homotopy perturbation method for quadratic riccati differential equation and comparison with adomian s decomposition method abstract in this paper a homotopy perturbation method is proposed to solve quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the proposed method is very effective and simple

he s homotopy perturbation method for a boundary layer equation in unbounded domain by means of he s homotopy perturbation method hpm an approximate solution of a boundary layer equation in unbounded domain is obtained comparison is made between the obtained results and those in open literature the results show that the method is very effective and simple

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

crank nicolson finite difference method for two dimensional diffusion with an integral condition a finite difference method which is based on the 5 5 crank nicolson cn scheme is developed for solving the heat equation in two dimensional space with an integral condition replacing one boundary condition the fully implicit method developed here is unconditionally stable and it has reasonable accuracy while the conditionally stable fully explicit schemes use less amount of central processor cpu time the unconditional stability of the scheme developed in this article for every diffusion number is significant some numerical tests are presented and the accuracy obtained and the cpu time required are reported error estimates derived in the maximum norm are tabulated

a computational method for solving one dimensional variable coefficient burgers equation abstract the burgers equation is a simple one dimensional model of the navier stoke equation in this paper the exact solution to one dimensional variable coefficient burgers equation is obtained in the reproducing kernel space w 2 3 the exact solution is represented in the form of series the n term approximation u n t x to exact solution u t x is proved to converge to the exact solution moreover the approximate error of u n t x is monotone decreasing some numerical examples have been studied to demonstrate the accuracy of the present method results obtained by the method have been compared with the exact solution of each example and are found to be in good agreement with each other

addendum to colored noise induced discontinuous transitions in symbiotic ecosystems a symbiotic ecosystem with gompertz self regulation and with adaptive competition between populations is studied by means of a n species lotka volterra stochastic model the influence of fluctuating environment on the carrying capacity of a population is modeled as a dichotomous noise the study is a follow up of previous investigations of symbiotic ecosystems subjected to the generalized verhulst self regulation phys rev e 69 061106 2004 65 051108 2002 in the framework of mean field approximation the behavior of the solutions of the self consistency equation for a stationary system is examined analytically in the full phase space of system parameters depending on the mutual interplay of symbiosis and competition of species variation of noise parameters amplitude correlation time can induce doubly unidirectional discontinuous transitions as well as single unidirectional discontinuous transitions of the mean population size

solving singular boundary value problems of higher even order in this paper we present a new algorithm to solve a kind of singular boundary value problems of higher even order based on the theory of numerical analysis in the reproducing kernel space w 2 2 m 3 0 1 singular boundary value problem of sixth order has been discussed for the problem u 6 x a x u 4 x b x u x c x u x f x 0

fully implicit finite differences methods for two dimensional diffusion with a non local boundary condition three new fully implicit methods which are based on the 5 5 crank nicolson method the 5 5 n h noye hayman implicit method and the 9 9 n h implicit method are developed for solving the heat equation in two dimensional space with non local boundary conditions the latter is fourth order while the others are second order while the implicit methods developed here like the scheme based on the standard implicit backward time centered space btcs method use a large amount of central processor cpu time the high accuracy of the new fourth order fully implicit scheme is significant like the btcs method the new methods are also unconditionally stable

solving singular two point boundary value problem in reproducing kernel space in this paper we present a new method for solving singular two point boundary value problem for certain ordinary differential equation having singular coefficients its exact solution is represented in the form of series in reproducing kernel space in the mean time the n term approximation u n x to the exact solution u x is obtained and is proved to converge to the exact solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other

the exact solution for solving a class nonlinear operator equations in the reproducing kernel space in this paper the author will introduce the sobolev space w 1 2 w n firstly and then give the reproducing properties of the space w 1 2 w n as these results of those a class of the nonlinear operator equation n i 1 m i j 1 a i ju f is transformed to the n dimensional linear operator equation au f finally we also give the exact solution of this nonlinear operator equation

solving singular nonlinear second order periodic boundary value problems in the reproducing kernel space abstract in this paper we present a new method for solving a singular nonlinear second order periodic boundary value problem its analytical solution is represented in the form of series in the reproducing kernel space in the mean time the n term approximation u n x to the analytical solution u x is obtained and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

second order boundary value problems with integral boundary conditions abstract in this paper we investigate the existence of positive solutions of nonlocal second order boundary value problems with integral boundary conditions

how to solve the equation aubu cu f in this paper we first discuss the problem of how to solve a continuous linear operator equation in separable hilbert space we give the necessary and sufficient condition for the existence of solution and the method to decide the existence for solutions of the equation is also given if the solution exists we give the representation and approxima of the minimal normal solution of the equation and formula are obtained we conduct some research of how to solve the operator equation aubu cu f we give the factorization method and characteristic value method to solve such an equation

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

nontrivial solutions of singular sublinear sturm liouville problems abstract the singular superlinear sturm liouville problems l x h x f x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators where l x p x x q x x and h x is allowed to be singular at both x 0 and x 1 in particular f is not necessary to be nonnegative the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory

a finite difference method for the singularly perturbed problem with nonlocal boundary condition the purpose of this paper is to present a finite difference method for numerical solutions of singularly perturbed boundary value problem for second order ordinary differential equation with nonlocal boundary condition by the method of integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form an exponentially fitted difference scheme on an uniform mesh is developed which is shown to be original e uniformly first order accurate in the discrete maximum norm for original problem numerical results are presented which illustrate the theoretical results

stepwise stability for the heat equation with a nonlocal constraint the authors prove the stepwise stability for a finite difference scheme for the heat equation with an integral constraint the resulting matrix is nonsymmetric and does not have the usual band structure the proof is based on the method of matrix analysis the eigenvalues of several matrices are found explicitly or their location described precisely this method relies upon the relationship of the characteristic polynomials of these matrices with orthogonal polynomials

solving a nonlinear system of second order boundary value problems in this paper a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems the analytical solution is represented in the form of series in the reproducing kernel space in the mean time the approximate solution un x is obtained by the n term intercept of the analytical solution and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

modified homotopy perturbation method for nonlinear equations and comparison with adomian decomposition method abstract in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method by using modified homotopy perturbation method the modified adomian decomposition method s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method appl math comput 145 2003 887 893 is compared with new algorithms some numerical illustrations are given to show the efficiency of algorithms

application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations perturbation methods depend on a small parameter which is difficult to be found for real life nonlinear problems to overcome this shortcoming two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article one is he s variational iteration method vim and the other is the homotopy perturbation method hpm the vim is to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the hpm deforms a difficult problem into a simple problem which can be easily solved nonlinear convective radiative cooling equation nonlinear heat equation porous media equation and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures comparison of the applied methods with exact solutions reveals that both methods are tremendously effective