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Discrete & Continuous Dynamical Systems - S

October 2020 , Volume 13 , Issue 10

Issue on Lie symmetries and other approaches

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Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving
Chaudry Masood Khalique, Muhammad Usman and Maria Luz Gandarais
2020, 13(10): i-ii doi: 10.3934/dcdss.2020415 +[Abstract](253) +[HTML](86) +[PDF](75.78KB)
Abstract:
Special issue dedicated to Professor David Paul Mason
Chaudry Masood Khalique, Muhammad Usman and Maria Luz Gandarais
2020, 13(10): iii-iv doi: 10.3934/dcdss.2020416 +[Abstract](175) +[HTML](83) +[PDF](2472.3KB)
Abstract:
Local meshless differential quadrature collocation method for time-fractional PDEs
Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz and Sakhi Zaman
2020, 13(10): 2641-2654 doi: 10.3934/dcdss.2020223 +[Abstract](646) +[HTML](314) +[PDF](2068.55KB)
Abstract:

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of \begin{document}$ \alpha \in (0,1) $\end{document} and \begin{document}$ \alpha\in(1,2) $\end{document}. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

Conservation laws and line soliton solutions of a family of modified KP equations
Stephen C. Anco, Maria Luz Gandarias and Elena Recio
2020, 13(10): 2655-2665 doi: 10.3934/dcdss.2020225 +[Abstract](599) +[HTML](303) +[PDF](435.49KB)
Abstract:

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation
Asim Aziz, Wasim Jamshed, Yasir Ali and Moniba Shams
2020, 13(10): 2667-2690 doi: 10.3934/dcdss.2020142 +[Abstract](985) +[HTML](405) +[PDF](1489.18KB)
Abstract:

In this numerical study, researchers explore the flow, heat transfer and entropy of electrically conducting hybrid nanofluid over the horizontal penetrable stretching surface with velocity slip conditions at the interface. The non-Newtonian fluid models lead to better understanding of flow and heat transfer characteristics of nanofluids. Therefore, non-Newtonian Maxwell mathematical model is considered for the hybrid nanofluid and the uniform magnetic field is applied at an angle to the direction of the flow. The Joule heating and thermal radiation impact are also considered in the simplified model. The governing nonlinear partial differential equations for hybrid Maxwell nanofluid flow, heat transfer and entropy generation are simplified by taking boundary layer approximations and then reduced to ordinary differential equations using suitable similarity transformations. The Keller box scheme is then adopted to solve the system of ordinary differential equations. The Ethylene glycol based Copper Ethylene glycol (\begin{document}$ Cu $\end{document}-\begin{document}$ EG $\end{document}) nanofluid and Ferro-Copper Ethylene glycol (\begin{document}$ Fe_3O_4-Cu $\end{document}-\begin{document}$ EG $\end{document}) hybrid nanofluids are considered to produce the numerical results for velocity, temperature and entropy profiles as well as the skin friction factor and the local Nusselt number. The main findings indicate that hybrid Maxwell nanofluid is better thermal conductor when compared with the conventional nanofluid, the greater angle of inclination of magnetic field offers greater resistance to fluid motion within boundary layer and the heat transfer rate act as descending function of nanoparticles shape factor.

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
María-Santos Bruzón, Elena Recio, Tamara-María Garrido and Rafael de la Rosa
2020, 13(10): 2691-2701 doi: 10.3934/dcdss.2020222 +[Abstract](589) +[HTML](269) +[PDF](399.15KB)
Abstract:

We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.

Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation
Zhijie Cao and Lijun Zhang
2020, 13(10): 2703-2717 doi: 10.3934/dcdss.2020218 +[Abstract](469) +[HTML](294) +[PDF](343.02KB)
Abstract:

In this paper, we study a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields. The Lie point symmetries of this equation are derived, according to which this equation is classified into four different kinds. Conservation laws for this equation are constructed by using the conservation theorem of Ibragimov.

Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations
Paul Eloe and Jaganmohan Jonnalagadda
2020, 13(10): 2719-2734 doi: 10.3934/dcdss.2020220 +[Abstract](500) +[HTML](258) +[PDF](316.83KB)
Abstract:

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance.

Exact and numerical solution of stochastic Burgers equations with variable coefficients
Stephanie Flores, Elijah Hight, Everardo Olivares-Vargas, Tamer Oraby, Jose Palacio, Erwin Suazo and Jasang Yoon
2020, 13(10): 2735-2750 doi: 10.3934/dcdss.2020224 +[Abstract](805) +[HTML](283) +[PDF](953.21KB)
Abstract:

We will introduce exact and numerical solutions to some stochastic Burgers equations with variable coefficients. The solutions are found using a coupled system of deterministic Burgers equations and stochastic differential equations.

Lagrangian dynamics by nonlocal constants of motion
Gianluca Gorni and Gaetano Zampieri
2020, 13(10): 2751-2759 doi: 10.3934/dcdss.2020216 +[Abstract](433) +[HTML](259) +[PDF](319.26KB)
Abstract:

A simple general theorem is used as a tool that generates nonlocal constants of motion for Lagrangian systems. We review some cases where the constants that we find are useful in the study of the systems: the homogeneous potentials of degree \begin{document}$ -2 $\end{document}, the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also prove a new result on explosion in the past for mechanical system with hydraulic (quadratic) fluid resistance and bounded potential.

Numerical study of an influenza epidemic dynamical model with diffusion
Mudassar Imran, Mohamed Ben-Romdhane, Ali R. Ansari and Helmi Temimi
2020, 13(10): 2761-2787 doi: 10.3934/dcdss.2020168 +[Abstract](687) +[HTML](327) +[PDF](4190.3KB)
Abstract:

In this paper, a deterministic model is formulated in the aim of performing a thorough investigation of the transmission dynamics of influenza. The main advantage of our model compared to existing models is that it takes into account the effects of hospitalization as well as the diffusion. The proposed model consisting of a dynamical system of partial differential equations with diffusion terms is numerically solved using fast and accurate numerical techniques for partial differential equations. Furthermore, the basic reproduction number that guarantees the local stability of disease-free steady state without diffusion term is calculated. Various numerical simulation for different values of the model input parameters are finally presented in order to show the effect of the effective contact rate on the steady state of the different population compartments.

A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation
Chaudry Masood Khalique and Letlhogonolo Daddy Moleleki
2020, 13(10): 2789-2802 doi: 10.3934/dcdss.2020169 +[Abstract](518) +[HTML](283) +[PDF](283.73KB)
Abstract:

This paper aims to study a generalized first extended (3+1)- dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \begin{document}$ ({G'}/{G})- $\end{document}expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.

Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system
Ben Muatjetjeja, Dimpho Millicent Mothibi and Chaudry Masood Khalique
2020, 13(10): 2803-2812 doi: 10.3934/dcdss.2020219 +[Abstract](512) +[HTML](266) +[PDF](285.71KB)
Abstract:

In this paper we perform Lie group classification of a generalized coupled (2+1)-dimensional hyperbolic system, viz., \begin{document}$ u_{tt}-u_{xx}-u_{yy}+f(v) = 0,\,v_{tt}-v_{xx}-v_{yy}+g(u) = 0, $\end{document} which models many physical phenomena in nonlinear sciences. We show that the Lie group classification of the system provides us with an eleven-dimensional equivalence Lie algebra, whereas the principal Lie algebra is six-dimensional and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary functions \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document}, the forms of which include, amongst others, the power and exponential functions. Finally, for three cases we carry out symmetry reductions for the coupled system.

On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor
Rehana Naz
2020, 13(10): 2813-2828 doi: 10.3934/dcdss.2020170 +[Abstract](520) +[HTML](277) +[PDF](447.78KB)
Abstract:

In this article, the sufficiency issues, first integrals and exact solutions for the Uzawa-Lucas model with unskilled labor are investigated. The sufficient conditions are established by utilizing Arrow's Sufficiency theorem. The non-negativeness conditions for the balanced growth path (BGP) are provided and growth rate is explicitly given in terms of parameters of the model. The first integrals are established by the partial Hamiltonian approach. Then first integrals are utilized to construct the exact solutions for all the variables. The growth rates of all variables and graphical representation of exact solutions are provided for the special case when the inverse of the intertemporal elasticity of substitution is the same as the share of physical capital.

First integrals of Hamiltonian systems: The inverse problem
Rehana Naz, Fazal M Mahomed and Azam Chaudhry
2020, 13(10): 2829-2840 doi: 10.3934/dcdss.2020121 +[Abstract](602) +[HTML](307) +[PDF](299.08KB)
Abstract:

There has, to date, been much focus on when a Hamiltonian operator or symmetry results in a first integral for Hamiltonian systems. Very little emphasis has been given to the inverse problem, viz. which operator arises from a first integral of a Hamiltonian system. In this note, we consider this problem with examples mainly taken from economic growth theory. We also provide an example from classical mechanics.

Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries
Rehana Naz and Imran Naeem
2020, 13(10): 2841-2851 doi: 10.3934/dcdss.2020122 +[Abstract](598) +[HTML](311) +[PDF](313.87KB)
Abstract:

We analyze the local conservation laws, auxiliary (potential) systems, potential symmetries and a class of new exact solutions for the Black-Scholes model time-dependent parameters (BST model). First, we utilize the computer package GeM to construct local conservation laws of the BST model for three different forms of multipliers. We obtain two conserved vectors for the second-order multipliers of form \begin{document}$ \Lambda(x,u,u_x,u_{xx}) $\end{document}. We define two potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable \begin{document}$ v $\end{document} or \begin{document}$ w $\end{document} and one couplet potential system involving both potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document}. Moreover, a spectral potential system is constructed by introducing a new potential variable \begin{document}$ p_\alpha $\end{document} which is a linear combination of potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document}. The potential symmetries of BST model are derived by computing the point symmetries of its potential systems. Both singlet potential systems provide three potential symmetries. The couplet potential system yields three potential symmetries and no potential symmetries exist for the spectral potential system. We utilize the potential symmetries of singlet potential systems to construct three new solutions of BST model.

On group analysis of optimal control problems in economic growth models
Gülden Gün Polat and Teoman Özer
2020, 13(10): 2853-2876 doi: 10.3934/dcdss.2020215 +[Abstract](1043) +[HTML](310) +[PDF](426.62KB)
Abstract:

The optimal control problems in economic growth theory are analyzed by considering the Pontryagin's maximum principle for both current and present value Hamiltonian functions based on the theory of Lie groups. As a result of these necessary conditions, two coupled first-order differential equations are obtained for two different economic growth models. The first integrals and the analytical solutions (closed-form solutions) of two different economic growth models are analyzed via the group theory including Lie point symmetries, Jacobi last multiplier, Prelle-Singer method, \begin{document}$ \lambda $\end{document}-symmetry and the mathematical relations among them.

Wave-propagation in an incompressible hollow elastic cylinder with residual stress
Moniba Shams
2020, 13(10): 2877-2904 doi: 10.3934/dcdss.2020123 +[Abstract](619) +[HTML](301) +[PDF](631.88KB)
Abstract:

A study is presented to observe the effect of residual stress on waves in an incompressible, hyper-elastic, thick and hollow cylinder of infinite length. The problem is based on the non-linear theory of infinitesimal deformations occurring after a finite deformation. A prototype model of strain energy function is used which adequately includes the effects of residual stress and deformation. The expressions for internal pressure and the axial load are calculated and graphical illustrations are presented. Analysis of infinitesimal wave propagation is carried for the axisymmetric case in the considered cylinder. Numerical solution is obtained in the undeformed configuration and analyzed for the two-point boundary-value problem. Dispersion curves are plotted for varying choice of parameters.

Space-time kernel based numerical method for generalized Black-Scholes equation
Marjan Uddin and Hazrat Ali
2020, 13(10): 2905-2915 doi: 10.3934/dcdss.2020221 +[Abstract](637) +[HTML](317) +[PDF](448.6KB)
Abstract:

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations
Lijun Zhang, Yixia Shi and Maoan Han
2020, 13(10): 2917-2926 doi: 10.3934/dcdss.2020217 +[Abstract](561) +[HTML](324) +[PDF](434.21KB)
Abstract:

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable \begin{document}$ u(x, t) $\end{document} which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation
Lijun Zhang, Peiying Yuan, Jingli Fu and Chaudry Masood Khalique
2020, 13(10): 2927-2939 doi: 10.3934/dcdss.2020214 +[Abstract](492) +[HTML](284) +[PDF](412.41KB)
Abstract:

The bounded traveling wave solutions of the Zakharov-Rubenchik equation are investigated by using the method of dynamical system theorems in this paper. After suitable transformations we find that the traveling wave equations of the Zakharov-Rubenchik equation are fully determined by a second-order singular ordinary differential equation (ODE) with three real coefficients which can be arbitrary constants. We derive abundant exact bounded periodic and solitary wave solutions of the Zakharov-Rubenchik equation via studying the bifurcations and exact solutions of the derived ODE.

A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation
Yufeng Zhang, Wen-Xiu Ma and Jin-Yun Yang
2020, 13(10): 2941-2948 doi: 10.3934/dcdss.2020167 +[Abstract](566) +[HTML](335) +[PDF](386.28KB)
Abstract:

We aim to generalize the (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation, passing the three-soliton test, to a new one which still has diverse solution structures. We add all second-order derivative terms to the HSI equation but demand the existence of lump solutions. Such lump solutions are formulated in terms of the coefficients, except two, in the resulting generalized HSI equation. As an illustrative example, a special completely generalized HSI equation is given, together with a lump solution, and three 3d-plots and contour plots of the lump solution are made to elucidate the characteristics of the presented lump solutions.

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