Approximate Analytical Solutions for Scattering States of D-dimensional Hulthen Potentials

Using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Schr6dinger equation of D-dimensional Hulthen potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of scattering states are attained. The normalized wave functions expressed in terms of hypergeometrie functions of scattering states on the ＂k/2π scale＂ and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solutions is discussed.

Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

Approximate Analytic Solution of Solitary Wave for a Class of Nonlinear Disturbed Long-Wave System

In this paper, the approximate expressions of the solitary wave solutions for a class of nonlinear disturbed long-wave system are constructed using the homotopie mapping method.

Analytical Solution for an Ultracold Trapped Ion at the Node of the Standing Wave Laser Without Rotating Wave Approximation

A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

An Analytic Approximate Solution of the SIR Model

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

Approximation analytical solutions for a unified plasma sheath model by double decomposition method

A anified plasma sheath model and its potential equation are proposed. Any higher-ordor-approximation analytical solutions for the unified plasma sheath potential equa tiop are derived by double decomposition method.

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Approximate analytical solutions for threedimensional ascent trajectory of a solid-fuel launch vehicle with time-varying mass flow rate

In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management,Approximate Analytical Solutions(AAS)for the threedimensional(3D)ascent flight states are derived,which are the only solutions capable of considering time-varying Mass Flow Rate(MFR)at present.The uneven MFR makes the thrust vary nonlinearly and thus increases the difficulty of the problem greatly.The AAS are derived based on a 3D Generalized Ascent Dynamics Model(GADM)with a normalized mass as the independent variable.To simplify some highly nonlinear terms in the GADM,several approximate functions are introduced carefully,while the errors of the approximations relative to the original terms are regarded as minor perturbations.Notably,a finite series with positive and negative exponents,called Exponent-Symmetry Series(ESS),is proposed for function approximation to decrease the highest exponent in the AAS so as to reduce computer round-off errors.To calculate the ESS coefficients,a method of seeking the Optimal Interpolation Points(OIP)is proposed using the leastsquares-approximation theory.Due to the artful design of the approximations,the GADM can be decomposed into two analytically solvable subsystems by a perturbation method,and thus the AAS are obtained successfully.Finally,to help implement the AAS,two indirect methods for measuring the remaining mass and predicting the burnout time in flight are put forward using information from accelerometers.Simulation results verify the superiority of the AAS under the condition of time-varying MFR.

Solution of Travelling Wave for Nonlinear Disturbed Long-Wave System

The approximate expressions of the travelling wave solutions for a class of nonlinear disturbed long-wave system are constructed using the generalized variational iteration method.

Large deflection of clamped circular plate and accuracy of its approximate analytical solutions

A different set of governing equations on the large deflection of plates are derived by the principle of virtual work(PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration.

The scattering states of the generalized Hulthén potential with an improved new approximate scheme for the centrifugal term

This paper finds the approximate analytical scattering state solutions of the arbitrary 1-wave Schrodinger equation for the generalized Hulthen potential by taking an improved new approximate scheme for the centrifugal term. The normalized analytical radial wave functions of the 1-wave SchrSdinger equation for the generalized Hulthen potential are presented and the corresponding calculation formula of phase shifts is derived. Some useful figures are plotted to show the improved accuracy of the obtained results and two special cases for the standard Hulthen potential and Woods-Saxon potential are also studied briefly.

New Analytical Study of the Effects Thermo-Diffusion, Diffusion-Thermo and Chemical Reaction of Viscous Fluid on Magneto Hydrodynamics Flow in Divergent and Convergent Channels

In this paper, the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied. The governing partial differential equation was transformed into a system of dimensionless non-similar coupled ordinary differential equation. The transformed conservations equations were solved by using new algorithm. Basically, this new algorithm depends mainly on the Taylor expansion application with the coefficients of power series resulting from integrating the order differential equation. Results obtained from new algorithm are compared with the results of numerical Range-Kutta fourth-order algorithm with help of the shooting algorithm. The comparison revealed that the resulting solutions were excellent agreement. Thermo-diffusion and diffusion-thermo effects were investigated to analyze the behavior of temperature and concentration profile. Also the influences of the first order chemical reaction and the rate of mass and heat transfer were studied. The computed analytical solution result for the velocity, temperature and concentration distribution with the effect of various important dimensionless parameters was analyzed and discussed graphically.

High precision approximate analytical solutions to ODE using LS-SVM

The problem of solving differential equations and the properties of solutions have always been an important content of differential equation study. In practical application and scientific research,it is difficult to obtain analytical solutions for most differential equations. In recent years,with the development of computer technology,some new intelligent algorithms have been used to solve differential equations. They overcome the drawbacks of traditional methods and provide the approximate solution in closed form( i. e.,continuous and differentiable). The least squares support vector machine( LS-SVM) has nice properties in solving differential equations. In order to further improve the accuracy of approximate analytical solutions and facilitative calculation,a novel method based on numerical methods and LS-SVM methods is presented to solve linear ordinary differential equations( ODEs). In our approach,a high precision of the numerical solution is added as a constraint to the nonlinear LS-SVM regression model,and the optimal parameters of the model are adjusted to minimize an appropriate error function. Finally,the approximate solution in closed form is obtained by solving a system of linear equations. The numerical experiments demonstrate that our proposed method can improve the accuracy of approximate solutions.

Approximate Analytical Solution of the Generalized Kolmogorov-Petrovsky-Piskunov Equation with Cubic Nonlinearity

In this paper,the approximate analytical oscillatory solutions to the generalized KolmogorovPetrovsky-Piskunov equation(gKPPE for short)are discussed by employing the theory of dynamical system and hypothesis undetermined method.According to the corresponding dynamical system of the bounded traveling wave solutions to the gKPPE,the number and qualitative properties of these bounded solutions are received.Furthermore,pulses(bell-shaped)and waves fronts(kink-shaped)of the gKPPE are given.In particular,two types of approximate analytical oscillatory solutions are constructed.Besides,the error estimations between the approximate analytical oscillatory solutions and the exact solutions of the gKPPE are obtained by the homogeneity principle.Finally,the approximate analytical oscillatory solutions are compared with the numerical solutions,which shows the two types of solutions are similar.

Analytical approximate solutions of AdS black holes in Einstein-Weyl-scalar gravity

We consider Einstein-Weyl gravity with a minimally coupled scalar field in four dimensional spacetime.Using the minimal geometric deformation(MGD)approach,we split the highly nonlinear coupled field equations into two subsystems that describe the background geometry and scalar field source,respectively.By considering the Schwarzschild-AdS metric as background geometry,we derive analytical approximate solutions of the scalar field and deformation metric functions using the homotopy analysis method(HAM),providing their analytical approximations to fourth order.Moreover,we discuss the accuracy of the analytical approximations,showing they are sufficiently accurate throughout the exterior spacetime.

Power Flow Analytical Solutions and Multi-dimensional Voltage Stability Boundaries Based on Multivariate Quotient-difference Method

This paper proposes a novel Multivariate Quotient-Difference(MQD)method to obtain the approximate analytical solution for AC power flow equations.Therefore,in the online environment,the power flow solutions covering different operating conditions can be directly obtained by plugging values into multiple symbolic variables,such that the power injections and consumptions of selected buses or areas can be independently adjusted.This method first derives a power flow solution through a Multivariate Power Series(MPS).Next,the MQD method is applied to transform the obtained MPS to a Multivariate Pad´e Approximants(MPA)to expand the Radius of Convergence(ROC),so that the accuracy of the derived analytical solution can be significantly increased.In addition,the hypersurface of the voltage stability boundary can be identified by an analytical formula obtained from the coefficients of MPA.This direct method for power flow solutions and voltage stability boundaries is fast for many online applications,since such analytical solutions can be derived offline and evaluated online by only plugging values into the symbolic variables according to the actual operating conditions.The proposed method is validated in detail on New England 39-bus and IEEE 118-bus systems with independent load variations in multi-regions.

Improved Analytical Approximations to the Scattering Solutions of the Schrdinger Equation with a Hyperbolical Potential

Using an improved approximate formula to the centrifugal term, we present arbitrary l-state scattering solutions of the hyperbolic potential. The approximate analytical formula of scattering phase shifts and normalized wavefunctions are presented. All data calculated by the above approximate analytical formula are compared with those obtained by using the numerical integration method in the scattering state cases. We find that this improved approximate formula is better than previous one since the calculated results are in good agreement with those exact ones.

Stepwise superposition approximation approach for analytical solutions with non-zero initial concentration using existing solutions of zero initial concentration in contaminate transport

Analytical solutions for contaminant transport are widely used for both theoretical and practical purposes.However,many existing solutions are obtained subject to an initial condition of zero concentration,which is often unrealistic in many practical cases.This article proposed a stepwise superposition approximation approach to solve the non-zero initial concentration problem for first-type and third-type boundary conditions by using the existing zero initial concentration solution.Theoretical examples showed that the approach was highly efficient if a proper superposition scheme with relative concentration increments was constructed.The key parameter that controlled the convergence speed was the time increment（△t） multiplied by the rate constant（λ）.The approach served also as an alternative way to make a convenient concentration calculation even if the non-zero initial concentration solution of a problem was known.

Derivation of Varying Specific Heat Gasdynamic Functions,Normal Shock Analytical Solution and its Improvements

A set of new gasdynamic functions with varying specific heat are derived for the first time.An original analytical solution of normal shock waves is worked out therewith.This solution is thereafter further improved by not involving total temperature.Illustrative examples of comparison are given,including also some approximate solutions to show the orders of their errors.