multigrid and adaptive algorithm for solving the nonlinear schrodinger equation abstract in this paper a conservative difference scheme for generalized nonlinear schrodinger equations is given we apply multigrid method and adaptive algorithm to solve the equations numerical results are presented and compared they demonstrate that the multigrid and adaptive algorithm are efficient and can considerably relax the restrict on step size of time which is caused by nonlinear iteration

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

connection coefficients between boas buck polynomial sets abstract in this paper a general method to express explicitly connection coefficients between two boas buck polynomial sets is presented as application we consider some generalized hypergeometric polynomials from which we derive some well known results including duplication and inversion formulas

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

difference schemes for solving the generalized nonlinear schrodinger equation this paper studies finite difference schemes for solving the generalized nonlinear schrodinger gnls equationiut uxx q u 2 u f x t u a new linearlized crank nicolson type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the gnls equation several schemes including crank nicolson type schemes hopscotch type schemes split step fourier scheme and pseudospectral scheme are adopted for solving three model problems of gnls equation which arise from many physical problems withq s s2 q s ln 1 s andq s 4s 1 s respectively the numerical results demonstrate that the linearized crank nicolson scheme is efficient and robust

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

exponential time differencing crank nicolson method with a quartic spline approximation for nonlinear schrodinger equations abstract this paper studies a central difference and quartic spline approximation based exponential time differencing crank nicolson etd cn method for solving systems of one dimensional nonlinear schrodinger equations and two dimensional nonlinear schrodinger equations a local extrapolation is employed to achieve a fourth order accuracy in time the numerical method is proven to be highly efficient and stable for long range soliton computations numerical examples associated with dirichlet neumann and periodic boundary conditions are provided to illustrate the accuracy efficiency and stability of the method proposed

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

connection problems via lowering operators the lowering operator associated with a polynomial set pn n 0 is an operator not depending on n and satisfying the relation pn npn 1 in this paper we express explicitly the connection coefficients between two polynomial sets using their corresponding lowering operators we obtain some well known results as particular cases including some duplication and addition formulas

higher order exponential time differencing scheme for system of coupled nonlinear schrodinger equations the coupled nonlinear schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics like propagation of pulses efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system in this paper we introduce a new version of the cox and matthews third order exponential time differencing runge kutta etd3rk scheme based on the 1 2 pade approximation to the exponential function in addition we present its local extrapolation form to improve temporal accuracy to the fourth order the developed scheme and its extrapolation are seen to be strongly stable which have ability to damp spurious oscillations caused by high frequency components in the solution a computationally efficient algorithm of the new scheme based on a partial fraction splitting technique is presented in order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear schrodinger equations and performed several numerical experiments on them the numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long range solitary solutions given by coupled nonlinear schrodinger equations and conserved the conserved quantities mass and energy exactly to at least five decimal places

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

linearization coefficients for boas buck polynomial sets abstract in this paper we express explicitly the linearization coefficients related to three boas buck polynomial sets using their corresponding generating functions we apply the obtained results to many classes of polynomials including some generalized hypergeometric polynomials the corresponding linearization coefficients will be expressed by means of terminating generalized kampe de feriet functions which for some cases can be reduced to hypergeometric functions or hypergeometric terms using with this purpose some known reduction formulae we obtain some well known results as particular cases as well as some new reduction formulae

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

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

analytic structure of solutions of the one dimensional burgers equation with modified dissipation we use the one dimensional burgers equation to illustrate the effect of replacing the standard laplacian dissipation term by a more general function of the laplacian of which hyperviscosity is the best known example in equations of hydrodynamics we analyze the asymptotic structure of solutions in the fourier space at very high wave numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing reynolds numbers from algebraic recursion relations involving iterated integrations we give a detailed analysis of their analytic structure for two different types of dissipation a hyperviscous and an exponentially growing dissipation term our results obtained in the limit of vanishing reynolds numbers are validated by high precision numerical simulations at non zero reynolds numbers we then study the bottleneck problem an intermediate asymptotics phenomenon which in the case of the burgers equation arises when ones uses dissipation terms such as hyperviscosity growing faster at high wave numbers than the standard laplacian dissipation term a linearized solution of the well known boundary layer limit of the burgers equation involving two numerically determined parameters gives a good description of the bottleneck region

d orthogonality of hermite type polynomials abstract in this paper we state a characterization theorem for d orthogonal polynomials of hermite type as application we solve two characterization problems related to gould hopper polynomials and charlier type polynomials various properties of the resulting generalized hypergeometric polynomials are singled out in particularly we derive the d dimensional functional vectors ensuring the d orthogonality of these polynomials the obtained weight functions are reduced to the well known weight functions of charlier and hermite polynomials for d 1 we also explicitly express the d dimensional functional vector associated with polynomials of laguerre type which are related to the obtained hermite type polynomials

operational rules and a generalized hermite polynomials in this paper we use operational rules associated with three operators corresponding to a generalized hermite polynomials introduced by szego to derive as far as we know new proofs of some known properties as well as new expansions formulae related to these polynomials

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

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

some connection and linearization problems for polynomials in and beyond the askey scheme the connection problem is considered in a hypergeometric function framework for i the two most general families of polynomials belonging to the askey scheme wilson and racah and ii some generalized laguerre and jacobi polynomials falling outside that scheme sister celine cohen and prabhakar jain which are relevant to the study of quantum mechanical systems and include as particular cases the generalizations of the classical families with sobolev type orthogonality in addition using the same method three new linearization like formulae for the gegenbauer polynomials are also derived a linearization formula that generalizes the m n case of dougall s formula the analogue of the m n case of nielsen s inverse linearization formula for hermite polynomials and a connection formula for the squares closed analytical formulae for the corresponding connection and linearization coe cients are given in terms of hypergeometric functions of unit argument which at times can be further simpli ed and expressed as single hypergeometric terms c 2001 published by elsevier science b v

an element free based solution for nonlinear schrodinger equations using the icvmls ritz method an improved complex variable moving least squares ritz icvmls ritz method is proposed for predicting numerical solution of the two dimensional nonlinear schrodinger equation in this element free solution procedure the icvmls approximation is employed to reduce the number of unknown coefficients in the trial function it follows by the ritz procedure to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear schrodinger equation the central differencing scheme and newton s algorithm are adopted to solve the nonlinear equation system numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element free icvmls ritz method by comparing the computed results with the available analytical solutions

generating functions via integral transforms abstract in this paper we use some integral transforms to derive for a polynomial sequence p n x n 0 generating functions of the type g x t n 0 n p n x t n starting from a generating function of type g x t n 0 p n x t n where n n 0 is a real numbers sequence independent on x and t that allows us to unify the treatment of a generating function problem for many well known polynomial sequences in the literature

conservation of phase space properties using exponential integrators on the cubic schrodinger equation the cubic nonlinear schrodinger nls equation with periodic boundary conditions is solvable using inverse spectral theory the nonlinear spectrum of the associated lax pair reveals topological properties of the nls phase space that are difficult to assess by other means in this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach the initial condition used is a perturbation of the unstable plane wave solution which is difficult to numerically resolve our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step while a lower order multisymplectic is not as accurate and too slow to compete

complex space singularities of 2d euler flow in lagrangian coordinates we show that for 2d space periodic incompressible flow the solution can be evaluated numerically in lagrangian coordinates with the same accuracy that is achieved in standard eulerian spectral methods this allows the determination of complex space lagrangian singularities lagrangian singularities are found to be closer to the real domain than eulerian singularities and seem to correspond to fluid particles which escape to complex infinity by the current time various mathematical conjectures regarding eulerian lagrangian singularities are presented

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

calculation of complex singular solutions to the 3d incompressible euler equations abstract this paper presents numerical computations of complex singular solutions to the 3d incompressible euler equations the euler solutions found here consist of a complex valued velocity field u that contains all positive wavenumbers u satisfies the usual euler equations but with complex initial data the real valued velocity u u u where u u is an approximate singular solution to the euler equations under moore s approximation the method for computing singular solutions is an extension of that in caflisch 1993 25 for axisymmetric flow with swirl but with several improvements that prevent the extreme magnification of round off error which affected previous computations this enables the first clean analysis of the singular surface in three dimensional complex space we find singularities in the velocity field of the form u 1 for near 3 2 and u log where 0 denotes the singularity surface the logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations

some results on quasi monomiality a polynomial set p n n 0 is called quasi monomial if and only if it is possible to define two operators p and m independent of n such thatp p n x np n 1 x andm p n x p n 1 x in this paper we show that every polynomial set is quasi monomial and we present some useful tools to explicitly express the p and m operators for some polynomial families given by their generating functions the obtained results are applied to boas buck polynomial sets

results for some inversion problems for classical continuous and discrete orthogonal polynomials explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials charlier meixner krawtchouck hahn hahn eberlein into the falling factorial basis are given the corresponding inversion problems are solved explicitly this is done by using a general algorithm recently developed by the authors which is also applied to this kind of inversion problem but relating the basis and the classical continuous orthogonal polynomials of jacobi laguerre hermite and bessel

solving the generalized nonlinear schrodinger equation via quartic spline approximation this paper is concerned with a new conservative finite difference method for solving the generalized nonlinear schrodinger gnls equation iut uxx f u 2 u 0 the numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation central difference and extrapolation formulae are used for approximating the neumann boundary conditions introduced both continuous and discrete energy conservation and the stability property are investigated the numerical method provides an efficient and reliable way for computing long time solitary solutions given by the gnls equation numerical examples are given to demonstrate our conclusions

on linearization and connection coefficients for generalized hermite polynomials abstract we consider the problem of finding explicit formulas recurrence relations and sign properties for both connection and linearization coefficients for generalized hermite polynomials most of the computations are carried out by the computer algebra system maple using appropriate algorithms

some remarks on li sinai type solutions of the burgers equation for a class of complex valued initial conditions we construct solutions for the inviscid burgers equations of the same type as those recently introduced by li and sinai obtained by very simple considerations these solutions correspond to stable distributions with 1 2 we furthermore show that solutions of non gaussian type can be constructed for the viscous burgers equation

the use of quartic splines in the numerical solution of a fourth order boundary value problem abstract this paper derives via uniform quartic polynomial splines a few new consistency relations connecting the quartic spline function values at midknots and the corresponding values of the second and fourth derivatives respectively it is shown how these relations may be used in an algorithm for computing approximations to the solution and its higher derivatives for a linear fourth order boundary value problem two numerical illustrations are also included to demonstrate the practical usefulness of our algorithm

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

solving connection and linearization problems within the askey scheme and its q analogue via inversion formulas for the polynomial families pn x n belonging to the askey scheme or to its q analogue the hypergeometric represen tation provides a natural expansion of the form pn x n 0 dm n m x where the expanding basism x is in general a product of pochhammer symbols or q shifted factorials in this paper we solve the corresponding inversion problem i e we compute the coe6cients im n in the expansionn x n m 0 im n pm x which are then used as a tool for solving any connection and linearization problem within the askey scheme and its q analogue extensions of this approach for polynomials outside these two schemes are also given c

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

local discontinuous galerkin methods for nonlinear schrodinger equations in this paper we develop a local discontinuous galerkin method to solve the generalized nonlinear schrodinger equation and the coupled nonlinear schrodinger equation l 2 stability of the schemes are obtained for both of these nonlinear equations numerical examples are shown to demonstrate the accuracy and capability of these methods

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

difference equations of q appell polynomials in this paper we study some properties of the q appell polynomials including the recurrence relations and the q difference equations which extend some known classical q 1 results we also provide the recurrence relations and the q difference equations for q bernoulli polynomials q euler polynomials q genocchi polynomials and for q hermite polynomials as special cases of q appell polynomials

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly