scholarly journals Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Haiyong Qin ◽  
Mostafa M. A. Khater ◽  
Raghda A. M. Attia
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Longitudinal Strain ◽  
Rational Functions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Strain Wave ◽  
Integer Order ◽  
Wave Solutions

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

scholarly journals ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

Fractals ◽  
2021 ◽  
pp. 2240018
Author(s):  
ARMANDO CIANCIO ◽  
GULNUR YEL ◽  
AJAY KUMAR ◽  
HACI MEHMET BASKONUS ◽  
ESIN ILHAN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Expansion Method ◽  
Research Paper ◽  
Trigonometric Function ◽  
Integrable Models ◽  
Hyperbolic Function ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Wave Solutions

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

A method for constructing solutions of homogeneous partial differential equations: localized waves

1992 ◽  
Vol 437 (1901) ◽  
pp. 673-692 ◽  
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Propagation Axis ◽  
Potential Applications ◽  
Klein Gordon ◽  
Localized Waves

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

scholarly journals Backstepping-based output regulation of ordinary differential equations cascaded by wave equation with in-domain anti-damping

2018 ◽  
Vol 41 (1) ◽  
pp. 246-262 ◽  
Author(s):  
Jianjun Gu ◽  
Chunqiu Wei ◽  
Junmin Wang
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
State Feedback ◽  
Solvability Condition ◽  
Reference Signal ◽  
Output Regulation ◽  
The Body ◽  
Output Regulation Problem ◽  
Feedback Regulator

Output regulation is considered in this paper for ordinary differential equations cascaded by a wave equation, in which both the body equations and the uncontrolled end are subject to disturbances. The disturbances are generated by an exosystem. A backstepping state-feedback regulator is first designed to force the output to track the reference signal. The design is based on solving cascaded regulator equations, and the solvability condition of the equations is characterized in terms of a transfer function and the eigenvalues of the exosystem. An observer-based output-feedback regulator is then designed to solve the output regulation problem. Finally, the regulator tracking performance is illustrated through numerical simulations.

Uniform progressing wave solutions of the wave equation

1971 ◽  
Vol 70 (3) ◽  
pp. 455-465
Author(s):  
Erich Zauderer
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Standard Form ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Wave Solutions ◽  
Different Types ◽  
Uniform Expansions

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.

scholarly journals F-Expansion Method and Its Application for Finding Exact Analytical Solutions of Fractional order RLW Equation

2021 ◽  
Author(s):  
Attia Rani ◽  
Qazi Mahmood Ul-Hassan ◽  
Muhammad Ashraf ◽  
Jamshad Ahmad
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Nonlinear Partial Differential Equation ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Partial Differential ◽  
Wave Solutions ◽  
Rlw Equation

Exact nonlinear partial differential equation solutions are critical for describing new complex characteristics in a variety of fields of applied science. The aim of this research is to use the F-expansion method to find the generalized solitary wave solution of the regularized long wave (RLW) equation of fractional order. Fractional partial differential equations can also be transformed into ordinary differential equations using fractional complex transformation and the properties of the modified Riemann–Liouville fractional-order operator. Because of the chain rule and the derivative of composite functions, nonlinear fractional differential equations (NLFDEs) can be converted to ordinary differential equations. We have investigated various set of explicit solutions with some free parameters using this approach. The solitary wave solutions are derived from the moving wave solutions when the parameters are set to special values. Our findings show that this approach is a very active and straightforward way of formulating exact solutions to nonlinear evolution equations that arise in mathematical physics and engineering. It is anticipated that this research will provide insight and knowledge into the implementation of novel methods for solving wave equations.

scholarly journals Explicit Travelling Wave Solutions to Nonlinear Partial Differential Equations Arise in Mathematical Physics and Engineering

2021 ◽  
Vol 2 (01) ◽  
pp. 58-63
Author(s):  
Muktarebatul Jannah ◽  
Tarikul Islam ◽  
Armina Akter
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Travelling Wave ◽  
Research Field ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Non Linear

To describe the interior phenomena of the mysterious problems around the real world, non-linear partial differential equations (NLPDEs) plays a substantial role, for which construction of analytic solutions of those is most important. This paper stands for a goal to find fresh and wide-ranging solutions to some familiar NLPDEs namely the non-linear cubic Klein-Gordon (cKG) equation and the non-linear Benjamin-Ono (BO) equation. A wave variable transformation is made use to convert the mentioned equations into ordinary differential equations. To acquire the desired precise exact travelling wave solutions to the above-stated equations, the rational -expansion method is employed. Consequently, three types of equipped solutions are successfully come out in the forms of hyperbolic, trigonometric and rational functions in a compatible way. To analyse the physical problems arisen relating to nonlinear complex dynamical systems, our obtained solutions might be most helpful. So far we know, these achieved solutions are different than those in the literature. The applied method is efficient and reliable which might further be used to find different and novel solutions to many other NLPDEs successfully in research field.

A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

2014 ◽  
Author(s):  
Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rational Functions ◽  
Special Treatment ◽  
Air Bearings ◽  
Fluid Forces ◽  
Stability Diagrams ◽  
Nonlinear Transient

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed in [1]. The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace Transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor lead to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However the method [1] requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors [2] showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in [2] was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach enhances also the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

Exact traveling wave solutions of partial differential equations with power law nonlinearity

2015 ◽  
Vol 4 (3) ◽  
Author(s):  
H. Aminikhah ◽  
B. Pourreza Ziabary ◽  
H. Rezazadeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Hyperbolic Function ◽  
Mathematical Tool ◽  
Partial Differential ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

AbstractIn this paper, we applied the functional variable method for four famous partial differential equations with power lawnonlinearity. These equations are included the Kadomtsev-Petviashvili, (3+1)-Zakharov-Kuznetsov, Benjamin-Bona-Mahony-Peregrine and Boussinesq equations. Various exact traveling wave solutions of these equations are obtained that include the hyperbolic function solutions and the trigonometric function solutions. The solutions shown that this method provides a very effective, simple and powerful mathematical tool for solving nonlinear equations in various fields of applied sciences.

scholarly journals New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-35 ◽  
Author(s):  
Jalil Manafian ◽  
Mehrdad Lakestani
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Function Solution ◽  
Wave Solutions ◽  
Nagumo Equation

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

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