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Article
Publication date: 30 January 2024

Abdul-Majid Wazwaz

The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.

Abstract

Purpose

The purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.

Design/methodology/approach

The newly developed Painlevé integrable equations have been handled by using Hirota’s direct method. The authors obtain multiple soliton solutions and other kinds of solutions for these six models.

Findings

The developed Hamiltonian models exhibit complete integrability in analogy with the original equation.

Research limitations/implications

The present study is to address these two main motivations: the study of the integrability features and solitons and other useful solutions for the developed equations.

Practical implications

The work introduces six Painlevé-integrable equations developed from a Hamiltonian model.

Social implications

The work presents useful algorithms for constructing new integrable equations and for handling these equations.

Originality/value

The paper presents an original work with newly developed integrable equations and shows useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 34 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 12 February 2024

Bahram Jalili, Milad Sadinezhad Fard, Yasir Khan, Payam Jalili and D.D. Ganji

The current analysis produces the fractional sample of non-Newtonian Casson and Williamson boundary layer flow considering the heat flux and the slip velocity. An extended sheet…

Abstract

Purpose

The current analysis produces the fractional sample of non-Newtonian Casson and Williamson boundary layer flow considering the heat flux and the slip velocity. An extended sheet with a nonuniform thickness causes the steady boundary layer flow’s temperature and velocity fields. Our purpose in this research is to use Akbari Ganji method (AGM) to solve equations and compare the accuracy of this method with the spectral collocation method.

Design/methodology/approach

The trial polynomials that will be utilized to carry out the AGM are then used to solve the nonlinear governing system of the PDEs, which has been transformed into a nonlinear collection of linked ODEs.

Findings

The profile of temperature and dimensionless velocity for different parameters were displayed graphically. Also, the effect of two different parameters simultaneously on the temperature is displayed in three dimensions. The results demonstrate that the skin-friction coefficient rises with growing magnetic numbers, whereas the Casson and the local Williamson parameters show reverse manners.

Originality/value

Moreover, the usefulness and precision of the presented approach are pleasing, as can be seen by comparing the results with previous research. Also, the calculated solutions utilizing the provided procedure were physically sufficient and precise.

Details

Multidiscipline Modeling in Materials and Structures, vol. 20 no. 2
Type: Research Article
ISSN: 1573-6105

Keywords

Open Access
Article
Publication date: 27 November 2023

J.I. Ramos and Carmen María García López

The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the…

198

Abstract

Purpose

The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions.

Design/methodology/approach

An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds.

Findings

The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.

Originality/value

The blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 34 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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