1-\alpha )},\,\infty }(\mu )\) is also presented. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0136-6 Issue No:Vol. 7, No. 2 (2017)

Authors:Zhonglong Zhao; Bo Han Abstract: Abstract On the basis of bilinear equation of a (3+1)-dimensional B-type KP equation, we construct the lump-type solutions by symbolic computation. The (3+1)-dimensional B-type KP equation can be used to describe the propagation of nonlinear waves in fluid dynamics. The lump solutions of three dimensionally reduced (2+1)-dimensional B-type KP equation are derived. The sufficient and necessary conditions to guarantee the analyticity, positiveness and localization of lump solutions are discussed. Figures are presented to illustrate the energy distribution of these lump wave solutions. PubDate: 2017-08-04 DOI: 10.1007/s13324-017-0185-5

Authors: Gegenhasi Abstract: Abstract In this paper, we derive the Grammian determinant solutions to the modified semi-discrete two-dimensional Toda lattice equation, and then construct the semi-discrete two-dimensional Toda lattice equation with self-consistent sources via source generation procedure. The algebraic structure of the resulting coupled modified differential–difference equation is clarified by presenting its Grammian determinant solutions and Casorati determinant solutions. As an application of the Grammian determinant and Casorati determinant solution, the explicit one-soliton and two-soliton solution of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources are given. We also construct another form of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources which is the Bäcklund transformation for the semi-discrete two-dimensional Toda lattice equation with self-consistent sources. PubDate: 2017-07-17 DOI: 10.1007/s13324-017-0184-6

Authors:W. M. Abd-Elhameed Abstract: Abstract In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type \(_4F_{3}(1)\) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method. PubDate: 2017-07-15 DOI: 10.1007/s13324-017-0183-7

Authors:Ramin Asadi; Mehdi Vatandoost; Yousef Bahrampour Abstract: Abstract It can be seen from some theorems proved by Penrose that when Strong Causality or Distinguishing fail, they fail along at least a segment of a null geodesic. In this paper, we investigate the behavior of the other causality conditions. In particular, we prove that when causal continuity and stable causality fail at some point, there is a segment of a null geodesic through that point along which the conditions fail. PubDate: 2017-07-12 DOI: 10.1007/s13324-017-0182-8

Authors:Hui Nie; Junyi Zhu; Xianguo Geng Abstract: Abstract The Gerdjikov–Ivanov equation is investigated by the Riemann–Hilbert approach and the technique of regularization. The trace formula and new form of N-soliton solution are given. The dynamics of the stationary solitons and non-stationary solitons are discussed. PubDate: 2017-06-19 DOI: 10.1007/s13324-017-0179-3

Authors:Jin-Yun Yang; Wen-Xiu Ma; Zhenyun Qin Abstract: Abstract Based on the Hirota bilinear form of the \((2+1)\) -dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. PubDate: 2017-06-17 DOI: 10.1007/s13324-017-0181-9

Authors:Jun-ichi Inoguchi; Marian Ioan Munteanu; Ana Irina Nistor Abstract: Abstract We study magnetic trajectories corresponding to contact magnetic fields in 3-dimensional quasi-Sasakian manifolds. We show that they are slant curves, that is their contact angles are constant. We prove that such magnetic curves are geodesics for a certain linear connection for which all four structure tensor fields are parallel. PubDate: 2017-06-13 DOI: 10.1007/s13324-017-0180-x

Authors:Heng Wang; Shuhua Zheng Abstract: Abstract By using the dynamical system approach, the exact travelling wave solutions for a system of coupled nonlinear electrical transmission lines are studied. Based on this method, the bifurcations of phase portraits of a dynamical system are given. The two-dimensional solitary wave solutions and periodic wave solutions on coupled nonlinear transmission lines are obtained. With the aid of Maple, the numerical simulations are conducted for solitary wave solutions and periodic wave solutions to the model equation. The results presented in this paper improve upon previous studies. PubDate: 2017-06-01 DOI: 10.1007/s13324-017-0178-4