Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 1602

Search results for: Gompertz equation

1602 On Hyperbolic Gompertz Growth Model (HGGM)

Authors: S. O. Oyamakin, A. U. Chukwu,


We proposed a Hyperbolic Gompertz Growth Model (HGGM), which was developed by introducing a stabilizing parameter called θ using hyperbolic sine function into the classical gompertz growth equation. The resulting integral solution obtained deterministically was reprogrammed into a statistical model and used in modeling the height and diameter of Pines (Pinus caribaea). Its ability in model prediction was compared with the classical gompertz growth model, an approach which mimicked the natural variability of height/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using goodness of fit tests and model selection criteria. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the compliance of the error term to normality assumptions while using testing the independence of the error term using the runs test. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic gompertz growth models better than the source model (classical gompertz growth model) while the results of R2, Adj. R2, MSE, and AIC confirmed the predictive power of the Hyperbolic Monomolecular growth models over its source model.

Keywords: height, Dbh, forest, Pinus caribaea, hyperbolic, gompertz

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1601 The Growth Curve of Gompertz Model in Body Weight of Slovak Mixed-Sex Goose Breeds

Authors: Cyril Hrncar, Jozef Bujko, Widya P. B. Putra


The growth curve of poultry is important to evaluate the farming management system. This study was aimed to estimate the growth curve of body weight in goose. The growth curve in this study was estimated with non-linear Gompertz model through CurveExpert 1.4. software. Three Slovak mixed-sex goose breeds of Landes (L), Pomeranian (P) and Steinbacher (S) were used in this study. Total of 28 geese (10 L, 8 P and 10 S) were used to estimate the growth curve. Research showed that the asymptotic weight (A) in those geese were reached of 5332.51 g (L), 6186.14 g (P) and 5048.27 g (S). Thus, the maturing rate (k) in each breed were similar (0.05 g/day). The weight of inflection was reached of 1960.48 g (L), 2274.32 g (P) and 1855.98 g (S). The time of inflection (ti) was reached of 25.6 days (L), 26.2 days (P) and 27.80 days (S). The maximum growth rate (MGR) was reached of 98.02 g/day (L), 113.72 g/day (P) and 92.80 g/day (S). Hence, the coefficient of determination (R2) in Gompertz model was 0.99 for each breed. It can be concluded that Pomeranian geese had highest of growth trait than the other breeds.

Keywords: body weight, growth curve, inflection, Slovak geese, Gompertz model

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1600 Anaerobic Co-Digestion of Duckweed (Lemna gibba) and Waste Activated Sludge in Batch Mode

Authors: Rubia Gaur, Surindra Suthar


The present study investigates the anaerobic co-digestion of duckweed (Lemna gibba) and waste activated sludge (WAS) of different proportions with acclimatized anaerobic granular sludge (AAGS) as inoculum in mesophilic conditions. Batch experiments were performed in 500 mL capacity reagent bottles at 300C temperature. Varied combinations of pre-treated duckweed biomass with constant volume of anaerobic inoculum (AAGS - 100 mL) and waste activated sludge (WAS - 22.5 mL) were devised into five batch tests. The highest methane generation was observed with batch study, T4. The Gompertz model fits well on the experimental data of the batch study, T4. The values of correlation coefficient were achieved relatively higher (R2 ≥ 0.99). The co-digestion without pre-treatment of both duckweed and WAS shows poor generation of methane gas.

Keywords: aquatic weed, biogas, biomass, Gompertz equation, waste activated sludge

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1599 Comparison of the Logistic and the Gompertz Growth Functions Considering a Periodic Perturbation in the Model Parameters

Authors: Avan Al-Saffar, Eun-Jin Kim


Both the logistic growth model and the gompertz growth model are used to describe growth processes. Both models driven by perturbations in different cases are investigated using information theory as a useful measure of sustainability and the variability. Specifically, we study the effect of different oscillatory modulations in the system's parameters on the evolution of the system and Probability Density Function (PDF). We show the maintenance of the initial conditions for a long time. We offer Fisher information analysis in positive and/or negative feedback and explain its implications for the sustainability of population dynamics. We also display a finite amplitude solution due to the purely fluctuating growth rate whereas the periodic fluctuations in negative feedback can lead to break down the system's self-regulation with an exponentially growing solution. In the cases tested, the gompertz and logistic systems show similar behaviour in terms of information and sustainability although they develop differently in time.

Keywords: dynamical systems, fisher information, probability density function (pdf), sustainability

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1598 Growth Curves Genetic Analysis of Native South Caspian Sea Poultry Using Bayesian Statistics

Authors: Jamal Fayazi, Farhad Anoosheh, Mohammad R. Ghorbani, Ali R. Paydar


In this study, to determine the best non-linear regression model describing the growth curve of native poultry, 9657 chicks of generations 18, 19, and 20 raised in Mazandaran breeding center were used. Fowls and roosters of this center distributed in south of Caspian Sea region. To estimate the genetic variability of none linear regression parameter of growth traits, a Gibbs sampling of Bayesian analysis was used. The average body weight traits in the first day (BW1), eighth week (BW8) and twelfth week (BW12) were respectively estimated as 36.05, 763.03, and 1194.98 grams. Based on the coefficient of determination, mean squares of error and Akaike information criteria, Gompertz model was selected as the best growth descriptive function. In Gompertz model, the estimated values for the parameters of maturity weight (A), integration constant (B) and maturity rate (K) were estimated to be 1734.4, 3.986, and 0.282, respectively. The direct heritability of BW1, BW8 and BW12 were respectively reported to be as 0.378, 0.3709, 0.316, 0.389, 0.43, 0.09 and 0.07. With regard to estimated parameters, the results of this study indicated that there is a possibility to improve some property of growth curve using appropriate selection programs.

Keywords: direct heritability, Gompertz, growth traits, maturity weight, native poultry

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1597 Validation of Escherichia coli O157:H7 Inactivation on Apple-Carrot Juice Treated with Manothermosonication by Kinetic Models

Authors: Ozan Kahraman, Hao Feng


Several models such as Weibull, Modified Gompertz, Biphasic linear, and Log-logistic models have been proposed in order to describe non-linear inactivation kinetics and used to fit non-linear inactivation data of several microorganisms for inactivation by heat, high pressure processing or pulsed electric field. First-order kinetic parameters (D-values and z-values) have often been used in order to identify microbial inactivation by non-thermal processing methods such as ultrasound. Most ultrasonic inactivation studies employed first-order kinetic parameters (D-values and z-values) in order to describe the reduction on microbial survival count. This study was conducted to analyze the E. coli O157:H7 inactivation data by using five microbial survival models (First-order, Weibull, Modified Gompertz, Biphasic linear and Log-logistic). First-order, Weibull, Modified Gompertz, Biphasic linear and Log-logistic kinetic models were used for fitting inactivation curves of Escherichia coli O157:H7. The residual sum of squares and the total sum of squares criteria were used to evaluate the models. The statistical indices of the kinetic models were used to fit inactivation data for E. coli O157:H7 by MTS at three temperatures (40, 50, and 60 0C) and three pressures (100, 200, and 300 kPa). Based on the statistical indices and visual observations, the Weibull and Biphasic models were best fitting of the data for MTS treatment as shown by high R2 values. The non-linear kinetic models, including the Modified Gompertz, First-order, and Log-logistic models did not provide any better fit to data from MTS compared the Weibull and Biphasic models. It was observed that the data found in this study did not follow the first-order kinetics. It is possibly because of the cells which are sensitive to ultrasound treatment were inactivated first, resulting in a fast inactivation period, while those resistant to ultrasound were killed slowly. The Weibull and biphasic models were found as more flexible in order to determine the survival curves of E. coli O157:H7 treated by MTS on apple-carrot juice.

Keywords: Weibull, Biphasic, MTS, kinetic models, E.coli O157:H7

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1596 Asymptotic Expansion of the Korteweg-de Vries-Burgers Equation

Authors: Jian-Jun Shu


It is common knowledge that many physical problems (such as non-linear shallow-water waves and wave motion in plasmas) can be described by the Korteweg-de Vries (KdV) equation, which possesses certain special solutions, known as solitary waves or solitons. As a marriage of the KdV equation and the classical Burgers (KdVB) equation, the Korteweg-de Vries-Burgers (KdVB) equation is a mathematical model of waves on shallow water surfaces in the presence of viscous dissipation. Asymptotic analysis is a method of describing limiting behavior and is a key tool for exploring the differential equations which arise in the mathematical modeling of real-world phenomena. By using variable transformations, the asymptotic expansion of the KdVB equation is presented in this paper. The asymptotic expansion may provide a good gauge on the validation of the corresponding numerical scheme.

Keywords: asymptotic expansion, differential equation, Korteweg-de Vries-Burgers (KdVB) equation, soliton

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1595 Development of Simple-To-Apply Biogas Kinetic Models for the Co-Digestion of Food Waste and Maize Husk

Authors: Owamah Hilary, O. C. Izinyon


Many existing biogas kinetic models are difficult to apply to substrates they were not developed for, as they are substrate specific. Biodegradability kinetic (BIK) model and maximum biogas production potential and stability assessment (MBPPSA) model were therefore developed in this study for the anaerobic co-digestion of food waste and maize husk. Biodegradability constant (k) was estimated as 0.11d-1 using the BIK model. The results of maximum biogas production potential (A) obtained using the MBPPSA model corresponded well with the results obtained using the popular but complex modified Gompertz model for digesters B-1, B-2, B-3, B-4, and B-5. The (If) value of MBPPSA model also showed that digesters B-3, B-4, and B-5 were stable, while B-1 and B-2 were unstable. Similar stability observation was also obtained using the modified Gompertz model. The MBPPSA model can therefore be used as alternative model for anaerobic digestion feasibility studies and plant design.

Keywords: biogas, inoculum, model development, stability assessment

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1594 An Analytical Method for Solving General Riccati Equation

Authors: Y. Pala, M. O. Ertas


In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati equation, analytical solution, proper solution, nonlinear

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1593 Image Transform Based on Integral Equation-Wavelet Approach

Authors: Yuan Yan Tang, Lina Yang, Hong Li


Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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1592 Second Order Solitary Solutions to the Hodgkin-Huxley Equation

Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis


Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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1591 Study of Cahn-Hilliard Equation to Simulate Phase Separation

Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa


An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

Keywords: Cahn-Hilliard equation, miscibility gap, phase separation, dimensional domains

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1590 Study and Solving Partial Differential Equation of Danel Equation in the Vibration Shells

Authors: Hesamoddin Abdollahpour, Roghayeh Abdollahpour, Elham Rahgozar


This paper we deal with an analysis of the free vibrations of the governing partial differential equation that it is Danel equation in the shells. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of the hinged shell. A new implementation of the new method is presented to obtain natural frequency and corresponding displacement on the shell. Our purpose is to enhance the ability to solve the mentioned complicated partial differential equation (PDE) with a simple and innovative approach. The results reveal that this new method to solve Danel equation is very effective and simple, and can be applied to other nonlinear partial differential equations. It is necessary to mention that there are some valuable advantages in this way of solving nonlinear differential equations and also most of the sets of partial differential equations can be answered in this manner which in the other methods they have not had acceptable solutions up to now. We can solve equation(s), and consequently, there is no need to utilize similarity solutions which make the solution procedure a time-consuming task.

Keywords: large amplitude, free vibrations, analytical solution, Danell Equation, diagram of phase plane

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1589 Modification of Rk Equation of State for Liquid and Vapor of Ammonia by Genetic Algorithm

Authors: S. Mousavian, F. Mousavian, V. Nikkhah Rashidabad


Cubic equations of state like Redlich–Kwong (RK) EOS have been proved to be very reliable tools in the prediction of phase behavior. Despite their good performance in compositional calculations, they usually suffer from weaknesses in the predictions of saturated liquid density. In this research, RK equation was modified. The result of this study shows that modified equation has good agreement with experimental data.

Keywords: equation of state, modification, ammonia, genetic algorithm

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1588 Exact Solutions of a Nonlinear Schrodinger Equation with Kerr Law Nonlinearity

Authors: Muna Alghabshi, Edmana Krishnan


A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.

Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method

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1587 Divergence Regularization Method for Solving Ill-Posed Cauchy Problem for the Helmholtz Equation

Authors: Benedict Barnes, Anthony Y. Aidoo


A Divergence Regularization Method (DRM) is used to regularize the ill-posed Helmholtz equation where the boundary deflection is inhomogeneous in a Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes the inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation. This ensures the existence, as well as, uniqueness of solution for the equation. The DRM restores all the three conditions of well-posedness in the sense of Hadamard.

Keywords: divergence regularization method, Helmholtz equation, ill-posed inhomogeneous Cauchy boundary conditions

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1586 Solution of the Nonrelativistic Radial Wave Equation of Hydrogen Atom Using the Green's Function Approach

Authors: F. U. Rahman, R. Q. Zhang


This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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1585 A Study of Non Linear Partial Differential Equation with Random Initial Condition

Authors: Ayaz Ahmad


In this work, we present the effect of noise on the solution of a partial differential equation (PDE) in three different setting. We shall first consider random initial condition for two nonlinear dispersive PDE the non linear Schrodinger equation and the Kortteweg –de vries equation and analyse their effect on some special solution , the soliton solutions.The second case considered a linear partial differential equation , the wave equation with random initial conditions allow to substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, when we shall show that the addition of a multiplicative noise term forbids the blow up of solutions under a very weak hypothesis for which we have finite time blow up of a solution in the deterministic case. Here we consider the problem of wave propagation, which is modelled by a nonlinear dispersive equation with noisy initial condition .As observed noise can also be introduced directly in the equations.

Keywords: drift term, finite time blow up, inverse problem, soliton solution

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1584 Exact Soliton Solutions of the Integrable (2+1)-Dimensional Fokas-Lenells Equation

Authors: Meruyert Zhassybayeva, Kuralay Yesmukhanova, Ratbay Myrzakulov


Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2+1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form from the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 +1)-dimensional Fokas-Lenells equation were found.

Keywords: Fokas-Lenells equation, integrability, soliton, the Hirota bilinear method

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1583 Chern-Simons Equation in Financial Theory and Time-Series Analysis

Authors: Ognjen Vukovic


Chern-Simons equation represents the cornerstone of quantum physics. The question that is often asked is if the aforementioned equation can be successfully applied to the interaction in international financial markets. By analysing the time series in financial theory, it is proved that Chern-Simons equation can be successfully applied to financial time-series. The aforementioned statement is based on one important premise and that is that the financial time series follow the fractional Brownian motion. All variants of Chern-Simons equation and theory are applied and analysed. Financial theory time series movement is, firstly, topologically analysed. The main idea is that exchange rate represents two-dimensional projections of three-dimensional Brownian motion movement. Main principles of knot theory and topology are applied to financial time series and setting is created so the Chern-Simons equation can be applied. As Chern-Simons equation is based on small particles, it is multiplied by the magnifying factor to mimic the real world movement. Afterwards, the following equation is optimised using Solver. The equation is applied to n financial time series in order to see if it can capture the interaction between financial time series and consequently explain it. The aforementioned equation represents a novel approach to financial time series analysis and hopefully it will direct further research.

Keywords: Brownian motion, Chern-Simons theory, financial time series, econophysics

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1582 Fixed Point Iteration of a Damped and Unforced Duffing's Equation

Authors: Paschal A. Ochang, Emmanuel C. Oji


The Duffing’s Equation is a second order system that is very important because they are fundamental to the behaviour of higher order systems and they have applications in almost all fields of science and engineering. In the biological area, it is useful in plant stem dependence and natural frequency and model of the Brain Crash Analysis (BCA). In Engineering, it is useful in the study of Damping indoor construction and Traffic lights and to the meteorologist it is used in the prediction of weather conditions. However, most Problems in real life that occur are non-linear in nature and may not have analytical solutions except approximations or simulations, so trying to find an exact explicit solution may in general be complicated and sometimes impossible. Therefore we aim to find out if it is possible to obtain one analytical fixed point to the non-linear ordinary equation using fixed point analytical method. We started by exposing the scope of the Duffing’s equation and other related works on it. With a major focus on the fixed point and fixed point iterative scheme, we tried different iterative schemes on the Duffing’s Equation. We were able to identify that one can only see the fixed points to a Damped Duffing’s Equation and not to the Undamped Duffing’s Equation. This is because the cubic nonlinearity term is the determining factor to the Duffing’s Equation. We finally came to the results where we identified the stability of an equation that is damped, forced and second order in nature. Generally, in this research, we approximate the solution of Duffing’s Equation by converting it to a system of First and Second Order Ordinary Differential Equation and using Fixed Point Iterative approach. This approach shows that for different versions of Duffing’s Equations (damped), we find fixed points, therefore the order of computations and running time of applied software in all fields using the Duffing’s equation will be reduced.

Keywords: damping, Duffing's equation, fixed point analysis, second order differential, stability analysis

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1581 A Novel Method for Solving Nonlinear Whitham–Broer–Kaup Equation System

Authors: Ayda Nikkar, Roghayye Ahmadiasl


In this letter, a new analytical method called homotopy perturbation method, which does not need small parameter in the equation is implemented for solving the nonlinear Whitham–Broer–Kaup (WBK) partial differential equation. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of exact solution has led us to significant consequences. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the HPM can be found widely applicable in engineering.

Keywords: homotopy perturbation method, Whitham–Broer–Kaup (WBK) equation, Modified Boussinesq, Approximate Long Wave

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1580 Operational Matrix Method for Fuzzy Fractional Reaction Diffusion Equation

Authors: Sachin Kumar


Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy.

Keywords: fractional PDE, fuzzy valued function, diffusion equation, Legendre polynomial, spectral method

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1579 A Posteriori Analysis of the Spectral Element Discretization of Heat Equation

Authors: Chor Nejmeddine, Ines Ben Omrane, Mohamed Abdelwahed


In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler's implicit scheme in time and spectral method in space. We propose two families of error indicators, both of which are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Keywords: heat equation, spectral elements discretization, error indicators, Euler

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1578 Approximate Solution to Non-Linear Schrödinger Equation with Harmonic Oscillator by Elzaki Decomposition Method

Authors: Emad K. Jaradat, Ala’a Al-Faqih


Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

Keywords: non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two-dimensional Schrodinger equation

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1577 Relativistic Energy Analysis for Some q Deformed Shape Invariant Potentials in D Dimensions Using SUSYQM Approach

Authors: A. Suparmi, C. Cari, M. Yunianto, B. N. Pratiwi


D-dimensional Dirac equations of q-deformed shape invariant potentials were solved using supersymmetric quantum mechanics (SUSY QM) in the case of exact spin symmetry. The D dimensional radial Dirac equation for shape invariant potential reduces to one-dimensional Schrodinger type equation by an appropriate variable and parameter change. The relativistic energy spectra were analyzed by using SUSY QM and shape invariant properties from radial D dimensional Dirac equation that have reduced to one dimensional Schrodinger type equation. The SUSY operator was used to generate the D dimensional relativistic radial wave functions, the relativistic energy equation reduced to the non-relativistic energy in the non-relativistic limit.

Keywords: D-dimensional dirac equation, non-central potential, SUSY QM, radial wave function

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1576 A Mathematical Equation to Calculate Stock Price of Different Growth Model

Authors: Weiping Liu


This paper presents an equation to calculate stock prices of different growth model. This equation is mathematically derived by using discounted cash flow method. It has the advantages of being very easy to use and very accurate. It can still be used even when the first stage is lengthy. This equation is more generalized because it can be used for all the three popular stock price models. It can be programmed into financial calculator or electronic spreadsheets. In addition, it can be extended to a multistage model. It is more versatile and efficient than the traditional methods.

Keywords: stock price, multistage model, different growth model, discounted cash flow method

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1575 Energy Conservation and H-Theorem for the Enskog-Vlasov Equation

Authors: Eugene Benilov, Mikhail Benilov


The Enskog-Vlasov (EV) equation is a widely used semi-phenomenological model of gas/liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H-theorem as well, it can be used to rigorously derive the so-called Maxwell construction which determines the parameters of liquid-vapor equilibria. Finally, we show that the EV model provides an accurate description of the thermodynamics of noble fluids, and there exists a version simple enough for use in applications.

Keywords: Enskog collision integral, hard spheres, kinetic equation, phase transition

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1574 Numerical Solution of Manning's Equation in Rectangular Channels

Authors: Abdulrahman Abdulrahman


When the Manning equation is used, a unique value of normal depth in the uniform flow exists for a given channel geometry, discharge, roughness, and slope. Depending on the value of normal depth relative to the critical depth, the flow type (supercritical or subcritical) for a given characteristic of channel conditions is determined whether or not flow is uniform. There is no general solution of Manning's equation for determining the flow depth for a given flow rate, because the area of cross section and the hydraulic radius produce a complicated function of depth. The familiar solution of normal depth for a rectangular channel involves 1) a trial-and-error solution; 2) constructing a non-dimensional graph; 3) preparing tables involving non-dimensional parameters. Author in this paper has derived semi-analytical solution to Manning's equation for determining the flow depth given the flow rate in rectangular open channel. The solution was derived by expressing Manning's equation in non-dimensional form, then expanding this form using Maclaurin's series. In order to simplify the solution, terms containing power up to 4 have been considered. The resulted equation is a quartic equation with a standard form, where its solution was obtained by resolving this into two quadratic factors. The proposed solution for Manning's equation is valid over a large range of parameters, and its maximum error is within -1.586%.

Keywords: channel design, civil engineering, hydraulic engineering, open channel flow, Manning's equation, normal depth, uniform flow

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1573 Exactly Fractional Solutions of Nonlinear Lattice Equation via Some Fractional Transformations

Authors: A. Zerarka, W. Djoudi


We use some fractional transformations to obtain many types of new exact solutions of nonlinear lattice equation. These solutions include rational solutions, periodic wave solutions, and doubly periodic wave solutions.

Keywords: fractional transformations, nonlinear equation, travelling wave solutions, lattice equation

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