Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

Riaz

et al. 2022

MCA

Self Cite

In this paper, a new approach to investigating the unsteady natural convection flow of viscous fluid over a moveable inclined plate with exponential heating is carried out. The mathematical modeling is based on fractional treatment of the governing equation subject to the temperature, velocity and concentration field. Innovative definitions of time fractional operators with singular and non-singular kernels have been working on the developed constitutive mass, energy and momentum equations. The fractionalized analytical solutions based on special functions are obtained by using Laplace transform method to tackle the non-dimensional partial differential equations for velocity, mass and energy. Our results propose that by increasing the value of the Schimdth number and Prandtl number the concentration and temperature profiles decreased, respectively. The presence of a Prandtl number increases the thermal conductivity and reflects the control of thickness of momentum. The experimental results for flow features are shown in graphs over a limited period of time for various parameters. Furthermore, some special cases for the movement of the plate are also studied and results are demonstrated graphically via Mathcad-15 software.

Section: Mathematical Modelmentioning

confidence: 99%

Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels

Riaz

et al. 2022

MCA

Self Cite

“…For example, the occurrence of homoclinic orbits, smooth heteroclinic orbits and periodic orbits for travelling wave models describes the periodic wave solutions, oscillatory travelling wave solutions, smooth kink wave solutions and smooth solitary solutions for considered PDEs, respectively. One can refer to [20][21][22][23][24] for the study of recent work in this field.…”

Section: Introductionmentioning

confidence: 99%

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Kumar

Mann

Kharbanda

2022

Preprint

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

Math Methods in App Sciences

“…The exact solutions of NLPDEs symbolically and physically reveal the mechanisms of many nonlinear complex phenomena. Several techniques have been developed for investigating exact wave structures of NLPDEs in the shape of soliton, traveling wave, and elliptic function solutions 1–30 . Muto et al have firstly submitted the nonlinear mathematical model of the interaction of deoxyribonucleic acid (DNA) with an external microwave field which was proposed as

ϕ_{t t} = C^{2} ϕ_{z z} - (ε / C^{2}) ϕ_{z z t t} + δ {(ϕ_{z}^{2})}_{z},

where ϕ ( z , t ) describes longitudinal displacements in DNA 31,32 .…”

Section: Introductionmentioning

confidence: 99%

Dynamics of exact soliton solutions in the double‐chain model of deoxyribonucleic acid

Bilal

Younas

Ren

2021

In this research, we study analytically the double‐chain model. The model consists of two long elastic homogeneous strands (or rods), which represent two polynucleotide chains of the deoxyribonucleic acid molecule, connected with each other by an elastic membrane (or some linear springs) representing the hydrogen bonds between the base pairs of the two chains. The new extended direct algebraic method and the generalized Kudryashov method are successfully utilized to discuss the exact soliton solutions to the double‐chain model of deoxyribonucleic acid that plays an important role in biology. The solutions obtained by these mechanisms can be divided into solitary, singular, kink, single wave, combine behavior as well as hyperbolic, plane wave, and trigonometric solutions with arbitrary parameters. Some solutions have been exemplified by graphics to understand the physical meaning of the DNA model. The accomplished solutions seem with all essential constraint conditions, which are obligatory for them to subsist. Hence, our techniques via fortification of symbolic computations provide an active and potent mathematical implement for solving diverse benevolent nonlinear wave problems. The results show that the system theoretically has extremely rich exact wave structures of biological relevance.