a new analytic algorithm of lane emden type equations an reliable ease to use analytic algorithm is provided for lane emden type equation which models many phenomena in mathematical physics and astrophysics this algorithm logically contains the well known adomian decomposition method different from all other analytic techniques this algorithm itself provides us with a convenient way to adjust convergence regions even without pade technique some applications are given to show its validity

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

stability and convergence of radial basis function finite difference method for the numerical solution of the reaction diffusion equations stability convergence and application of radial basis function finite difference rbf fd scheme is studied for solving the reaction diffusion equations rdes we show that the explicit rbf fd method is stable and stability condition depends on the shape parameter of related radial basis function the generalized multiquadric gmq is applied as radial basis function and weight coefficients are explicitly presented for equispaced node distribution also two methods are presented to compute the optimal shape parameter the combination of these methods with the gmq fd method will produce two efficient algorithms for numerical solution of rdes the variable gmq fd vgmq fd and the constant gmq fd cgmq fd we test the scheme on traveling wave and compare its accuracy with the conventional finite difference method fdm

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

a generalization of the method of upper and lower solutions for discontinuous first order problems with nonlinear boundary conditions in this paper we prove the existence of extremal solutions for a discontinuous bvp between a lower and an upper solution which may be discontinuous in a finite set of points furthermore we present a monotone method with which we can explicitly construct a pair of sequences that converge to the extremal solutions despite the discontinuities of the right hand side of the equation

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

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

schrodinger equation as recurrences iii partitioning approach for pt ii see ibid vol 17 no 8 p 1603 10 1984 in the generalised lanczos basis the author partitions the hamiltonian to tridiagonal form and mutatis mutandis apply the ideas of i and ii it solves the schrodinger eigenvalue problem again the resulting method modifies and complements the formalism and simplifies its use and interpretation as an example of application the unique asymptotical effective hamiltonian is derived also for the anharmonic oscillators the convergence of its continued fractional definition is proved and the compact formula for the higher order asymptotic corrections is found

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

the decomposition method for ordinary differential equations with discontinuities in this paper the decomposition method is applied to the initial value problem ivp for ordinary differential equations with discontinuities for both linear and nonlinear cases

pade oscillators and a new formulation of perturbation theory being guided by the problem of bound states in potentials close to their pade approximants a new rayleigh schrodinger type perturbation theory is developed the unperturbed system is understood here in a broader sense its solutions are not needed but merely the related nondiagonal unperturbed propagator r in particular all the chain models h0 es0 h0 s0 band matrices with arbitrary perturbations are then perturbatively solvable with r constructed in terms of auxiliary matrix continued fraction fn alternatively a generalized unperturbed spectrum fn may be required as an input the algebraically constructed asymptotics of the fn s play this role in our pade examples due to s i the sturmians may also be constructed in the test evaluations of the binding energies and or couplings the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

the decomposition method for continuous population models for single and interacting species the adomian decomposition method is applied to continuous population models for both single and interacting species in comparison with existing techniques the decomposition method is highly effective in terms of accuracy and rapid convergence analytical and numerical studies are presented

a new modification of adomian decomposition method for solving a kind of evolution equation in this paper we propose a new modification of adomian decomposition method for solving a kind of evolution equation the modification will improve the convergence of the series solution the validity of the modified technique is verified through illustrative examples

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

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

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases