brief paper the asymptotic behavior of stochastically perturbed di sir epidemic models with saturated incidences in this paper we consider a class of di sir epidemic models with saturated incidences and parameter perturbation we investigate the asymptotic behavior according to the perturbation and the reproductive number r 0 when the perturbation is large the infective in every group decays exponentially to zero while the susceptible converges weakly to stationary distribution regardless of the magnitude of r 0 when the perturbation is small we get the same exponential stability and weak convergence if r 0 1 and we use a new class of stochastic lyapunov functions to obtain the ergodicity and positive recurrence if r 0 1

soliton solutions of the kadomtsev petviashvili ii equation we study a general class of line soliton solutions of the kadomtsev petviashvili ii kpii equation by investigating the wronskian form of its tau function we show that in addition to the previously known line soliton solutions of kpii this class also contains a large variety of multisoliton solutions many of which exhibit nontrivial spatial interaction patterns we also show that in general such solutions consist of unequal numbers of incoming and outgoing line solitons from the asymptotic analysis of the tau function we explicitly characterize the incoming and outgoing line solitons of this class of solutions we illustrate these results by discussing several examples

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

slow scale split step tau leap method for stiff stochastic chemical systems abstract tau leaping is a family of algorithms for the approximate simulation of the discrete state continuous time markov chains a motivation for the development of such methods can be found for instance in the fields of chemical kinetics and systems biology it is known that the dynamical behavior of biochemical systems is often intrinsically stiff representing a serious challenge for their numerical approximation the naive extension of stiff deterministic solvers to stochastic integration often yields numerical solutions with either impractically large relaxation times or incorrectly resolved covariance in this paper we propose a splitting heuristic which helps to resolve some of these issues the proposed integrator contains a number of unknown parameters which are estimated for each particular problem from the moment equations of the corresponding linearized system we show that this method is able to reproduce the exact mean and variance of the linear scalar test equation and demonstrates a good accuracy for the arbitrarily stiff systems at least in the linear case the numerical examples for both linear and nonlinear systems are also provided and the obtained results confirm the efficiency of the considered splitting approach

reversible equivalent monomolecular tau leaping methods provide for efficient and approximate time stepping of chemical reaction systems modeled by continuous time discrete state stochastic dynamics we investigate the application of leaping methods for small number and stiff systems i e systems whose dynamics involve different time scales and have some molecular species present in very small numbers specifically in the range 0 to 10 we propose a new explicit leaping scheme reversible equivalent monomolecular tau remm which shows considerable promise in the simulation of such systems the remm scheme is based on the fact that the exact solution of the two prototypical monomolecular reversible reactions s1 s2 and s 0 as a function of time takes a simple form involving binomial and or poisson random variables the remm method involves approximating bimolecular reversible reactions by suitable monomolecular reversible reactions as well as considering each reversible pair of reactions in the system to be operating in isolation during the time step we illustrate the use of the remm method through a number of biologically motivated examples and compare its performance to those of the implicit and trapezoidal implicit algorithms in most cases considered remm appears to perform better than these two methods while having the important advantage of being computationally faster due to the explicit nature of the method furthermore when stepsize is increased the remm exhibits a more robust performance than the implicit or the trapezoidal implicit for small number stiff problems

compact structures for variants of the generalized kdv and the generalized kp equations in this paper we discuss mathematical variants in higher dimensions of the generalized kdv and the generalized kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the compact and the noncompact dispersive structures are examined in a general way

tau leaping of stiff stochastic chemical systems via local central limit approximation stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems it is not only important for a time stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately in the context of stochastic chemical systems time stepping methods are known as tau leaping well known existing tau leaping methods have shortcomings in this regard the implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance on the other hand the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems we propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system s email protected s 2 the proposed method uses a central limit approximation cla locally over the tau leaping interval and is referred to as the email protected the cla predicts the mean and covariance as solutions of certain differential equations odes and for efficiency we solve these using a single time step of a suitable low order method we perform a mean covariance stability analysis of various possible low order schemes to determine the best scheme numerical experiments presented show that email protected performs favorably for stiff systems and that the email protected is also able to capture bimodal distributions unlike the cla itself the proposed email protected method uses a split implicit step to compute the mean update we also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit euler method as applied to the fluid limit differential equation

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

a mathematical approach to hiv infection dynamics in order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as hiv infection process and aids disease progression it is efficient to introduce a general class of time dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition an attempt is then made to combine the implicit approach usually adapted for convective diffusion operators and explicit approach more suited to treat continuous type operators representing various physiological interactions resulting in a semi implicit product formula decomposing the operators in this way and considering their individual properties it is seen that approximation solvability of the original model is verified under suitable conditions once appropriate terms are formulated to describe treatment by antiretroviral therapy the time dependence of the reaction terms appears and such product formula is useful for generating approximate numerical solutions to the governing equations with this knowledge a continuous model for hiv disease progression is formulated and physiological interpretations are provided the abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the hiv virus in the human body and its reaction to treatment by antiretroviral therapy the product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with hiv1 and is applied to perform numerical simulations based on the model of the hiv infection process and disease progression finally the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

numerical methods for the solution of the third and fifth order dispersive korteweg de vries equations abstract two numerical methods are proposed for the solution of the third and fifth order korteweg de vries equations the first method is derived using central differences to replace the space derivatives with a predictor corrector time stepping and the second method by linearizing the implicit corrector scheme in which the solution is then found by solving a linear algebraic system at each time step rather than a nonlinear algebraic system which is more usual an important advantage to be gained from the use of the linearized implicit method over the predictor corrector method which is optimally stable is the ability to vary the mesh length the methods are analysed with respect to stability criteria and numerical dispersion numerical results portraying a single soliton solution and the interaction of more than one soliton are reported for the third order korteweg de vries equation numerical results for the fifth order korteweg de vries equation using the linearized implicit method are also reported

asymptotic behavior and stability of a stochastic model for aids transmission a stochastic model for aids transmission is presented in this paper the existence and uniqueness of its solution is proved and the solution belongs to an interval satisfying the actual condition furthermore the disease equilibrium can be demonstrated to be existent and both its almost surely exponential stability and its pth moment exponential stability can be concluded based on parameter analysis the first quadrant can be integrated into several parts according to the comparison between almost surely exponential stability and pth moment exponential stability and the potential meaning of the stability is further demonstrated finally based on the above theoretical analysis some conditions for aids to decrease and then to die out are obtained by computer simulation

the lie group method based on radial basis functions for solving nonlinear high dimensional generalized benjamin bona mahony burgers equation in arbitrary domains abstract the aim of this paper is to introduce a new numerical method for solving the nonlinear generalized benjamin bona mahony burgers gbbmb equation this method is combination of group preserving scheme gps with radial basis functions rbfs which takes advantage of two powerful methods one as geometric numerical integration method and the other meshless method thus we introduce this method as the lie group method based on radial basis functions lg rbfs in this method we use kansas approach to approximate the spatial derivatives and then we apply gps method to approximate first order time derivative one of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions to demonstrate this point we solve nonlinear gbbmb equation on various geometric domains in one two and three dimension spaces the results of numerical experiments are compared with analytical solutions and the method presented in dehghan et al 2014 to confirm the accuracy and efficiency of the presented method

exact solutions of compact and noncompact structures for the kp bbm equation the tanh and the sine cosine methods are used for a reliable treatment of the 2 1 dimensional kp bbm equation generalized forms of the kp bbm equation are investigated as well a variety of exact travelling wave solutions of distinct physical structures compactons solitons solitary patterns and periodic solutions are formally derived

an application of the modified adomian decomposition method for n 1 dimensional sine gordon field abstract based upon the modified adomian decomposition method a scheme is developed to obtain approximate jacobi elliptic function solutions of the n 1 dimensional sine gordon field with initial conditions as a result we derive the approximate solution and exact jacobi elliptic function solution of the n 1 dimensional sine gordon field with initial conditions moreover the approximate solution is compared with the exact solution with n 2 and with different modulus k 0 25 0 5 and 1

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

the cauchy problem and stability of solitary wave solutions for rlw kp type equations abstract the kadomtsev petviashvilli kp equation ut ux uux uxxx x euyy 0 arises in various contexts where nonlinear dispersive waves propagate principally along the x axis but with weak dispersive effects being felt in the direction parallel to the y axis perpendicular to the main direction of propagation we propose and analyze here a class of evolution equations of the form ut ux upux lut x euyy 0 which provides an alternative to eq in the same way the regularized long wave equation is related to the classical korteweg de vries kdv equation the operator l is a pseudo differential operator in the x variable p 1 is an integer and e 1 after discussing the underlying motivation for the class a local well posedness theory for the initial value problem is developed with assumptions on l and p that include conditions appertaining to models of interesting physical phenomenon the solutions defined locally in time t are shown to be smoothly extendable to the entire time axis in the particularly interesting case where l x2 and e 1 possesses travelling wave solutions u x y t c x ct y provided c 1 and 0 0 4 3 and c 1 and for 4 3 if c 4p 4 p the paper concludes with commentary on extensions of the present theory to more than two space dimensions

the long time behavior of di sir epidemic model with stochastic perturbation in this paper we present a di sir epidemic model with two categories stochastic perturbations the long time behavior of the two stochastic systems is studied mainly we show how the solution goes around the infection free equilibrium and the endemic equilibrium of deterministic system under different conditions

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

finite difference approximate solutions for the two dimensional burgers system abstract finite difference approximate solutions for the two dimensional burgers system which models a turbulent flow in a channel implicit euler method is applied to obtain approximate solutions existence of solutions is shown by using leray schauder fixed point theorem stability and uniqueness of the solution are also shown by judicious applications of the discrete gronwall s inequality and energy methods

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

numerical simulation of kadomtsev petviashvili benjamin bona mahony equations using finite difference method in this paper the finite difference method is employed to solve kadomtsev petviashvili benjamin bona mahony ii kp bbm ii partial differential equations the time and space variable are discretized by the crank nicholson method and the central difference scheme respectively the consistence and stability are also proved some examples are investigated to verify the efficiency of the present method

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

bifurcations of travelling wave solutions for the generalized kp bbm equation abstract by using the bifurcation theory of dynamical systems to the generalized kadomtsov petviashvili benjamin bona mahony equation the existence of solitary wave solutions compactons solution non smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given some exact explicit parametric representations of the above waves are determined

finite difference discretization of the kuramoto sivashinsky equation we analyze a crank nicolson type finite difference scheme for the kuramoto sivashinsky equation in one space dimension with periodic boundary conditions we discuss linearizations of the scheme and derive second order error estimates

a finite difference method for the korteweg de vries and the kadomtsev petviashvili equations a linearized implicit finite difference method for the korteweg de vries equation is proposed and straightforwardly extended to the kadomtsev petviashvili equation we investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable the numerical experiments for the korteweg de vries and the kadomtsev petviashvili equations are carried out with various conditions numerical results for the collision of two lump type solitary wave solutions to the kadomtsev petviashvili equation are also reported

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

solutions of the kp equation associated to rank three commuting differential operators over a singular elliptic curve abstract we give a method for finding explicit solutions of the kadomtsev petviashvili kp equation which are associated to rank three commuting ordinary differential operators explicit formulae for the coefficients of rank three commuting differential operators are used to reduce the kp equation to a nonlinear partial differential equation in only two variables this equation which is obtained in a similar way to the rank two krichever novikov kn equation is easier to solve than kp and some solutions are given

fractional diffusion equations by the kansa method this study makes the first attempt to apply the kansa method in the solution of the time fractional diffusion equations in which the multiquadrics and thin plate spline serve as the radial basis function in the discretization formulation the finite difference scheme and the kansa method are respectively used to discretize time fractional derivative and spatial derivative terms the numerical solutions of one and two dimensional cases are presented and discussed which agree well with the corresponding analytical solution