multisymplectic geometry and multisymplectic preissman scheme for the kp equation the multisymplectic structure of the kp equation is obtained directly from the variational principal using the covariant de donder weyl hamilton function theories we reformulate the kp equation to the multisymplectic form which was proposed by bridges from the multisymplectic equation we can derive a multisymplectic numerical scheme of the kp equation which can be simplified to the multisymplectic 45 points scheme

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

new application of adomian decomposition method on fokker planck equation adomian decomposition method has been used for finding the explicit and numerical solutions of the reduced fokker planck equation in a force field the obtained results are compared with those found by other theories such as lie group theory the behaviour of the approximate distribution functions are shown graphically the results of lie group method confirm the correctness of those obtained by adomian decomposition method

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

novel implicit explicit local conservative schemes for the regularized long wave equation and convergence analysis abstract two implicit and two explicit schemes preserving the local momentum conservation laws exactly on any time space region are proposed for the regularized long wave equation with appropriate boundary conditions the schemes will be energy and mass preserving globally combining with the momentum conservation laws we obtain the priori estimates of the numerical solution and the error estimates in l norm for the proposed implicit schemes numerical experiments show the excellent performance of the proposed schemes and the numerical results coincide with the theoretical ones

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

a new explicit multisymplectic scheme for the regularized long wave equation in this paper we derive a new ten point multisymplectic scheme for the regularized long wave equation from its bridges multisymplectic form the new scheme is an explicit scheme in the sense that it does not need iteration we discuss some properties of the new scheme the performance and the efficiency of the new scheme are illustrated by solving several test examples the obtained results are presented and compared with previous methods numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well

some linearly and non linearly implicit schemes for the numerical solutions of the regularized long wave equation in this paper we present some linearly implicit and non linearly implicit schemes for investigating the numerical solutions of the regularized long wave equation numerical experiments indicate that all the present schemes can give satisfactory numerical solutions meanwhile numerical results also confirm the excellent conservations of three invariants in long time computations

analytical solution of the bagley torvik equation by adomian decomposition method abstract the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi d controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of bagley torvik equation r l bagley p j torvik on the appearance of the fractional derivative in the behavior of real materials asme j appl mech 51 1984 294 298 i podlubny fractional differential equations academic press san diego ca usa 1999 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with those available in the work of podlubny i podlubny fractional differential equations academic press san diego ca usa 1999 a good agreement of the results is observed

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

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

application of a lumped galerkin method to the regularized long wave equation in this paper a lumped galerkin method based on quadratic b spline finite elements is used to find numerical solutions of the one dimensional regularized long wave rlw equation with a variant of initial and boundary conditions the obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation results are compared with published numerical solutions a linear stability analysis of the scheme is also investigated

a computational method for the regularized long wave rlw equation a numerical solution of the regularized long wave rlw equation based on collocation method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves interaction of solitary waves with different amplitudes are shown the three invariants of the motion are evaluated to determine the conservation properties of the system the temporal evaluation of a maxwellian initial pulse is then studied and then we prove that the number of solitons which are generated from maxwellian initial condition are determined

a conservative weighted finite difference scheme for regularized long wave equation in this paper a new finite difference method for initial boundary value problem of regularized long wave rlw equation is considered two energy conservative finite difference schemes are proposed and the prior estimate of the solution is obtained existence and uniqueness of numerical solution are derived it is proved that the finite difference schemes are convergent and stable numerical experiment results show that the theory is accurate and the schemes are efficient and reliable

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

adomian decomposition method for solving fractional nonlinear differential equations in this article we have discussed a new application of adomian decomposition method on time fractional nonlinear fractional differential equations three models with fractional time derivative of order a 0 a 1 are considered and solved by means of adomian decomposition method the behaviour of adomian solutions and the effects of different values of a are investigated numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

a modified adomian method for system of nonlinear differential equations in this paper a modified adomian method is introduced to solve some systems of nonlinear differential equations the method can be mechanized in maple and a procedure is written to solve the approximate analytic solution of the systems some examples will be presented to illustrate the effectiveness of the method and the procedure a comparison between the program and the original adomian method s program is drawn

numerical implementation of the multisymplectic preissman scheme and its equivalent schemes we analyze the multisymplectic preissman scheme for the kdv equation with the periodic boundary condition and show that the unconvergence of the widely used iterative methods to solve the resulting nonlinear algebra system of the preissman scheme is due to the introduced potential function a artificial numerical condition is added to the periodic boundary condition the added boundary condition makes the numerical implementation of the multisymplectic preissman scheme practical and is proved not to change the numerical solutions of the kdv equation based on our analysis we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the preissman scheme because of less computing cost and less computer storages by eliminating the auxiliary variables we also derive two schemes for the kdv equation one is a 12 point scheme and the other is an 8 point scheme as the byproducts we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property numerical experiments on soliton collisions are also provided to confirm our conclusion and to show the benefits of the multisymplectic schemes with comparison of the spectral method and zabusky kruskal scheme

non perturbative analytical solution of the general lotka volterra three species system the lotka volterra equations are solved for the general three species system using the adomian decomposition comparisons are shown for a prey predator model whose exact solution is available as well as for perturbative solution by murty and rao

approximate analytical solutions of general lotka volterra equations on presente une solution analytique approchee des equations de lotka volterra generales en utilisant des transformations de laplace et une methode de perturbation

adomian decomposition method for solving the diffusion convection reaction equations abstract adomian decomposition method is used to solve the explicit and numerical solutions of three types of the diffusion convection reaction decr equations the calculations are carried out for three different types of the dcre such as the black scholes equation used in financial market option pricing fokker planck fp equation for lazer filed and fokker planck equation from plasma physics the behaviour of the approximate solutions of the distribution functions is shown graphically and compared with that obtained by other theories such as the variational iteration method

local momentum preserving algorithms for the grlw equation abstract in this paper four local momentum preserving lmp algorithms for the grlw equation are studied the proposed algorithms which conserve the local momentum conservation in any local time space region are proved to be momentum and mass preserving globally with appropriate boundary conditions two of the algorithms are explicit and the other two are implicit we make nonlinear convergence and stability analysis for the implicit algorithms numerical experiments confirm the long time preservation of the proposed algorithms and verify the theoretical analysis

conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion abstract a conservative difference scheme is given for a model of nonlinear dispersive waves convergence and stability of the scheme are proved by means of this scheme we explore numerically the relationship between the boundary data and the amplitudes and number of solitary waves it produces

the decomposition method for stiff systems of ordinary differential equations in this paper the decomposition method is applied to initial value problems for systems of ordinary differential equations in both linear and nonlinear cases focusing our interest in stiff problems

convenient analytic recurrence algorithms for the adomian polynomials abstract in this article we present four analytic recurrence algorithms for the multivariable adomian polynomials as special cases we deduce the four simplified results for the one variable adomian polynomials these algorithms are comprised of simple orderly and analytic recurrence formulas which do not require time intensive operations such as expanding regrouping parametrization and so on they are straightforward to implement in any symbolic software and are shown to be very efficient by our verification using mathematica 7 0 we emphasize that from the summation expressions a n k 1 n u n k for the multivariable adomian polynomials and an k 1 n f k u 0 c n k for the one variable adomian polynomials we obtain the recurrence formulas for the u n k and the c n k these provide a theoretical basis for developing new algorithmic approaches such as for parallel computing in particular the recurrence process of one particular algorithm for the one variable adomian polynomials does not involve the differentiation operation but significantly only the arithmetic operations of multiplication and addition are involved precisely c n 1 u n n 1 and c n k 1 n j 0 n k j 1 u j 1 c n 1 j k 1 2 k n we also discuss several other algorithms previously reported in the literature including the adomian rach recurrence algorithm 1 and this author s index recurrence algorithm 23 36