on solving systems of multi pantograph equations via spectral tau method abstract the current manuscript focuses on solving systems of multi pantograph equations the spectral tau method is applied for solving systems of multi pantograph equations with shifted jacobi polynomials as basis functions the convergence analysis of the proposed technique is also investigated we introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods

on the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations an analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by phillips and karageorghis 27 1990 823 a new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given the tensor product of ultraspherical polynomials is used to approximate a function of more than one variable formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

an accurate solution of parabolic equations by expansion in ultraspherical polynomials abstract an ultraspherical expansion technique is applied to numerically obtain the solution of the third boundary value problem for a linear partial differential equation in a one space variable the method depends basically on the fact that an expansion in a series of ultraspherical polynomials c n for the function and its space derivatives occuring in the partial differential equation is assumed the coefficients of expansion are then determined by reducing the partial differential equation with its boundary and initial conditions to a system of ordinary differential equations for these coefficients this system may be solved analytically or numerically in a step by step manner the expansion for the function enables one to get the sought for approximation for any possible value of the parameter 1 2 numerical experiments show that more accurate results may be obtained by taking to be small and negative the method in its present form may be considered as a generalization of that of dew and scraton the extension of the method to the polar type equations u t p 2 u x 2 k u x x is also considered

the coefficients of differentiated expansions and derivatives of ultraspherical polynomials abstract a formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by karageorghis and phillips in their recent report 5 formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given the special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are considered an application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described

on the connection coefficients and recurrence relations arising from expansions in series of laguerre polynomials a formula expressing the laguerre coefficients of a general order derivative of an infinitely differentiable function in terms of its original coefficients is proved and a formula expressing explicitly the derivatives of laguerre polynomials of any degree and for any order as a linear combination of suitable laguerre polynomials is deduced a formula for the laguerre coefficients of the moments of one single laguerre polynomial of certain degree is given formulae for the laguerre coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its laguerre coefficients are also obtained a simple approach in order to build and solve recursively for the connection coefficients between jacobi laguerre and hermite laguerre polynomials is described an explicit formula for these coefficients between jacobi and laguerre polynomials is given of which the ultra spherical polynomials of the first and second kinds and legendre polynomials are important special cases an analytical formula for the connection coefficients between hermite and laguerre polynomials is also obtained

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

a note on the chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function an expression for the chebyshev coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary bessel polynomials abstract a formula expressing explicitly the derivatives of ordinary bessel polynomials of any degree and for any order in terms of the ordinary bessel polynomials themselves is proved another explicit formula which expresses the ordinary bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original ordinary bessel coefficients is also given a formula for the ordinary bessel coefficients of the moments of one single ordinary bessel polynomial of certain degree is proved formula for the ordinary bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ordinary bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between generalized bessel and ordinary bessel polynomials is described explicit formula for these coefficients between jacobi and ordinary bessel polynomials is given of which the ultraspherical polynomials and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre ordinary bessel and hermite ordinary bessel are also developed

a fast and well conditioned spectral method a spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions the method leads to matrices that are al

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

shifted jacobi spectral galerkin method for solving fractional order initial value problems abstract in this paper the shifted jacobi spectral galerkin method is introduced to deal with fractional order initial value problems fivps in the proposed method the exact solution of the fivp is approximated by using shifted jacobi polynomials on each subinterval of the total time the main advantage of the proposed method is that the rate of convergence in l 2 norm depends on the local smoothness of solution this enables the proposed method to work for nonsmooth solutions the second merit is that the flexibility between the length of sub interval and the degree of polynomial significantly enhances the numerical accuracy we derive the spectral galerkin approximation formula for the volterra integral equation of the second kind which is equivalent to the fivp error analysis in l 2 norm for the case 0 in shifted jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is lipschitz continuous in the second argument numerical illustrations for multi order linear and nonlinear fivps are demonstrated to confirm the convergence rate

jacobi spectral galerkin method for the integrated forms of second order differential equations abstract efficiently direct solvers based on the jacobi galerkin method for the integrated forms of second order elliptic equations in one and two space variables are presented they are based on appropriate base functions for the galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted the homogeneous dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the jacobi polynomials the direct solution algorithm here developed for the homogeneous dirichlet problem in two dimensions relies upon a diagonalization process numerical results indicating the high accuracy and effectiveness of these algorithms are presented

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

interface stabilised finite element method for moving domains and free surface flows abstract a recently developed stabilised finite element method which draws upon features of both continuous and discontinuous finite element methods is generalised for the incompressible navier stokes equations on moving domains and for free surface problems the approach allows the natural incorporation of upwinding of the advective flux and the formulation is stable when using equal order lagrange basis functions for the velocity and pressure fields particular attention is paid to the formulation of a fractional step algorithm for moving domain problems numerous applications to environmental free surface flows are presented ranging from simple test cases to simulations of sophisticated laboratory experiments it is shown that the formulation which contains no flow dependent stabilisation parameters is able to model a wide range of flow conditions stably without user intervention and it is shown that negligible numerical dissipation is introduced the physics of the considered numerical examples involves sensitive free surface wave behaviour which has been accurately captured due to the stability of the model together with negligible introduced numerical dissipation

the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function abstract a formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given formulae for the ultraspherical coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ultraspherical coefficients are also obtained the corresponding formulae for the important special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are deduced two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients in the sense of lewanowicz are discussed comparisons with the results obtained by optimal algorithm of lewanowicz 1976 are also made

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

recurrence relations for connection coefficients between q orthogonal polynomials of discrete variables in the non uniform lattice x s q2s we obtain the structure relations for q orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q orthogonal polynomials an explicit example is also given

efficient spectral petrov galerkin methods for the integrated forms of third and fifth order elliptic differential equations using general parameters generalized jacobi polynomials abstract this article analyzes some algorithms for solving numerically the integrated forms of third and fifth order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual petrov galerkin method two new families of general parameters generalized jacobi polynomials are introduced and used for this purpose numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented

shifted jacobi spectral galerkin method for solving hyperbolic partial differential equations abstract herein we propose a numerical scheme to solve spectrally hyperbolic partial differential equations hpdes using galerkin method and approximate the solutions using double shifted jacobi polynomials the main characteristic behind this approach is that it reduces such problems to those of solving systems of algebraic equations which greatly simplifies the problem the validity and efficiency of the proposed method are investigated and verified through several examples

efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of al salam carlitz i polynomials two formulae expressing explicitly the derivatives and moments of al salam carlitz i polynomials of any degree and for any order in terms of al salam carlitz i themselves are proved two other formulae for the expansion coefficients of general order derivatives d p f x and for the moments xd p f x of an arbitrary function f x in terms of its original expansion coefficients are also obtained application of these formulae for solving q difference equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between al salam carlitz i polynomials and any system of basic hypergeometric orthogonal polynomials belonging to the q hahn class is described

a new algorithm for the solution of nonlinear two dimensional volterra integro differential equations of high order abstract in this manuscript a new numerical approach is developed for approximating the solution of the one dimensional and two dimensional volterra integro differential equations 2d vides of high order new operational matrices of integration based on shifted jacobi polynomials are presented and used in combination with the collocation method to convert the 2d vide into a system of algebraic equations in addition the convergence analysis of the suggested approach is investigated the efficiency of the algorithm is demonstrated by means of several examples and the results are compared with those given using other numerical schemes

efficient spectral galerkin algorithms for direct solution of second order equations using ultraspherical polynomials it is well known that spectral methods tau galerkin collocation have a condition number of o n4 n is the number of retained modes of polynomial approximations this paper presents some efficient spectral algorithms which have a condition number of o n2 based on the ultraspherical galerkin methods for second order elliptic equations in one and two space variables the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with specially structured matrices that can be efficiently inverted the complexities of the algorithms are a small multiple of nd 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions

an accurate solution of the poisson equation by the chebyshev tau method abstract a new tau method is presented for the two dimensional poisson equation comparison of the results for the test problem u x y sin 4 x sin 4 y with those computed by haidvogel and zang using the matrix diagonalization method and dang vu and delcarte using the chebyshev collocation method indicates that our method would be more accurate

quick construction of recurrence relations for the jacobi coefficients abstract recurrence relations are given for the jacobi series coefficients of functions which satisfy linear differential equations of first to third order with polynomial coefficients

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques