scholarly journals A Partial Lagrangian Approach to Mathematical Models of Epidemiology

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
I. Naeem ◽  
F. M. Mahomed
Keyword(s):  
Mathematical Models ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Order Equation ◽  
Second Order ◽  
Lagrangian Approach ◽  
Hiv Model ◽  
First Order ◽  
Partial Lagrangian

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

First integrals and exact solutions of some compartmental disease models

2019 ◽  
Vol 74 (4) ◽  
pp. 293-304
Author(s):  
Burhan Ul Haq ◽  
Imran Naeem
Keyword(s):  
Hamiltonian System ◽  
Exact Solutions ◽  
First Integrals ◽  
Second Order ◽  
Disease Models ◽  
Disease Dynamics ◽  
Hamiltonian Approach ◽  
First Order ◽  
Hamiltonian Operators ◽  
Second Order Equations

AbstractThe notions of artificial Hamiltonian (partial Hamiltonian) and partial Hamiltonian operators are used to derive the first integrals for the first order systems of ordinary differential equations (ODEs) in epidemiology, which need not be derived from standard Hamiltonian approaches. We show that every system of first order ODEs can be cast into artificial Hamiltonian system $\dot{q}=\frac{{\partial H}}{{\partial p}}$, $\dot{p}=-\frac{{\partial H}}{{\partial q}}+\Gamma(t,\;q,\;p)$ (see [1]). Moreover, the second order equations and the system of second order ODEs can be written in the form of artificial Hamiltonian system. Then, the partial Hamiltonian approach is employed to derive the first integrals for systems under consideration. These first integrals are then utilized to find the exact solutions of models from the epidemiology for a distinct class of population. For physical insights, the solution curves of the closed-form expressions obtained are interpreted in order for readers understand the disease dynamics in a much deeper way. The effects of various pertinent parameters on the prognosis of the disease are observed and discussed briefly.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

Improved Approach to the Streamline Curvature Method in Turbomachinery

1987 ◽  
Vol 109 (3) ◽  
pp. 213-217 ◽  
Author(s):  
S. Abdallah ◽  
R. E. Henderson
Keyword(s):  
Flow Field ◽  
Velocity Gradient ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
First Order ◽  
Streamline Curvature ◽  
Stators And Rotors ◽  
Streamline Curvature Method

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

Kinetics of Glyphosate Adsorption on Composite Resin from Aqueous Solution

2013 ◽  
Vol 803 ◽  
pp. 157-160
Author(s):  
Zhen Zhen Kong ◽  
Dong Mei Jia ◽  
Su Wen Cui
Keyword(s):  
Experimental Data ◽  
Aqueous Solution ◽  
Composite Resin ◽  
Order Equation ◽  
Second Order ◽  
First Order ◽  
Basic Resin ◽  
Pseudo Second Order ◽  
Kinetics Of ◽  
Best Fit

The composite weakly basic resin (D301Fe) was prepared and examined using scanning electron microscopy and Fourier transform infrared spectroscopy. The adsorption kinetics of glyphosate from aqueous solution onto composite weakly basic resin (D301Fe) were investigated under different conditions. The experimental data was analyzed using various adsorption kinetic models like pseudo-first order, the pseudo-second order, the Elovich and the parabolic diffusion models to determine the best-fit equation for the adsorption of glyphosate onto D301Fe. The results show that the pseudo-second order equation fitted the experimental data well and its adsorption was chemisorption-controlled.

A Comparative Study on the Removal of 2,2',4,4'-Tetrabrominated Diphenyl Ether (BDE-47) by Four Different Methods

2014 ◽  
Vol 700 ◽  
pp. 211-215
Author(s):  
Yi Miao Lin ◽  
Ling Yun Li ◽  
Ji Wei Hu ◽  
Ming Yi Fan ◽  
Chao Zhou ◽  
...  
Keyword(s):  
Exchange Resin ◽  
Diphenyl Ether ◽  
Ion Exchange Resin ◽  
Order Equation ◽  
Second Order ◽  
Order Kinetic ◽  
First Order ◽  
Atomic Force ◽  
Phase Reduction ◽  
Pseudo Second Order

The zero-valent iron (ZVI) particles were synthesized by the aqueous phase reduction, and the tapping mode image of atomic force microscope (AFM) showed that the diameter of the ZVI particles was in the range of 90 nm - 400 nm. By comparison of the debromination of BDE-47 by sunlight, ZVI, ZVI impregnated activated carbon (ZVI/AC) and ZVI impregnated ion exchange resin (ZVI/IER), the debromination effect was found to descend in the following order: ZVI/IER > ZVI/AC > ZVI > sunlight. Second order and first order kinetic models were used for the fitting of the debromination data of BDE-47. Results show that the debromination data of BDE-47 by the sunlight, ZVI, ZVI/AC and ZVI/IER in the current study are generally best described by the pseudo first order equation. Meanwhile, the debromination data of BDE-47 by the ZVI and ZVI/IER can also be described by the pseudo second order equation.

NEUTRINO OSCILLATION IN DENSE MEDIUMS

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3483-3492 ◽  
Author(s):  
Z. Y. LAW ◽  
A. H. CHAN ◽  
C. H. OH
Keyword(s):  
Neutrino Oscillation ◽  
Standard Treatment ◽  
Neutral Current ◽  
Matter Density ◽  
Order Equation ◽  
Second Order ◽  
Dense Medium ◽  
First Order ◽  
Probability Current ◽  
Current Interaction

It is found that a term normally discarded in the standard treatment of the MSW effect might be relevant in the case of non-adiabatic varying matter density, leading to a second order field equation, instead of the usual first order "Schrodinger equation". This leads to dispersion relation that gives rise to the possibility of neutrino trapping in a dense medium as well as the coupling of neutrino oscillation to neutral current interaction. This is found to be in agreement with previous results1. The corresponding conserved probability current is derived for this second order equation, and applied to the case of 2-flavor neutrino oscillation in a dense medium. The results in this work might be applicable to the oscillation of neutrinos in dense astrophysical medium.

EXACT SOLUTIONS OF THE DIRAC EQUATION WITH A COULOMB PLUS SCALAR POTENTIAL IN 2 + 1 DIMENSIONS

2002 ◽  
Vol 11 (06) ◽  
pp. 483-489 ◽  
Author(s):  
SHI-HAI DONG ◽  
XIAO-YAN GU ◽  
ZHONG-QI MA ◽  
SHISHAN DONG
Keyword(s):  
Dirac Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Coulomb Potential ◽  
Scalar Potential ◽  
Second Order ◽  
First Order ◽  
Second Order Differential Equations

The exact solutions of the (2+1)-dimensional Dirac equation with a Coulomb potential and a scalar one are analytically presented by studying the second-order differential equations obtained from a pair of coupled first-order ones. The eigenvalues are studied in some detail.

scholarly journals On wave matrices, and some properties of the wave equation

1936 ◽  
Vol 157 (890) ◽  
pp. 1-27 ◽  
Keyword(s):  
Wave Equation ◽  
Wave Theory ◽  
Order Equation ◽  
Second Order ◽  
Order Operator ◽  
First Order ◽  
Second Order Wave Equation

1—Wave matrices became important in wave theory as the result of the use of them made by Dirac to express the operator of the second order wave equation as the square of a linear one, and hence obtain a first order equation. Thus, p 2 representing the second order operator, the equation p 2 Ψ = 0, may be factorized, and written (∑ E α p α ) (∑ E α p α ) Ψ = 0, (α = 1, 2, . . . , n ), giving the first order equation ∑ E α p α Ψ = 0, (1) if the p α commute with themselves and with the E α , and if the E α are matrix roots of +1 or of —1, which satisfy E α E β = — E β E a (β ≠ α). (2)

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