Search results

The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper is to employ the modified (MDTM) differential transform method (DTM) to obtain approximate periodic solutions of two cases of nonlinear jerk equation.

The approach is based on MDTM that is developed by combining DTM, Laplace transform and Padé approximant.

Comparison of results obtained by MDTM with those obtained by numerical solutions indicates the excellent accuracy of solution.

The MDTM is extended to determining approximate periodic solution of third-order nonlinear differential equations.

The purpose of this study is to investigate the magneto-hydrodynamics boundary layer Falkner–Skan flow over a flat plate numerically by using the Runge–Kutta method featuring shooting technique and analytically via a new modified analytical technique called improved generalized Adomian decomposition method (improved-GDM).

It is well established that the generalized decomposition method (GDM) (Yong-Chang et al., 2008), which uses a new kind of decomposition strategy for the nonlinear function, has proved its efficiency and superiority when compared to the standard ADM method. In this investigation, based on the idea of improved-ADM method developed by Lina and Song (Song and Wang, 2013), the authors proposed a new analytical algorithm of computation named improved-GDM. Thereafter, the proposed algorithm is tested by solving the nonlinear problem of the hydro-magnetic boundary layer flow over a flat plate.

The proposed improved generalized decomposition method (I-GDM) introduces a convergence-control parameter “ω’’ into the GDM, which accelerates the convergence of solution and reduces considerably the computation time. In fact, the key of this method is mainly based on the best selection of the convergence-control parameter ω.

The paper presents a new efficient algorithm of computation that can be considered as an alternative for solving the nonlinear initial boundary layer value problems. Obtained results show clearly the accuracy of the proposed method.

The purpose of this paper is to propose a Tau method for solving nonlinear Blasius equation which is a partial differential equation on a flat plate.

The operational matrices of derivative and product of modified generalized Laguerre functions are presented. These matrices together with the Tau method are then utilized to reduce the solution of the Blasius equation to the solution of a system of nonlinear equations.

The paper presents the comparison of this work with some well‐known results and shows that the present solution is highly accurate.

This paper demonstrates solving of the nonlinear Blasius equation with an efficient method.

This paper aims to traitted the combined effects of ferromagnetic particles and magnetic field on mixed convection in the Falkner Skan equation using analytical solution by the Duan–Rach method.

Visualization and grouping of effects of various physical parameters such as electrical conductivity of ferro-particles (electrical conductivity calculated using Maxwell model), ferro fluid volume fraction for Magnetite-Fe₃O₄-water and magnetic field represented by the Hartmann number in a set of third- and second-order nonlinear coupled ordinary differential equations. This set of equations is analytically processed using the Duan–Rach Approach (DRA).

Obtained DRA results are validated using a numerical solution (Runge–Kutta–Fehlberg-based shooting method). The main objective of this research is to analyze the influence of physical parameters, in particular electrical conductivity, Ferrofluid volume fraction in the case of Magnetite-Fe₃O₄-water, in addition to the types of solid nanoparticles and Hartmann number on dynamic and thermal distributions (velocity/temperature). Results of the comparison between the numerical solution (Runge–Kutta–Fehlberg-based shooting method) and the analytical solution (DRA) show that the DRA data are in good agreement with numerical data and available literature.

The study uses Runge–Kutta–Fehlberg-based shooting method) and the analytical solution (DRA) to investigate the effect of mixed convection, in the presence of Ferro particles (Magnetite-Fe₃O₄) in a basic fluid (water for example) and subjected to an external magnetic field on the Falkner–Skan system.

The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C² is a nonlinear autonomous function.

The authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs.

All the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results.

The authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.

The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM).

In this study, the jamming transition is presented as a result of spontaneous deviations of headway and velocity that is caused by the acceleration/breaking rate to be higher than the critical value. Dissipative dynamics of traffic flow can be represented within the framework of the Lorenz scheme based on the car-following model in the one-lane highway. Through this paper, an analytical approximation for the solution is calculated via ADM that leads to a solution for headway deviation as a function of time.

A highly nonlinear differential equation having no exact solution due to JTP is considered and headway deviation is obtained implementing a number of different initial conditions. The results are discussed and compared with the available data in the literature and numerical solutions obtained from a built-in numerical function of the mathematical software used in the study. The advantage of using ADM for the problem is presented in the study and discussed on the basis of the results produced by the applied method.

This is the first study to apply ADM to JTP.

This paper aims to study the impacts of viscous dissipation, thermal radiation and Joule heating on squeezing flow current and the heat transfer mechanism for a magnetohydrodynamic (MHD) nanofluid flow in parallel disks during a suction/blowing process.

First, the governing momentum/energy equations are transformed into a non-dimensional form and then the obtained equations are solved by modified Adomian decomposition method (ADM), known as Duan–Rach approach (DRA).

The effect of the radiation parameter, suction/blowing parameter, magnetic parameter, squeezing number and nanoparticles concentration on the heat transfer and flow field are investigated in the results. The results show that the fluid velocity increases with increasing suction parameter, while the temperature profile decreases with increasing suction parameter.

A complete analysis of the MHD fluid squeezed between two parallel disks by considering Joule heating, thermal radiation and adding different nanoparticles using the novel method called DRA is addressed.

The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.

Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.

This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.

This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear instabilities at two convective Froude numbers.

The simulations start from a small perturbation to a base flow with a hyperbolic-tangent velocity profile. The subsequent development of the shear instabilities is studied from the simulations using a number of flux-limiting schemes, including the second-order MINMOD, the third-order ULTRA-QUICK and the fifth-order WENO schemes for the spatial interpolation of the nonlinear fluxes. The fourth-order Runge-Kutta method advances the simulation in time.

The simulations determine two parameters: the fractional growth rate of the linear instabilities; and the vorticity thickness of the first nonlinear peak. Grid refinement using 32, 64, 128, 256 and 512 nodes over one wave length determines the exact values by extrapolation and the computational error for the parameters. It also determines the overall order of convergence for each of the flux-limiting schemes used in the numerical simulations.

The four-digit accuracy of the numerical simulations presented in this paper are comparable to analytical solutions. The development of this reliable numerical simulation method has paved the way for further study of the instabilities in shear flows that radiate waves.

Initial value problems for the one‐dimensional third‐order dispersion equations are investigated using the reliable Adomian decomposition method (ADM).

The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms.

It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non‐homogeneous equations.

The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data.

The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory.

The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively‐based exact solutions are found.