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Discrete & Continuous Dynamical Systems - S

2018 , Volume 11 , Issue 4

Special issue on Lie symmetries, conservation laws and other approaches in solving nonlinear differential equations

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PREFACE
2018, 11(4): i-ii doi: 10.3934/dcdss.201804i +[Abstract](114) +[HTML](23) +[PDF](75.2KB)
Abstract:
Exact solutions of nonlinear partial differential equations
Barbara Abraham-Shrauner
2018, 11(4): 577-582 doi: 10.3934/dcdss.2018032 +[Abstract](137) +[HTML](36) +[PDF](284.25KB)
Abstract:

Tests for determination of which nonlinear partial differential equations may have exact analytic nonlinear solutions of any of two types of hyperbolic functions or any of three types of Jacobian elliptic functions are presented. The Power Index Method is the principal method employed that extends the calculation of the power index for the most nonlinear terms to all terms in the nonlinear partial differential equations. An additional test is the identification of the net order of differentiation of each term in the nonlinear differential equations. The nonlinear differential equations considered are evolution equations. The tests extend the homogeneous balance condition that is necessary to conditions that may only be sufficient but are very simple to apply. Superposition of Jacobian elliptic functions is also presented with the introduction of a new basis that simplifies the calculations.

Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection
Noreen Sher Akbar , Dharmendra Tripathi and  Zafar Hayat Khan
2018, 11(4): 583-594 doi: 10.3934/dcdss.2018033 +[Abstract](254) +[HTML](74) +[PDF](543.46KB)
Abstract:

The present study analyzes the heat energy transfer in nano fluids flow through the porous stretching surface. Cattanneo-Christov heat flux model is employed to study the heat energy transfer. Darcy law is used to discuss the flow characteristics over the different types of permeable sheets with suction and injection. Nanofluids is considered as water based single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs) nanofluids. A comparative study for SWCNT and MWCNT is also made. Governing equations are transformed into set of ordinary differential equations using similarity transformations. The computational results are obtained by using Runge-Kutta fourth order method along with shooting technique. Numerical and graphical results are presented to discuss the effects of various physical parameters on velocity profile, temperature profile, Nusselt number, Sherwood number and skin friction coefficient for different type of nanoparticles for suction and injection cases. Stream lines and isotherms are also plotted for three different cases viz. permeable sheet with suction, impermeable sheet and permeable sheet with injection. A comparative analysis with existing results is tabulated which validate that the numerical results of present study have good correlation with existing results. The outcomes of the results show that skin friction coefficient is more for SWCNT in caparison of MWCNT and the boundary layer thickness is maximum for permeable stretching sheet with suction parameter.

Exact solution of magnetohydrodynamic slip flow and heat transfer over an oscillating and translating porous plate
Yasir Ali and  Arshad Alam Khan
2018, 11(4): 595-606 doi: 10.3934/dcdss.2018034 +[Abstract](334) +[HTML](27) +[PDF](1273.29KB)
Abstract:

Objective of this paper is to study natural convection MHD flow past over a moving porous plate with heat source in the porous medium. The motion of the plate is translating as well as oscillating and embedded in the porous medium. The exact solution of the governing equations, of the flow and heat transfer for this model is obtained. To study heat flux for our model we use Nusselt number. Comparisons of effects of magnetic parameter \begin{document}$M$\end{document}, translation \begin{document}$a$\end{document} and heat source parameter \begin{document}$S$\end{document} on velocity and temperature profile is given. The effects of some other physical parameters like Prandtl number \begin{document}$P_r$\end{document}, Grashof number for heat transfer \begin{document}$G_r$\end{document}, Permeability parameter \begin{document}$K_p$\end{document}, is presented graphically on the distributions of velocity and temperature. It is concluded that the fluid motion in the boundary layer increases with increase of \begin{document}$a$\end{document}, \begin{document}$S$\end{document}, \begin{document}$G_r$\end{document} and \begin{document}$K_P$\end{document}. Whereas opposite behavior is observed for \begin{document}$M$\end{document} and \begin{document}$P_r$\end{document}. The heat source parameter increases the temperature of fluid and on the other hand cooling effects occur due to \begin{document}$P_r$\end{document} and \begin{document}$v_0$\end{document}.

Conservation laws and symmetries of time-dependent generalized KdV equations
Stephen Anco , Maria Rosa and  Maria Luz Gandarias
2018, 11(4): 607-615 doi: 10.3934/dcdss.2018035 +[Abstract](183) +[HTML](28) +[PDF](335.08KB)
Abstract:

A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries is derived. The physical meaning of the conservation laws and the symmetries is discussed.

Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity
Asim Aziz and  Wasim Jamshed
2018, 11(4): 617-630 doi: 10.3934/dcdss.2018036 +[Abstract](354) +[HTML](79) +[PDF](485.57KB)
Abstract:

In this paper, unsteady magnetohydrodynamic (MHD) boundary layer slip flow and heat transfer of power-law nanofluid over a nonlinear porous stretching sheet is investigated numerically. The thermal conductivity of the nanofluid is assumed as a function of temperature and the partial slip conditions are employed at the boundary. The nonlinear coupled system of partial differential equations governing the flow and heat transfer of a power-law nanofluid is first transformed into a system of nonlinear coupled ordinary differential equations by applying a suitable similarity transformation. The resulting system is then solved numerically using shooting technique. Numerical results are presented in the form of graphs and tables and the effect of the power-law index, velocity and thermal slip parameters, nanofluid volume concentration parameter, applied magnetic field parameter, suction/injection parameter on the velocity and temperature profiles are examined from physical point of view. The boundary layer thickness decreases with increase in strength of applied magnetic field, nanoparticle volume concentration, velocity slip and the unsteadiness of the stretching surface. Whereas thermal boundary layer thickness increase with increasing values of magnetic parameter, nanoparticle volume concentration and velocity slip at the boundary.

Symmetries and conservation laws of a KdV6 equation
María Santos Bruzón and  Tamara María Garrido
2018, 11(4): 631-641 doi: 10.3934/dcdss.2018038 +[Abstract](159) +[HTML](28) +[PDF](324.34KB)
Abstract:

In the present work we make an analysis of the Korteweg-de Vries of sixth order. We apply the classical Lie method of infinitesimals and the nonclassical method, due to Bluman and Cole, to deduce new symmetries of the equation which cannot be obtained by Lie classical method. Moreover, we obtain ten different conservation laws depending on the parameters and we conclude that potential symmetries project on the infinitesimals corresponding to the classical symmetries.

Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach
Azam Chaudhry and  Rehana Naz
2018, 11(4): 643-654 doi: 10.3934/dcdss.2018039 +[Abstract](72) +[HTML](23) +[PDF](328.54KB)
Abstract:

In this paper, we present a dynamic picture of the two sector Lucas-Uzawa model with logarithmic utility preferences and homogeneous technology as was proposed by Bethmann [3] for a Robinson Crusoe economy. We use a newly developed partial Hamiltonian approach to derive a new set of closed-form solutions for the model with logarithmic utility preferences and homogeneous technology. Unlike the previous literature, our model yields three distinct closed-form solutions to the model. We establish the growth rates of all the variables which fully describe the dynamics of the model. Even though the first closed-form solution provides static growth rates and the other two provide dynamic growth rates, in the long run all the closed-form solutions approach the same static balanced growth path.

Conditional symmetries of nonlinear third-order ordinary differential equations
Aeeman Fatima , F. M. Mahomed and  Chaudry Masood Khalique
2018, 11(4): 655-666 doi: 10.3934/dcdss.2018040 +[Abstract](102) +[HTML](30) +[PDF](370.36KB)
Abstract:

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

Symmetry analysis of a Lane-Emden-Klein-Gordon-Fock system with central symmetry
Igor Freire and  Ben Muatjetjeja
2018, 11(4): 667-673 doi: 10.3934/dcdss.2018041 +[Abstract](120) +[HTML](76) +[PDF](304.5KB)
Abstract:

In this paper we introduce a sort of Lane-Emden system derived from the Klein-Gordon-Fock equation with central symmetry. Point symmetries are obtained and, since the system can be derived as the Euler-Lagrange equation of a certain functional, a Noether symmetry classification is also considered and conservation laws are derived from the point symmetries.

Nonlocal and nonvariational extensions of Killing-type equations
Gianluca Gorni and  Gaetano Zampieri
2018, 11(4): 675-689 doi: 10.3934/dcdss.2018042 +[Abstract](90) +[HTML](22) +[PDF](409.72KB)
Abstract:

The Killing-like equation and the inverse Noether theorem arise in connection with the search for first integrals of Lagrangian systems. We generalize the theory to include "nonlocal" constants of motion of the form \begin{document} $N_0+∈t N_1\, dt$ \end{document}, and also to nonvariational Lagrangian systems \begin{document} $\frac{d}{dt}\partial_{\dot q}L-\partial_qL=Q$ \end{document}. As examples we study nonlocal constants of motion for the Lane-Emden system, for the dissipative Maxwell-Bloch system and for the particle in a homogeneous potential.

A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation
Mudassar Imran , Youssef Raffoul , Muhammad Usman and  Chi Zhang
2018, 11(4): 691-705 doi: 10.3934/dcdss.2018043 +[Abstract](102) +[HTML](31) +[PDF](443.36KB)
Abstract:

In this work, we consider an ordinary differential equation obtained from a damped externally excited Korteweg de Vries-Kuramoto Sivashinsky (KdV-KS) type equation using traveling coordinates. We also include controls and delays and use an asymptotic perturbation method to analyze the stability of the traveling wave solutions. The existence of bounded solutions is presented as well. We consider the primary resonance defined by the detuning parameter. External-excitation and frequency-response curves are shown to exhibit jump and hysteresis phenomena (discontinuous transitions between two stable solutions) for the KdV-KS type equation. We have obtained the existence of the bounded solutions of the system obtained from an ordinary differential equation associated with the KdV-KS equation and also show the global stability for a special case when there is no external force.

Conservation laws by symmetries and adjoint symmetries
Wen-Xiu Ma
2018, 11(4): 707-721 doi: 10.3934/dcdss.2018044 +[Abstract](95) +[HTML](31) +[PDF](399.51KB)
Abstract:

Conservation laws are fomulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not require the existence of a Lagrangian for a given system, and the presented examples include computations of conserved densities for the heat equation, Burgers' equation and the Korteweg-de Vries equation.

Characterization of partial Hamiltonian operators and related first integrals
Rehana Naz and  Fazal M. Mahomed
2018, 11(4): 723-734 doi: 10.3934/dcdss.2018045 +[Abstract](109) +[HTML](25) +[PDF](349.47KB)
Abstract:

We focus on partial Hamiltonian systems for the characterization of their operators and related first integrals. Firstly, it is shown that if an operator is a partial Hamiltonian operator which yields a first integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a first integral. Thirdly, characterization of partial Hamiltonian operators and related first integral conditions are provided for the partial Hamiltonian system. Applications to mechanics are presented to illustrate the theory.

New conservation forms and Lie algebras of Ermakov-Pinney equation
Özlem Orhan and  Teoman Özer
2018, 11(4): 735-746 doi: 10.3934/dcdss.2018046 +[Abstract](58) +[HTML](21) +[PDF](355.71KB)
Abstract:

In this study, we investigate first integrals and exact solutions of the Ermakov-Pinney equation. Firstly, the Lagrangian for the equation is constructed and then the determining equations are obtained based on the Lagrangian approach. Noether symmetry classification is implemented, the first integrals, conservation laws are obtained and classified. This classification includes Noether symmetries and first integrals with respect to different choices of external potential function. Furthermore, the time independent integrals and analytical solutions are obtained by using the modified Prelle-Singer procedure as a different approach. Additionally, for the investigation of conservation laws of the equation, we present the mathematical connections between the λ-symmetries, Lie point symmetries and the modified Prelle-Singer procedure. Finally, new Lagrangian and Hamiltonian forms of the equation are determined.

Differential invariants of a generalized variable-coefficient Gardner equation
Rafael de la Rosa and  María Santos Bruzón
2018, 11(4): 747-757 doi: 10.3934/dcdss.2018047 +[Abstract](63) +[HTML](32) +[PDF](388.92KB)
Abstract:

In this paper, we consider a generalized variable-coefficient Gardner equation. By using the equivalence group of this equation, we derive the differential invariants of first order and the corresponding invariant equations. We employ these differential invariants and invariant equations to find the most general subclass of variable-coefficient Gardner equations which can be mapped into a specific constant-coefficient equation by means of an equivalence transformation. Furthermore, differential invariants are applied to obtain exact solutions.

Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs
Lijun Zhang and  Chaudry Masood Khalique
2018, 11(4): 759-772 doi: 10.3934/dcdss.2018048 +[Abstract](83) +[HTML](29) +[PDF](412.75KB)
Abstract:

In this paper, by using dynamical system theorems we study the bifurcation of a second-order ordinary differential equation which can be obtained from many nonlinear partial differential equations via traveling wave transformation and integrations. We present all the bounded exact solutions of this second-order ordinary differential equation which contains four parameters by normalization and classification. As a result, one can obtain all possible bounded exact traveling wave solutions including soliatry waves, kink and periodic wave solutions of many nonlinear wave equations by the formulas presented in this paper. As an example, all bounded traveling wave solutions of the modified regularized long wave equation are obtained to illustrate our approach.

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