Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method

The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions

Solitary-wave solutions of the Degasperis-Procesi equation by means of the homotopy analysis method

The homotopy analysis method is applied to the Degasperis-Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

M-fractional derivative under interval uncertainty: theory, properties and applications

In the recent years some efforts were made to propose simple and well-behaved fractional derivatives that inherit the classical properties from the first order derivative. In this regards, the truncated M-fractional derivative for α-differentiable function was recently introduced that is a generalization of four fractional derivatives presented in the literature and has their important features. In this research, we aim to generalize this novel and effective derivative under interval uncertainty. The concept of interval truncated M-fractional derivative is introduced and some of the distinguished properties of this interesting fractional derivative such as Rolle’s and mean value theorems, are developed for the interval functions. In addition, the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative are also investigated. Finally, we present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w-increasing (w-decreasing) by solving a number of test problems

Effects of Thermocapillarity and Thermal Radiation on Flow and Heat Transfer in a Thin Liquid Film on an Unsteady Stretching Sheet

This paper examines the effects of thermocapillarity and thermal radiation on the boundary layer flow and heat transfer in a thin film on an unsteady stretching sheet with nonuniform heat source/sink. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and then are solved by using the homotopy analysis method (HAM). The effects of the radiation parameter, the thermocapillarity number, and the temperature-dependent parameter in this study are discussed and presented graphically via velocity and temperature profiles

On Series Solutions for MHD Plane and Axisymmetric Flow Near a Stagnation Point

This investigation presents a mathematical model describing the momentum, heat and mass transfer characteristics of magnetohydrodynamic MHD flow and heat generating/absorbing fluid near a stagnation point of an isothermal two-dimensional body of an axisymmetric body. The fluid is electrically conducting in the presence of a uniform magnetic field. The series solution is obtained for the resulting coupled nonlinear differential equation. Homotopy analysis method HAM is utilized in obtaining the solution. Numerical values of the skin friction coefficient and the wall heat transfer coefficient are also computed

Boundary layer flow of nanofluid over an exponentially stretching surface

The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically. The transport equations include the effects of Brownian motion parameter and thermophoresis parameter. The highly nonlinear coupled partial differential equations are simplified with the help of suitable similarity transformations. The reduced equations are then solved analytically with the help of homotopy analysis method (HAM). The convergence of HAM solutions are obtained by plotting h-curve. The expressions for velocity, temperature and nanoparticle volume fraction are computed for some values of the parameters namely, suction injection parameter α, Lewis number Le, the Brownian motion parameter Nb and thermophoresis parameter Nt

Flow and heat transfer in a nanofluid thin film over an unsteady stretching sheet

This study analyzes the heat transfer of a thin film flow on an unsteady stretching sheet in nanofluids. Three different types of nanoparticles are considered; copper Cu, alumina Al2O3 and titania TiO2 with water as the base fluid. The governing equations are simplified using similarity transformations. The resulting coupled nonlinear differential equations are solved by the Homotopy Analysis Method (HAM). The analytical series solutions are presented and the numerical results obtained are tabulated. In particular, it shows that the heat transfer rate decreases when nanoparticles volume fraction increases

Entropy Generation Analysis for Stagnation Point Flow in a Porous Medium over a Permeable Stretching Surface

This paper presents entropy generation analysis for stagnation point flow in a porous medium over a permeable stretching surface with heat generation/absorption and convective boundary condition. We have used Von Karman transformations to transform the governing equations into ordinary differential equations.Thevelocity, temperature and concentration profiles obtained using the Homotopy Analysis Method. The HAM is a valid mathematical tool for most of non-linear problems in science and engineering. Finally we have computed the entropy generation number. The effect of the Prandtl number, Brinkman number, Reynolds number, suction/injection parameter, Biot number, Lewis number, Brownian motion parameter, thermophoresisparameterand constant parameters on velocity, concentration and temperature profiles are analyzed. Moreover the influences of the Reynolds number and Brinkman number on the entropy generation are presented.The entropy generation number increases with increasing the Brinkman and Reynolds number

Formulas for the amplitude of the van der Pol limit cycle through the homotopy analysis method

The limit cycle of the van der Pol oscillator, x¨+ε(x2−1)x˙+x=0, is studied in the plane (x,x˙) by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter ε is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.The Gobierno of Navarra, Spain, Res. 07/05/2008 is acknowledged by its financial support