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.

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.

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 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.

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.

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.

Continuum states of modified Morse potential

Using a proper approximation scheme to the centrifugal term, we study any l-wave continuum states of the Schrodinger equation for the modified Morse potential. The normalised analytical radial wave functions are presented, and a corresponding calculation formula of phase shifts is derived. It is shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some numerical results are calculated to show the accuracy of our results.

Differential transform method for solving Richards' equation

An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.

MOVING BOUNDARY PROBLEM FOR DIFFUSION RELEASE OF DRUG FROM A CYLINDER POLYMERIC MATRIX

An approximate analytical solution of moving boundary problem for diffusion release of drug from a cylinder polymeric matrix was obtained by use of refined integral method. The release kinetics has been analyzed for non-erodible matrices with perfect sink condition. The formulas of the moving boundary and the fractional drug release were given. The moving boundary and the fractional drug release have been calculated at various drug loading levels, mid the calculated results were in good agreement with those of experiments. The comparison of the moving boundary in spherical, cylinder, planar matrices has been completed. An approximate formula for estimating the available release time was presented. These results are useful for the clinic experiments. This investigation provides a new theoretical tool for studying the diffusion release of drug from a cylinder polymeric matrix and designing the controlled released drug.

Reduced Differential Transform Method for Solving Nonlinear Biomathematics Models

In this paper,we study the approximate solutions for some of nonlinear Biomathematics models via the e-epidemic SI1I2R model characterizing the spread of viruses in a computer network and SIR childhood disease model.The reduced differential transforms method(RDTM)is one of the interesting methods for finding the approximate solutions for nonlinear problems.We apply the RDTM to discuss the analytic approximate solutions to the SI1I2R model for the spread of virus HCV-subtype and SIR childhood disease model.We discuss the numerical results at some special values of parameters in the approximate solutions.We use the computer software package such as Mathematical to find more iteration when calculating the approximate solutions.Graphical results and discussed quantitatively are presented to illustrate behavior of the obtained approximate solutions.

Mathematical Model of a Hyperbolic Hydraulic Fracture with Tortuosity

The aim of the research is to study the propagation of a hydraulic fracture with tortuosity due to contact areas between touching asperities on opposite crack walls. The tortuous fracture is replaced by a model symmetric partially open fracture with a hyperbolic crack law and a modified Reynolds flow law. The normal stress at the crack walls is assumed to be proportional to the half-width of the model fracture. The Lie point symmetry of the nonlinear diffusion equation for the fracture half-width is derived and the general form of the group invariant solution is obtained. It was found that the fluid flux at the fracture entry cannot be prescribed arbitrarily, because it is determined by the group invariant solution and that the exponent n in the modified Reynolds flow power law must lie in the range 2 < n < 5. The boundary value problem is solved numerically using a backward shooting method from the fracture tip, offset by 0 < δ ≪ 1 to avoid singularities, to the fracture entry. The numerical results showed that the tortuosity and the pressure due to the contact regions both have the effect of increasing the fracture length. The spatial gradient of the half-width was found to be singular at the fracture tip for 3 < n < 5, to be finite for the Reynolds flow law n = 3 and to be zero for 2 < n < 3. The thin fluid film approximation breaks down at the fracture tip for 3 < n < 5 while it remains valid for increasingly tortuous fractures with 2 < n < 3. The effect of the touching asperities is to decrease the width averaged fluid velocity. An approximate analytical solution for the half-width, which was found to agree well with the numerical solution, is derived by making the approximation that the width averaged fluid velocity increases linearly with distance along the fracture.

Subharmonic resonance of single-degree-of-freedom piecewise-smooth nonlinear oscillator

Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example,the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated.The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented.By changing the solving process of Krylov-Bogoliubov-Mitropolsky(KBM)asymptotic method for subharmonic resonance of smooth nonlinear system,the classical KBM method is extended to piecewise-smooth nonlinear system.The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved,and the stability of the subharmonic resonance steady-statesolution is also analyzed.It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness.Through a demonstration example,the accuracy of approximate analytical solution is verified by numerical solution,and they have good consistency.Based on the approximate analytical solution,the infuences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail.The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system.And it provides an effective reference for the study of subharmonicr esonance of other piecewise-smooth systems.