This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented.
Optical fiber communication system is one of the core supporting systems of the modern internet age, and studies on the ultrashort optical pulses are at the forefront of fiber optics, modern optics and optical engineering. Hereby, symbolic computation on the recently-proposed generalized higher-order variable-coefficient Hirota equation is performed, for certain ultrashort optical pulses propagating in a nonlinear inhomogeneous fiber. For the complex envelope function associated with the optical-pulse electric field in the fiber, an auto-Bäcklund transformation is worked out, along with a family of the analytic solutions. Both our Bäcklund transformation and analytic solutions depend on the optical-fiber variable coefficients which represent the effects of the first-order dispersion, second-order dispersion, third-order dispersion, Kerr nonlinearity, time delaying, phase modulation and gain/loss. Relevant constraints among those coefficients are also presented. We expect that the work could be of some use for the fiber-optics investigations.
In this work, we investigate the uniqueness of solutions for a class of nonlinear boundary value problems for fractional differential equations. The main novelty of this work is that the Lipschitz constant is related to the first eigenvalues corresponding to the relevant operators. Our analysis relies on the -positive operator.
In this paper, the -dimensional B-type Kadomtsev–Petviashvili (BKP) equation is investigated, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion. With the aid of the binary Bell polynomial, its bilinear formalism is succinctly constructed, based on which, the soliton wave solution is also obtained. Furthermore, by means of homoclinic breather limit method, its rogue waves and homoclinic breather waves are derived, respectively. Our results show that rogue wave can come from the extreme behavior of the breather solitary wave for -dimensional nonlinear wave fields.
In this paper, the homoclinic breather limit method is employed to find the breather wave and the rational rogue wave solutions of the ( )-dimensional Ito equation. Moreover, based on its bilinear form, the solitary wave solutions of the equation are also presented with a detailed derivation. The dynamic behaviors of breather waves, rogue waves and solitary waves are analyzed with some graphics, respectively. The results imply that the extreme behavior of the breather solitary wave yields the rogue wave for the ( )-dimensional Ito equation.
In this letter, the local fractional similarity solution is addressed for the non-differentiable diffusion equation. Structuring the similarity transformations via the rule of the local fractional partial derivative operators, we transform the diffusive operator into a similarity ordinary differential equation. The obtained result shows the non-differentiability of the solution suitable to describe the properties and behaviors of the fractal content.
Under investigation in this work is a generalized ( )-dimensional Kadomtsev–Petviashvili equation, which can describe many nonlinear phenomena in fluid dynamics. By virtue of Bell’s polynomials, an effective and straightforward way is presented to explicitly construct its bilinear form and soliton solutions. Furthermore, based on the bilinear formalism, a direct method is employed to explicitly construct its rogue wave solutions with an ansätz function. Finally, the interaction phenomena between rogue waves and solitary waves are presented with a detailed derivation. The results can be used to enrich the dynamical behavior of higher dimensional nonlinear wave fields.
In this paper, a discrete Hirota equation is analytically investigated. The -fold Darboux transformation (DT) is constructed based on the Lax pair for the equation. In addition, the discrete -soliton solutions under the vanishing background are derived. Especially, the dynamic features of one-soliton and two-soliton solutions are displayed through figures.
In the present paper, the prolongation technique and Painlevé analysis are performed to a two-component Korteweg–de Vries system. It is proved that this system is both Lax integrable and P-integrable. By embedding the prolongation algebra in the algebra, the 3×3 Lax representation of the system is derived. Moreover, the auto-Bäcklund transformation and some exact solutions for the two-component Korteweg–de Vries system are proposed explicitly, and it is shown that this system owns solitary wave solutions which demonstrate fission and fusion behaviors.
In this paper, we establish the uniqueness of positive solution for a fractional model of turbulent flow in a porous medium by using the fixed point theorem of the mixed monotone operator. An example is also given to illustrate the application of the main result.
In this paper, a new integral transform operator, which is similar to Fourier transform, is proposed for the first time. As a testing example, an application to the one-dimensional heat-diffusion problem is discussed. The result demonstrates accuracy and efficiency of the present technology to find the analytical solution for the heat-transfer problem. (C) 2016 Elsevier Ltd. All rights reserved.
The Newton iteration is basic for solving nonlinear optimization problems and studying parameter estimation algorithms. In this letter, a maximum likelihood estimation algorithm is developed for estimating the parameters of Hammerstein nonlinear controlled autoregressive autoregressive moving average (CARARMA) systems by using the Newton iteration. A simulation example is provided to show the effectiveness of the proposed algorithm.
In this paper, we firstly introduce a concept of delayed Mittag-Leffler type matrix function, an extension of Mittag-Leffler matrix function for linear fractional ODEs, which help us to seek explicit formula of solutions to fractional delay differential equations by using the variation of constants method. Secondly, we present the finite time stability results by virtue of delayed Mittag-Leffler type matrix. Finally, an example is given to illustrate our theoretical results.
Block-oriented Hammerstein systems consist of a nonlinear static block followed by a linear dynamic block. For the identification of a complex class of multi-input multi-output (MIMO) Hammerstein systems with different types of coefficients: a matrix coefficient and scalar coefficients, it is difficult to express this class of complex Hammerstein systems as a regression identification model in all parameters of the nonlinear part and the linear part in which the standard least squares method can be easily applied to implement parameter estimation. By the matrix transformation, this paper reframes an MIMO Hammerstein system with different types of coefficients into two models, each of which is expressed as a regression form in the parameters of the nonlinear part or in the parameters of the linear part. Then a hierarchical extended least squares algorithm is applied to these two models to alternatively estimate the parameters of the nonlinear part and the linear part.
In this paper, the higher-order nonlinear Schrödinger equation, which can be widely used to describe the dynamics of the ultrashort pulses in optical fibers, is under investigation. By means of the modified Darboux transformation, the hierarchies of breather wave and rogue wave solutions are generated from the trivial solution. Furthermore, the main characteristics of the breather and rogue waves are graphically discussed. The results show that the extreme behavior of the breather wave yields the rogue wave for the higher-order nonlinear Schrödinger equation.
With the generalized bilinear operators based on a prime number , a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.
We are concerned with the uniqueness of solutions for the following nonlinear fractional boundary value problem: where denotes the standard Riemann–Liouville fractional derivative. Our analysis relies on the theory of linear operators and the norm.
Based on the nonlocal nonlinear Schrödinger equation that governs phenomenologically the propagation of laser beams in nonlocal nonlinear media, we theoretically investigate the propagation of sinh-Gaussian beams (ShGBs). Mathematical expressions are derived to describe the beam propagation, the intensity distribution, the beam width, and the beam curvature radius of ShGBs. It is found that the propagation behavior of ShGBs is variable and closely related to the parameter of sinh function (PShF). If the PShF is small, the transverse pattern of ShGBs keeps invariant during propagation for a proper input power, which can be regarded as solitons. If the PShF is large, it varies periodically, which is similar to the evolution of temporal higher-order solitons in nonlinear optical fiber. Numerical simulations are carried out to illustrate the typical propagation characteristics.
Under investigation in this paper is a coupled nonlinear Schrödinger (NLS) equation, which describes nonlinear pulse propagation in optical fibers by retaining terms up to the next leading asymptotic order. Based on the Lax pair of the coupled NLS equation, we construct the determinant representation of the -fold Darboux transformation(DT). Furthermore, by using the obtained -fold DT, we obtain its higher-order soliton, breather and rogue wave solutions. Finally, the dynamic characteristics of these solutions are discussed.
In this paper, a coupled Hirota system with higher-order effects is analytically investigated. The results show that the breather solutions can be converted into some types of nonlinear localized and periodic solutions on the plane-wave background. The exact relations for the conversions are presented, which depend on the higher-order effects, the background frequency and the eigenvalue. Via some graphic illustrations, the collisions between these nonlinear waves in the second-order conversions are displayed.