a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

a finite element method for equal width equation abstract a numerical solution of the equal width ew equation is obtained using space splitting technique and quadratic b spline galerkin finite element method solitary wave motion interaction of two solitary waves wave undulation and wave generation are studied using the proposed method comparisons are made with analytical solutions and with some spline finite element method calculations at selected times accuracy and efficiency are discussed by computing the numerical conserved laws and l 2 l error norms

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

a calculation with a biorthogonal wavelet transformation the use of a biorthogonal basis for continuous wavelet transformations is explored thus relaxing the so called admissibility condition on the analyzing wavelet as an application the eigenvalues and corresponding radial eigenfunctions of the hamiltonian of relativistic hydrogen like atoms are determined

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

a two dimensional bi cubic b spline finite element used in a study of mhd duct flow abstract a two dimensional bi cubic b spline finite element is proposed and used within a galerkin approach to model magneto hydrodynamic duct flow excellent agreement is obtained between the numerical results and the analytic solution even for large values of the applied magnetic field

application of galerkin s method to equal width wave equation the non linear equal width equation is solved by galerkin s method using linear finite elements a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found employing the crank nicolson approach including a product approximation the three invariants of the motion are calculated to determine the conservation properties of the system l 2 and l error norms are used to measure differences between the exact and numerical solutions the simulations of solitary wave motion are used to determine the properties of the algorithm finally the development of an undular bore is studied

the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

fourth order compact and energy conservative difference schemes for the nonlinear schrodinger equation in two dimensions abstract in this paper a fourth order compact and energy conservative difference scheme is proposed for solving the two dimensional nonlinear schrodinger equation with periodic boundary condition and initial condition and the optimal convergent rate without any restriction on the grid ratio at the order of o h 4 2 in the discrete l 2 norm with time step and mesh size h is obtained besides the standard techniques of the energy method a new technique and some important lemmas are proposed to prove the high order convergence in order to avoid the outer iteration in implementation a linearized compact and energy conservative difference scheme is derived numerical examples are given to support the theoretical analysis

a petrov galerkin method for solving the generalized equal width gew equation the generalized equal width gew equation is solved numerically by the petrov galerkin method using a linear hat function as the test function and a quadratic b spline function as the trial function product approximation has been used in this method a linear stability analysis of the scheme shows it to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient finally the maxwellian initial condition pulse is studied

the classification of the single traveling wave solutions to the generalized equal width equation abstract the complete discrimination system for polynomial is applied to the generalized equal width ew equation to obtain the classifications of single traveling wave solutions in particular we construct the corresponding solutions for the concrete parameters to show that each solution in the classification can be realized

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

computing the moments of high dimensional solutions of the master equation derived from the markov character only the master equation of chemical reactions is an accurate stochastic description of quite general systems in chemistry exact solutions of this equation are rare and the most frequently used approximative solution method is to write down the corresponding set of reaction rate equations in many cases this approximation is not valid or only partially so as stochastic effects caused by the natural noise present in the full description of the problem are poorly captured in this paper it is shown how a certain set of higher order equations can be derived it is shown by theory and example that stochastic effects are better captured using this technique while still maintaining the computational advantages of the reaction rate approach

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

a solver for the stochastic master equation applied to gene regulatory networks an important driver of gene regulatory networks is noise arising from the stochastic nature of interactions of genes their products and regulators thus such systems are stochastic and can be modelled by the chemical master equations a major challenge is the curse of dimensionality which occurs when one attempts to integrate these equations while stochastic simulation techniques effectively address the curse many repeated simulations are required to provide precise information about stationary points bifurcation phenomena and other properties of the stochastic processes an alternative way to address the curse of dimensionality is provided by sparse grid approximations the sparse grid methodology is applied and the application demonstrated to work efficiently for up to 10 proteins as sparse grid methods have been developed for the approximation of smooth functions a variant for infinite sequences had to be developed together with a multiresolution analysis similar to haar wavelets error bounds are provided which confirm the effectiveness of sparse grid approximations for smooth high dimensional probability distributions

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

the meshless kernel based method of lines for solving the equal width equation the equal width equation governs nonlinear wave phenomena like waves in shallow water here it is solved numerically by the method of lines using a somewhat unusual setup there is no linearization of the nonlinear terms no error in handling the starting approximation and there are boundary conditions only at infinity to achieve a space discretization of high accuracy with only few trial functions meshless translates of radial kernels are used in the numerical examples the motion of solitary waves the interaction of two and three solitary waves the generation of wave undulation the maxwell initial condition and the clash of two colliding solitary waves are simulated our numerical results compare favourably with results of earlier papers using other techniques

solitary waves of the equal width wave equation a numerical solution of the equal width wave equation based on galerkin s method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves the interaction of two solitary waves is seen to cause the creation of a source for solitary waves usually these are of small magnitude but when the amplitudes of the two interacting waves are equal and opposite the source produces trains of solitary waves whose amplitudes are of the same order as those of the initiating waves the three invariants of the motion are evaluated to determine the conservation properties of the system finally the temporal evolution of a maxwellian initial pulse is studied for small ut uux uxxt 0 only positive waves are formed and the behaviour mimics that of the kdv and filw equations for larger values of d both positive and negative solitary waves are generated

simulations of solitons using quadratic spline finite elements abstract a numerical solution to the korteweg de vries equation is obtained using the bubnov galerkin method with quadratic spline finite elements the element matrices are set up algebraically using reduce a linear stability analysis of the scheme shows the method to be unconditionally stable classical problems concerning the development motion and interaction of solitons are used to validate the method the numerical scheme is compared with published algorithms and shown to be both accurate and efficient

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

stability satisfied numerical approximates to the non analytical solutions of the cubic schrodinger equation abstract the time dependent complex schrodinger equation with cubic non linearity is solved by constructing differential quadrature algorithm based on sinc functions reduction the equation to a coupled system of real equations enables to approximate the space derivative terms by the proposed method the resultant ordinary differential equation system is integrated with respect to the time variable by using various explicit methods of lower and higher orders some initial boundary value problems containing some analytical and non analytical initial data are solved for experimental illustrations the computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solutions exist the two conserved quantities are calculated by using the numerical results in all cases the matrix stability analysis is implemented to control the time step size

collocation method using quartic b spline for the equal width ew equation the equal width ew equation is solved numerically by giving a new algorithm based on collocation method using quartic b splines at the mid knots points as element shape also we use the fourth runge kutta method for solving the system of first order ordinary differential equations instead of finite difference method our test problems including the migration and interaction of solitary waves are used to validate the algorithm which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evaluation of a maxwellian initial pulse is then studied

numerical studies on the split step finite difference method for nonlinear schrodinger equations in this study we use the split step finite difference method to solve various nonlinear schrodinger equations including coupled ones we study the numerical accuracy of the method detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear schrodinger equations

a conservative spectral collocation method for the nonlinear schrdinger equation in two dimensions in this study we present a conservative fourier spectral collocation fsc method to solve the two dimensional nonlinear schrdinger nls equation we prove that the proposed method preserves the mass and energy conservation laws in semi discrete formulations using the spectral differentiation matrices the nls equation is reduced to a system of nonlinear ordinary differential equations odes the compact implicit integration factor ciif method is later developed for the nonlinear odes in this approach the storage and cpu cost are significantly reduced such that the use of ciif method becomes attractive for two dimensional nls equation numerical results are presented to demonstrate the conservation accuracy and efficiency of the method

solitary waves induced by the boundary forced ew equation the development of a train of ew solitary waves induced by boundary forcing is studied using a petrov galerkin finite element scheme with shape functions taken as quadratic b spline functions a linear recurrence relationship for the numerical scheme is obtained via a crank nicholson approach involving a product approximation results are compared with those for the rlw and kdv equations

analytical and computational approaches on solitary wave solutions of the generalized equal width equation abstract in this article firstly numerical solutions of the generalized equal width gew equation have been obtained by a petrov galerkin finite element method using cubic b spline base functions as element shape functions and quadratic b spline base functions as the weight functions in order to prove the practicability and robustness of the numerical algorithm the error norms l2 l and three invariants i1 i2 and i3 are computed a linear stability analysis based on a fourier method states that the numerical scheme is unconditionally stable secondly we have proposed the modified extended tanh function method with the riccati differential equation which is a convenient and an effective method for getting the exact traveling wave solutions of the equation motion of single solitary wave is examined using the present methods the obtained results are indicated both in tabular and graphical form

multi scenario modelling of uncertainty in stochastic chemical systems uncertainty analysis has not been well studied at the molecular scale despite extensive knowledge of uncertainty in macroscale systems the ability to predict the effect of uncertainty allows for robust control of small scale systems such as nanoreactors surface reactions and gene toggle switches however it is difficult to model uncertainty in such chemical systems as they are stochastic in nature and require a large computational cost to address this issue a new model of uncertainty propagation in stochastic chemical systems based on the chemical master equation is proposed in the present study the uncertain solution is approximated by a composite state comprised of the averaged effect of samples from the uncertain parameter distributions this model is then used to study the effect of uncertainty on an isomerization system and a two gene regulation network called a repressilator the results of this model show that uncertainty in stochastic systems is dependent on both the uncertain distribution and the system under investigation

a petrov galerkin method for equal width equation abstract the equal width equation is solved numerically by petrov galerkin method using linear hat function and quadratic b spline function as trial and test functions respectively product approximation has been used in this method a linear stability analysis of the scheme is shown to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient the maxwellian initial condition pulse and the development of an undular bore are also studied

nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales we present an efficient numerical algorithm for simulating chemical kinetic systems with multiple time scales this algorithm is an improvement of the traditional stochastic simulation algorithm ssa also known as gillespie s algorithm it is in the form of a nested ssa and uses an outer ssa to simulate the slow reactions with rates computed from realizations of inner ssas that simulate the fast reactions the algorithm itself is quite general and seamless and it amounts to a small modification of the original ssa our analysis of such multi scale chemical kinetic systems allows us to identify the slow variables in the system derive effective dynamics on the slow time scale and provide error estimates for the nested ssa efficiency of the nested ssa is discussed using these error estimates and illustrated through several numerical examples

hybrid framework for the simulation of stochastic chemical kinetics abstract stochasticity plays a fundamental role in various biochemical processes such as cell regulatory networks and enzyme cascades isothermal well mixed systems can be modelled as markov processes typically simulated using the gillespie stochastic simulation algorithm ssa 25 while easy to implement and exact the computational cost of using the gillespie ssa to simulate such systems can become prohibitive as the frequency of reaction events increases this has motivated numerous coarse grained schemes where the fast reactions are approximated either using langevin dynamics or deterministically while such approaches provide a good approximation when all reactants are abundant the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions become significant this is particularly problematic when using such methods to compute statistics of extinction times for chemical species as well as simulating non equilibrium systems such as cell cycle models in which a single species can cycle between abundance and scarcity in this paper a hybrid jump diffusion model for simulating well mixed stochastic kinetics is derived it acts as a bridge between the gillespie ssa and the chemical langevin equation for low reactant reactions the underlying behaviour is purely discrete while purely diffusive when the concentrations of all species are large with the two different behaviours coexisting in the intermediate region a bound on the weak error in the classical large volume scaling limit is obtained and three different numerical discretisations of the jump diffusion model are described the benefits of such a formalism are illustrated using computational examples

hybrid method for the chemical master equation the chemical master equation is solved by a hybrid method coupling a macroscopic deterministic description with a mesoscopic stochastic model the molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules the macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation the probability distribution is obtained by the stochastic simulation algorithm due to gillespie the equations are coupled via a summation over the mesoscale variables this summation is approximated by quasi monte carlo methods the error in the approximations is analyzed the hybrid method is applied to three chemical systems from molecular cell biology

b spline finite element studies of the non linear schrodinger equation abstract the non linear schrodinger equation is solved numerically by a b spline finite element method the approach used is based on collocation of cubic b splines over spatial finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range time integration of the resulting system of ordinary differential equations is effected using a crank nicolson approximation standard problems are used to validate the algorithm comparisons are made with published numerical and analytical solutions the proposed method exhibits good conservation properties and performs well the development of temporally recurrent states from spatially periodic initial conditions is examined through computer experiments the birth of mobile as well as static solitons out of maxwellian initial data is observed

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

explicit finite difference methods for the ew and rlw equations an extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal width ew and regularized long wave rlw equations is reported such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing two of the methods presented here make use of three point fourth order accurate finite difference formulae for the first and second order spatial derivatives two methods are based on the analytical solution of second order ordinary differential equations which have locally exponential solutions and the fourth technique is a standard finite difference scheme a linear stability analysis of the four methods is presented it is shown that for the ew and rlw equations a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations the latter are reported to be more accurate than exponential techniques that employ second order accurate approximations and these in turn are more accurate than the standard explicit method

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

least squares finite element method for the advection diffusion equation the space time least squares finite element methods are constructed for the advection diffusion equation by using both linear shape functions and quadratic b spline shape functions two test problems are studied to demonstrate the accuracy of the present methods results of the two schemes have been compared

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

solving chemical master equations by adaptive wavelet compression solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods we present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step the step size is chosen adaptively according to estimates of the temporal and spatial approximation errors numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles

an efficient spectral collocation difference method for two dimensional schrodinger equation with neumann boundary conditions abstract based on the second order cosine spectral differentiation matrix deduced in this paper we present an efficient and high accurate numerical scheme for solving two dimensional schrodinger equation with neumann boundary conditions the crank nicolson finite difference scheme is utilized in temporal discretization and the cosine spectral collocation method is employed for the approximation of laplacian operator the new scheme conserves the mass and energy in discrete level and is implemented by a fast algorithm in terms of the relations between the cosine spectral differentiation matrix and fast cosine transform more importantly this strategy can be extended to address the problems with high even order derivatives in space numerical examples and applications are listed to confirm the validity and high accuracy of the method

hybrid discrete continuum algorithms for stochastic reaction networks direct solutions of the chemical master equation cme governing stochastic reaction networks srns are generally prohibitively expensive due to excessive numbers of possible discrete states in such systems to enhance computational efficiency we develop a hybrid approach where the evolution of states with low molecule counts is treated with the discrete cme model while that of states with large molecule counts is modeled by the continuum fokker planck equation the fokker planck equation is discretized using a 2nd order finite volume approach with appropriate treatment of flux components the numerical construction at the interface between the discrete and continuum regions implements the transfer of probability reaction by reaction according to the stoichiometry of the system the performance of this novel hybrid approach is explored for a two species circadian model with computational efficiency gains of about one order of magnitude

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

numerical studies on nonlinear schrodinger equations by spectral collocation method with preconditioning in this study we use the spectral collocation method with preconditioning to solve various nonlinear schrodinger equations to reduce round off error in spectral collocation method we use preconditioning we study the numerical accuracy of the method the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

a numerical solution of the equal width wave equation by a lumped galerkin method in this paper the equal width wave ew equation is solved by a numerical technique based on a lumped galerkin method using quadratic b spline finite elements to investigate the motion of a single solitary wave development of two solitary waves interaction and an undular bore the obtained results are compared with published numerical solutions a linear stability analysis of the method is also investigated

two completely explicit and unconditionally convergent fourier pseudo spectral methods for solving the nonlinear schrodinger equation abstract this paper aims to construct and analyze two new fourier pseudo spectral fps methods for the general nonlinear schrodinger nls equation the two fps methods have two merits unconditional convergence and complete explicitness in the practical computation further more by introducing a modified mass functional and a modified energy functional the two fps methods are proved to preserve the total mass and energy in the discrete sense besides the standard energy method the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique without any restriction on the grid ratio and initial value we establish the optimal error estimate of the two fps methods for solving the general nls equation while previous work just is valid for the cubic nls equation and requires small initial value for the focusing case these two fps methods are proved to be spectrally accurate in space and second order accurate in time respectively the analysis framework can be used to prove the unconditional convergence of many other fourier pseudo spectral methods for solving the nls type equations we investigate the effect of the nonlinear term on the progression simulation of the plane wave the conservation of the invariants and the effect of initial data on the blow up solution via different parameters numerical results are reported to show the accuracy and efficiency of the proposed methods

semi implicit operator splitting pade method for higher order nonlinear schrodinger equations abstract we propose a semi implicit finite difference operator splitting pade ospd method for solving the higher order nonlinear schrodinger equation which describes the optical soliton wave propagation in fibers the method achieves fourth order of accuracy in space and has been proven to be stable by linear stability analysis numerical experiments and comparisons are investigated to show the advantages and effectivity of the ospd method and some interesting collision behaviors are also observed

error estimates of exponential wave integrator fourier pseudospectral methods for the nonlinear schrodinger equation abstract in this article several exponential wave integrator fourier pseudospectra methods are developed for solving the nonlinear schrodinger equation in two dimensions these numerical methods are based on the fourier pseudospectral discretization for spatial derivative first and then utilizing four types of numerical integration formulas including the gautschi type trapezoidal middle rectangle and simpson rules to approximate the time integral in phase space the resulting numerical methods cover two explicit and an implicit schemes and can be implemented effectively thanks to the fast discrete fourier transform additionally the error estimates of two explicit schemes are established by virtue of the mathematical induction and standard energy method finally extensive numerical results are reported to show the efficiency and accuracy of the proposed new methods and confirm our theoretical analysis

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

numerical solution of burgers equation by cubic hermite collocation method abstract in this paper numerical solution of the non linear burgers equation are obtained by using cubic hermite collocation method chcm the advantage of the method is continuity of the dependent variable and its derivative throughout the solution range a linear stability analysis shows that the numerical scheme based on crank nicolson approximation in time is unconditionally stable this method is applied on some test problems with different choice of collocation points to validate the accuracy of the method the obtained numerical results show that the method is efficient robust and reliable even for high reynolds numbers for which the exact solution fails moreover the method can be applied to a wide class of nonlinear partial differential equations

an adaptive wavelet method for the chemical master equation an adaptive wavelet method for the chemical master equation is constructed the method is based on the representation of the solution in a sparse haar wavelet basis the time integration by rothe s method and an iterative procedure which in each time step selects those degrees of freedom which are essential for propagating the solution the accuracy and efficiency of the approach is discussed and the performance of the adaptive wavelet method is demonstrated by numerical examples