Ordinary Differential Equations and Applied Dynamics Commons^™

On The Peakon And Soliton Solutions Of An Integrable Pde With Cubic Nonlinearities, Rossen Ivanov, Tony Lyons

Conference papers

The interest in the singular solutions (peakons) has been inspired by the Camassa-Holm (CH) equation and its peakons. An integrable peakon equation with cubic nonlinearities was first discovered by Qiao. Another integrable equation with cubic nonlinearities was introduced by V. Novikov . We investigate the peakon and soliton solutions of the Qiao equation.

On The Numerical Solution Of Linear Fredholm-Volterra İntegro Differential Difference Equations With Piecewise İntervals, Mustafa Gülsu, Yalçın Öztürk

Applications and Applied Mathematics: An International Journal (AAM)

The numerical solution of a mixed linear integro delay differential-difference equation with piecewise interval is presented using the Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving a mixed linear integro delay differential difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms a mixed linear integro delay differential-difference equations and the given conditions into a matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system …

On Stability Of Dynamic Equations On Time Scales Via Dichotomic Maps, Veysel F. Hatipoğlu, Zeynep F. Koçak, Deniz Uçar

Applications and Applied Mathematics: An International Journal (AAM)

Dichotomic maps are used to check the stability of ordinary differential equations and difference equations. In this paper, this method is extended to dynamic equations on time scales; the stability and asymptotic stability to the trivial solution of the first order system of dynamic equations are examined using dichotomic and strictly dichotomic maps. This method, in dynamic equations, also involves Lyapunov’s direct method.

Investigation Of Nonlinear Problems Of Heat Conduction In Tapered Cooling Fins Via Symbolic Programming, Hooman Fatoorehchi, Hossein Abolghasemi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, symbolic programming is employed to handle a mathematical model representing conduction in heat dissipating fins with triangular profiles. As the first part of the analysis, the Modified Adomian Decomposition Method (MADM) is converted into a piece of computer code in MATLAB to seek solution for the mentioned problem with constant thermal conductivity (a linear problem). The results show that the proposed solution converges to the analytical solution rapidly. Afterwards, the code is extended to calculate Adomian polynomials and implemented to the similar, but more generalized, problem involving a power law dependence of thermal conductivity on temperature. The …

G-Strands, Darryl Holm, Rossen Ivanov, James Percival

Articles

A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …

Cyclic Universe With An Inflationary Phase From A Cosmological Model With Real Gas Quintessence, Rossen Ivanov, Emil Prodanov

Articles

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

Applying Differential Transform Method To Nonlinear Partial Differential Equations: A Modified Approach, Marwan T. Alquran

Applications and Applied Mathematics: An International Journal (AAM)

This paper proposes another use of the Differential transform method (DTM) in obtaining approximate solutions to nonlinear partial differential equations (PDEs). The idea here is that a PDE can be converted to an ordinary differential equation (ODE) upon using a wave variable, then applying the DTM to the resulting ODE. Three equations, namely, Benjamin-Bona-Mahony (BBM), Cahn-Hilliard equation and Gardner equation are considered in this study. The proposed method reduces the size of the numerical computations and use less rules than the usual DTM method used for multi-dimensional PDEs. The results show that this new approach gives very accurate solutions.

Solving Singular Boundary Value Problems Using Daftardar-Jafari Method, H. Jafari, M. Ahmadi, S. Sadeghi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we apply the suggested iterative method by Daftardar and Jafari hereafter called Daftardar-Jafari method for solving singular boundary value problems. In the implementation of this new method, one does not need the computation of the derivative of the so-called Adomian polynomials. The method is quite efficient and is practically well suited for use in these problems. Two illustrative examples has been presented.

Modeling And Analysis Of The Spread Of An Infectious Disease Cholera With Environmental Fluctuations, Manju Agarwal, Vinay Verma

Applications and Applied Mathematics: An International Journal (AAM)

A nonlinear delayed mathematical model with immigration for the spread of an infectious disease cholera with carriers in the environment is proposed and analyzed. It is assumed that all susceptible are affected by carrier population density. The carrier population density is assumed to follow the logistic model and grows due to conducive human population density related factors. The model is analyzed by stability theory of differential equations and computer simulation. Both the disease-free (DFE), (CFE) and endemic equilibria are found and their stability investigated. Bifurcation analyses about endemic equilibrium are also carried out analytically using the theory of differential equations. …

Oscillation Of Neutral Partial Dynamic Equations, Deniz Uçar, Yaşar Bolat

Applications and Applied Mathematics: An International Journal (AAM)

This paper is concerned with the oscillation of solutions of a certain more general neutral type dynamic equation. We establish within the necessary and sufficient conditions for the oscillation of its solutions.

An Approximate Solution Of The Mathieu Fractional Equation By Using The Generalized Differential Transform Method (Gdtm), H. S. Najafi, S. R. Mirshafaei, E. A. Toroqi

Applications and Applied Mathematics: An International Journal (AAM)

The generalized differential transform method (GDTM) is a powerful tool for solving fractional equations. In this paper we solve the Mathieu fractional equation by this method. The approximate solutions obtained are compared with the exact solution. We also show that if both differential orders decrease, we can still have an approximate solution in the different interval of p.

A Mathematical Model For Dengue Fever In A Virgin Environment, Jason K. Bowman

Senior Honors Projects

Dengue is a mosquito-borne viral infection found in tropical and subtropical regions around the world. The disease was named in 1779 and the first recorded epidemic of it occurred simultaneously on three continents within the following decade. Dengue is characterized by flu-like symptoms and, while its symptoms are generally reported as quite unpleasant, is rarely fatal. However, in some cases patients can contract a more serious form of the disease, known as Dengue Hemorrhagic Fever, which is far more dangerous. The World Health Organization estimates that today over 2.5 billion people are at risk for Dengue (over 40% of the …

Design Of Orbital Maneuvers With Aeroassisted Cubesatellites, Stephanie Clark

Graduate Theses and Dissertations

Recent advances within the field of cube satellite technology has allowed for the possible development of a maneuver that utilizes a satellite's Low Earth Orbit (LEO) and increased atmospheric density to effectively use lift and drag to implement a noncoplanar orbital maneuver. Noncoplanar maneuvers typically require large quantities of propellant due to the large delta-v that is required. However, similar maneuvers using perturbing forces require little or no propellant to create the delta-v required. This research reported here studied on the effects of lift on orbital changes, those of noncoplanar types in particular, for small satellites without orbital maneuvering thrusters. …

Periodic Solutions And Positive Solutions Of First And Second Order Logistic Type Odes With Harvesting, Cody Alan Palmer

UNLV Theses, Dissertations, Professional Papers, and Capstones

It was recently shown that the nonlinear logistic type ODE with periodic harvesting has a bifurcation on the periodic solutions with respect to the parameter ε:

u' = f (u) - ε h (t).

Namely, there exists an ε0 such that for 0 < ε < ε0 there are two periodic solutions, for ε = ε0 there is one periodic solution, and for ε >ε0 there are no periodic solutions, provided that....

In this paper we look at some numerical evidence regarding the behavior of this threshold for various types of harvesting terms, in particular we find evidence in the negative or a conjecture regarding the behavior of this threshold value.

Additionally, we look at analagous steady states for the reaction-diusion …

R₀ Analysis Of A Spatiotemporal Model For A Stream Population, H. W. Mckenzie, Y. Jin, Jon T. Jacobsen, M. A. Lewis

All HMC Faculty Publications and Research

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R₀ for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays …

Singular Solutions Of Coss-Coupled Epdiff Equations: Waltzing Peakons And Compacton Pairs, Colin Cotter, Darryl Holm, Rossen Ivanov, James Percival

Conference papers

We introduce EPDiff equations as Euler-Poincare´ equations related to Lagrangian provided by a metric, invariant under the Lie Group Diff(Rⁿ). Then we proceed with a particular form of EPDiff equations, a cross coupled two-component system of Camassa-Holm type. The system has a new type of peakon solutions, 'waltzing' peakons and compacton pairs.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov-Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schrodinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …