No abstract is provided for this article.
Purpose The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact. Design/methodology/approach A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales. Findings Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found. Originality/value The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.
The nano/microelectromechanical systems (N/MEMS) have gained considerable attention in the past few decades for their attractive properties such as small size, high reliability, batch fabrication, and low power consumption. Carbon‐based nanostructures such as one‐dimensional carbon nanotubes (CNTs) and two‐dimensional graphene are the key materials in many N/MEMS and very attractive due to their unique properties and ultra‐small dimensions especially the large surface area‐to‐volume ratio make them prime candidates for sensing applications in N/MEMS. Therefore, the main objective of this manuscript is to analyze the dynamic behavior of a graphene N/MEMS. The mathematical model of this system uses constitutive stress–strain law and the electrostatic Coulomb force for the restoring force of the spring and the driving force of the mass attached to the spring, respectively. The modeled system is then solved by employing the variational iteration method (VIM) accompanied by the techniques of the Laplace transform. This coupling of VIM and Laplace transform (Laplace‐based variational iteration method [LVIM]) not only suggests an easier approach to determine the Lagrange multiplier used in the VIM but also provides approximate nonlinear frequency and approximate solution of graphene N/MEMS in an efficient way. Moreover, the LVIM also approximates the pull‐in threshold, a critical phenomenon associated with N/MEMS, in terms of model parameters. Finally, to verify the obtained findings, the results are compared with those achieved numerically as well.
Time-fractional differential equations can accurately describe heat conduction in fractal media, such as wool fibers, goose down and polar bear hair. The fractional complex transform is used to convert time-fractional heat conduction equations with the modified Riemann-Liouville derivative into ordinary differential equations, and exact solutions can be easily obtained. The solution process is straightforward and concise.
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The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of the equation.
Single-walled carbon nanotubes (SWNTs) and multi-walled nanotubes (MWNTs) are gaining appeal in mechanical engineering and industrial applications due to their direct influence on enhancing the thermal conductivity of base fluids. With such intriguing properties of carbon nanotubes in mind, our goal in this work is to investigate radiation effects on the flow of carbon nanotube suspended nanofluids in the presence of a magnetic field past a stretched sheet impacted by slip state. CNTs flow and heat transmission are frequently modelled in practice using nonlinear differential equation systems. This system has been precisely solved, and an accurate analytical expression for the fluid velocity in terms of an exponential function has been derived, while the temperature distribution is stated in terms of a confluent hypergeometric function. The impact of the radiation parameter, slip parameter, sloid volume fraction, magnetic parameter, Eckart and Prandtl numbers on the velocity, temperature, and heat transfer rate profiles are demonstrated using a parametric analysis. When compared to the two types of nanoparticles (Cooper and Silver) in earlier published articles, temperature profiles for single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs) are revealed to be particularly sensitive to radiation, solid volume fraction, and slip parameters. Nanomechanical gears, nanosensors, nanocomposite materials, resonators, and thermal materials are only a few of the present problem's technical applications.
The fractional complex transform is used to analytically deal with fractional differential equations. Two examples are given to elucidate the solution procedure, showing it is extremely accessible to nonmathematicians.
This paper suggests a fractal two-phase fluid model for the polymer melt filling process to deal effectively with the unsmooth front interface. An infinitesimal fluid element model in a fractal space is proposed to establish the governing equations according to the conservation laws in fluid mechanics, the fractal divergence and fractal Laplace operator are defined. The unsmooth interface is solved numerically, and fibers’ motion properties on the interface are also elucidated. Moreover, the distribution of fibers on the interface at different stages shows the fractal property of the fibers’ motion. However, the motion of fibers is affected by the flow of macroscopic polymer melt, and the fiber orientation in the interface shows a certain statistical regularity. Based on the characters of fiber orientation, the fractal interface can be used for the optimal design of the polymer melt filling process.
No abstract is provided for this article.
This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.
Higher-order boundary value problems have been widely studied in thermal science, though there are some analytical methods available for such problems, the barycentric rational interpolation collocation method is proved in this paper to be the most effective as shown in three examples.
Article Determination of Limit Cycle by Hamiltonian Approach for Strongly Nonlinear Oscillators was published on December 1, 2010 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 11, issue 12).
Bubble elecrospinning is an excellent and novelty method to fabricate nanofibres with high throughout. This paper investigates experimentally the effects of solution concentration and its viscosity on diameter and morphology of bubble electrospun polyvinyl alcohol (PVA) nanofibres, where pure water is used as a solvent for environmental consideration. The experiment shows that the as spun fibres change their morphology from few beads, smooth surfaces and finally to larger diameter fibres when PVA solution concentration increases gradually. An allometric relation between the fibre diameter and the solution concentration is obtained. The minimal diameter is as small as 38·3 nm.
The homotopy perturbation technique does not depend upon a small parameter in the equation. By the homotopy technique in topology, a homotopy is constructed with an imbedding parameter p∈[0,1], which is considered as a “small parameter”. Some examples are given. The approximations obtained by the proposed method are uniformly valid not only for small parameters, but also for very large parameters.
Variational approach to various nonlinear equations is briefly introduced, mathematical models for electrospinning are also reviewed, a simple scaling approach to predicting the diameter of the electrospun fiber is given, and experimental verification is provided.
In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.
The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion.
