solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

volterra method for the radial schrodinger equation a new volterra type integral equation method for the numerical solution of the radial schrodinger equation is investigated the method carried out in configuration space is based on the conversion of differential equations into a system of volterra type integral equations together with the application of a spectral type clenshaw curtis quadrature through numerical examples the volterra type integral equation method is shown to be more efficient and more accurate than the integral equation method based on the fredholm formulation j comput phys 134 1997 134 and better than finite difference methods accompanying c code for the volterra type method is available upon request

on matrices related with fibonacci and lucas numbers in this paper we obtain some new results on matrices related with fibonacci numbers and lucas numbers also we derive the relation between pell numbers and its companion sequence by using our representations

a fast direct solver for quasi periodic scattering problems abstract we consider the numerical solution of the scattering of time harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium in two dimensions boundary integral formulations are ideal since they reduce the problem to n unknowns on the obstacle boundary however for complex geometries and or higher frequencies the resulting dense linear system becomes large ruling out dense direct methods and often ill conditioned despite being 2nd kind rendering fast multipole based iterative schemes also inefficient we present an integral equation based solver with o n complexity which handles such ill conditioning using recent advances in fast direct linear algebra to invert hierarchically the isolated obstacle matrix this is combined with a recent periodizing scheme that is robust for all incident angles including wood s anomalies based upon the free space green s function kernel the resulting solver is extremely efficient when multiple incident angles are needed as occurs in many applications our numerical tests include a complicated obstacle several wavelengths in size with n 10 5 and solution error of 10 10 where the solver is 66 times faster per incident angle than a fast multipole based iterative solution and 600 times faster when incident angles are chosen to share bloch phases

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

navier stokes solver using green s functions i channel flow and plane couette flow numerical solvers of the incompressible navier stokes equations have reproduced turbulence phenomena such as the law of the wall the dependence of turbulence intensities on the reynolds number and experimentally observed properties of turbulence energy production in this article we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher reynolds numbers than is currently possible every time step of a navier stokes solver in effect solves a linear boundary value problem the use of green s functions leads to numerical solvers which are highly accurate in resolving the boundary layer which is a source of delicate but exceedingly important physical effects at high reynolds numbers the use of green s functions brings with it a need for careful quadrature rules and a reconsideration of time steppers we derive and implement green s function based solvers for the channel flow and plane couette flow geometries the solvers are validated by reproducing turbulence phenomena in good agreement with earlier simulations and experiment

monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities abstract a new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities some comparison results are established an existence and enclosing theorem for the solutions is given in terms of upper and lower solutions a new monotone iterative scheme is proposed the numerical example is given

some properties of fuzzy groups on definit des sous groupes flous normaux des sous groupes flous normaux de niveau et leurs homomorphismes et on donne certaines de leurs proprietes

some identities for the generalized fibonacci numbers and the generalized lucas numbers in this paper we study some properties of the generalized fibonacci numbers and the generalized lucas numbers these numbers are equal to the total numbers of k independent sets in special graphs we give some identities for the generalized fibonacci numbers and the generalized lucas numbers which can be useful also in problems of counting of k independent sets in graphs

structured condition number for multiple right hand side linear systems with parameterized quasiseparable coefficient matrix abstract in this paper we consider the structured perturbation analysis for multiple right hand side linear systems with parameterized coefficient matrix especially we present the explicit expressions for structured condition numbers for multiple right hand sides linear systems with 1 1 quasiseparable coefficient matrix in the quasiseparable and the givens vector representations in addition the comparisons of these two condition numbers between themselves and with respect to unstructured condition number are investigated moreover the effective structured condition number for multiple right hand sides linear systems with 1 1 quasiseparable coefficient matrix is proposed the relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the givens vector representations are also studied numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones

some identities of the generalized fibonacci and lucas sequences abstract the purpose of this paper is to study generalized fibonacci and lucas sequences we first introduce generalized lucas sequences section 2 contains a list of elementary relationships about generalized fibonacci and lucas sequences in section 3 we give a generalization of the binet s formulas of generalized fibonacci lucas sequences and its applications section 4 is devote to derive many identities and congruence relations for generalized fibonacci lucas sequences by using operator method

fredholm integral equation method for the integro differential schrodinger equation a new method based on the clenshaw curtis quadrature for the numerical solution of the integro differential schrodinger equation is investigated the method shows that it converges quickly and the truncation errors decrease faster than any power of the inverse number of the chebyshev support points discretization technique is presented in detail accompanying c code for the fredholm type method is available upon request

spectral integration of linear boundary value problems spectral integration was deployed by orszag and co workers 1977 1980 1981 to obtain stable and efficient solvers for the incompressible navier stokes equation in rectangular geometries since then several variations of spectral integration have appeared in the literature in this article we derive yet more versions of spectral integration these new versions of spectral integration rely exclusively on banded matrices as opposed to banded matrices bordered with dense rows in addition we derive a factored form of spectral integration which relies only on bi and tri diagonal matrices key properties such as the accuracy of spectral integration even when green s functions are not resolved by the underlying grid and the accuracy of spectral integration in spite of ill conditioning of underlying linear systems are investigated timed comparisons show that reducing spectral integration to bi and tri diagonal systems leads to significant speed ups

integral equation method for coupled schrodinger equations a new integral equation method for the numerical solution of the radial schrodinger equation in one dimension developed by the authors 1997 j comput phys 134 134 is extended to systems of coupled schrodinger equations with both positive and negative channel energies the method carried out in configuration space is based on the conversion of differential equations into a system of integral equations together with the application of a spectral type clenshaw curtis quadrature an accompanying general multichannel fortran code is available upon request

on k fibonacci numbers of arithmetic indexes in this paper we study the sums of k fibonacci numbers with indexes in an arithmetic sequence say an r for fixed integers a and r this enables us to give in a straightforward way several formulas for the sums of such numbers

fast inversion algorithms for a class of structured operator matrices we study linear complexity inversion algorithms for diagonal plus semiseparable operator matrices applications to integral and differential equations and block matrices are obtained comparison of different algorithms is performed complexity choice of parameters and results of numerical experiments are analyzed

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

on the k generalized fibonacci numbers and high order linear recurrence relations in this paper using elementary symmetric function and complete homogeneous symmetric function we obtain a determinant formula for the k generalized fibonacci sequence the relationship between the k generalized fibonacci sequence and the order k linear homogeneous recurrence relation has been investigated and a general solution for the latter is also derived furthermore we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k generalized fibonacci numbers

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

a common generalization of convolved generalized fibonacci and lucas polynomials and its applications the purpose of this paper is to give a common generalization of convolved generalized fibonacci and lucas polynomials and some recurrence relations are established as applications we obtain some computational formulas of mixed multiple sums for h x fibonacci polynomials and lucas polynomials

on the order k generalized lucas numbers in this paper we give a new generalization of the lucas numbers in matrix representation also we present a relation between the generalized order k lucas sequences and fibonacci sequences

on the representation of k generalized fibonacci and lucas numbers abstract in this paper we give some determinantal and permanental representations of k generalized fibonacci and lucas numbers we obtain the binet s formula for these sequences by using our representations

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

accelerated monotone iterative methods for a boundary value problem of second order discrete equations abstract an accelerated monotone iterative method for a boundary value problem of second order discrete equations is presented this method leads to an existence comparison theorem as well as a computational algorithm for the solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations

a fast chebyshev method for simulating flexible wing propulsion abstract we develop a highly efficient numerical method to simulate small amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid we allow the wing s elastic modulus and mass density to vary arbitrarily with an eye towards optimizing these distributions for propulsive performance the method to determine the wing kinematics is based on chebyshev collocation of the 1d beam equation as coupled to the surrounding 2d fluid flow through small amplitude analysis of the euler equations with trailing edge vortex shedding the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1d wing kinematics a class of semi analytical solutions permits fast evaluation of this operator with o n log n operations where n is the number of collocation points on the wing this is in contrast to the minimum o n 2 cost of a direct 2d fluid solver the coupled wing fluid problem is thus recast as a pde with nonlocal operator which we solve using a preconditioned iterative method these techniques yield a solver of near optimal complexity o n log n allowing one to rapidly search the infinite dimensional parameter space of all possible material distributions and even perform optimization over this space

on the numerical solution of two point boundary value problems ii abstract in a recent paper greengard and rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two point boundary value problems for second order ordinary differential equations in this paper we extend the method to systems of ordinary differential equations after reducing the system of differential equations to a system of second kind integral equations we discretize the latter via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n p squared n cubed operations with n the number of nodes on the interval p the desired order of convergence and n the number of equations in the system thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference of finite element methods we in addition present a newton method for solving boundary value problems for nonlinear first order systems in which each newton iterate is the solution of a second kind integral equation the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems kr

fuzzy groups and group homomorphisms abstract in this paper we study the effect of group homomorphisms on the chains of level subgroups of fuzzy groups we prove necessary and sufficient conditions under which the chains of level subgroups of homomorphic images pre images of an arbitrary fuzzy group can be obtained from that of the fuzzy group these theorems also provide us the method to find the chains of level subgroups of homomorphic images and pre images of arbitrary fuzzy groups

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

homomorphisms of poset valued algebras the notion of a homomorphism for fuzzy algebraic structures having arbitrary possibly different partially ordered sets or lattices as co domains is introduced basic properties of homomorphisms embeddings and isomorphisms are investigated and fundamental homomorphism and isomorphism theorems are proved

generalisations of some results of das abstract in this paper we present generalisations of some results of p s das 1981 to the case of an arbitrary fuzzy group

numerical solution of the orr sommerfeld equation using the viscous green function and split gaussian quadrature we continue our study of the construction of numerical methods for solving two point boundary value problems using green functions building on the successful use of split gauss type quadrature schemes here we adapt the method for eigenvalue problems in particular the orr sommerfeld equation of hydrodynamic stability theory use of the green function for the viscous part of the problem reduces the fourth order ordinary differential equation to an integro differential equation which we then discretize using the split gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators as the latter are only second order the resulting discrete equations are much more stable than those obtained from the original differential equation this permits us to obtain results for the standard test problem plane poiseuille flow at unit streamwise wavenumber and reynolds number 10000 that we believe are the most accurate to date

the integral equation method for a steady kinematic dynamo problem with only a few exceptions the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method however the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics the paper is intended to facilitate the use of integral equation solvers in dynamo theory in concrete the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter for the case of spherical geometry three examples of the function r with steady and oscillatory solutions are considered a convergence rate proportional to the inverse squared of the number of grid points is achieved based on this method a convergence accelerating strategy is developed and the convergence rate is improved remarkably typically quite accurate results can be obtained with a few tens of grid points in order to demonstrate its suitability for the treatment of dynamos in other than spherical domains the method is also applied to x2 dynamos in rectangular boxes the magnetic fields and the electric potentials for the first eigenvalues are visualized

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

volterra type integral equation method for the radial schrodinger equation single channel case a new volterra type integral equation method for the numerical solution of the single channel radial schrodinger equation is investigated the method carried out in configuration space is based on the conversion of differential equations into a system of volterra type integral equations together with the application of a spectral type clenshaw curtis quadrature through numerical examples the volterra type integral equation method is shown to be more efficient and more accurate than the integral equation method based on the fredholm formulation 1 and better than finite difference methods accompanying c code for the volterra type method is available upon request

factorizations and representations of the backward second order linear recurrences we show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second order linear recurrences also considering how to the negatively subscripted terms of second order linear recurrences can be connected to chebyshev polynomials by determinants of these matrices we give factorizations and representations of these numbers

a class of generalized tribonacci sequences applied to counting problems abstract generalized tribonacci numbers with the third order linear recurrence with constant coefficients t k n t k n 1 t k n 2 k t k n 3 for n 2 are investigated for some sets of the initial triples t0 t1 t2 in particular generating functions the binet formula and the limit of ratio of consecutive terms t k n 1 t k n are discussed these numbers are related to numbers of path graphs colorings with k 2 colors or equivalently to counting of q ary sequences of length n for q k 2 satisfying requirements which follow the problem of degeneration in the ising model with the second neighbor interactions it is shown that the results obtained can be considered as the base for considerations of cycle graph colorings cyclic q ary sequences these are counting problems so t0 t1 t2 and k should be natural numbers but these sequences can be considered for any real numbers the special cases k 0 1 lead to the fibonacci and the usual tribonacci numbers respectively so the results can be applied to binary and ternary sequences

existence of fuzzy subgroups of every level cardinality up to a abstract in this note we present two groups which have fuzzy subgroups of any desired finite level cardinality we also give an example of a fuzzy subgroup having infinite level cardinality

nystrom clenshaw curtis quadrature for integral equations with discontinuous kernels a new highly accurate numerical approximation scheme based on a gauss type clenshaw curtis quadrature for fredholm integral equations of the second kind x t ab k t s x s ds y t whose kernel k t s is either discontinuous or not smooth along the main diagonal is presented this scheme is of spectral accuracy when k t s is infinitely differentiable away from the diagonal t s relation to the singular value decomposition is indicated application to integro differential schrodinger equations with nonlocal potentials is given

a levinson like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices in this paper we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix the solver we will derive is based on the levinson algorithm which is used for solving strongly nonsingular toeplitz systems in the first part an o p2n solver for a semiseparable matrix of semiseparability rank p is derived and in a second part we derive an o l2n solver for a band matrix with bandwidth 2l 1 both solvers are constructed in a similar way firstly a yule walker like equation needs to be solved and secondly this solution is used for solving a linear equation with an arbitrary right hand side finally a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices the overall complexity of this solver is 6 l p 2n plus lower order terms in the final section numerical experiments are performed attention is paid to the timing and the accuracy of the described methods

the generalized order k fibonacci pell sequence by matrix methods in this paper we consider the usual and generalized order k fibonacci and pell recurrences then we define a new recurrence which we call generalized order k f p sequence also we present a systematic investigation of the generalized order k f p sequence we give the generalized binet formula some identities and an explicit formula for sums of the generalized order k f p sequence by matrix methods further we give the generating function and combinatorial representations of these numbers also we present an algorithm for computing the sums of the generalized order k pell numbers as well as the pell numbers themselves

truncated lucas sequence and its period abstract we prove some theorems concerning the truncated lucas sequences we also state some conjectures concerning the period mod m of truncated lucas sequences where the integer m is at least 2 in addition we present some computer generated results that tend to support these conjectures

a fast direct solver for boundary value problems on locally perturbed geometries abstract many applications including optimal design and adaptive discretization techniques involve solving several boundary value problems on geometries that are local perturbations of an original geometry this manuscript presents a fast direct solver for boundary value problems that are recast as boundary integral equations the idea is to write the discretized boundary integral equation on a new geometry as a low rank update to the discretized problem on the original geometry using the sherman morrison formula the inverse can be expressed in terms of the inverse of the original system applied to the low rank factors and the right hand side numerical results illustrate for problems where perturbation is localized the fast direct solver is three times faster than building a new solver from scratch

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

an integral equation approach to the incompressible navier stokes equations in two dimensions we present a collection of methods for solving the incompressible navier stokes equations in the plane that are based on a pure stream function formulation the advantages of this approach are twofold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided the disadvantage is that the solution of a nonlinear fourth order partial differential equation is required by recasting this partial differential equation as an integral equation we avoid the ill conditioning which hampers finite difference and finite element methods in this environment by using fast algorithms for the evaluation of volume integrals we are able to solve the equations using o m or o m log m operations where m is the number of points in the discretization of the domain

bivariate fibonacci like p polynomials abstract in this article we study the bivariate fibonacci and lucas p polynomials p 0 is integer from which specifying x y and p bivariate fibonacci and lucas polynomials bivariate pell and pell lucas polynomials jacobsthal and jacobsthal lucas polynomials fibonacci and lucas p polynomials fibonacci and lucas p numbers pell and pell lucas p numbers and chebyshev polynomials of the first and second kind are obtained afterwards we obtain some properties of the bivariate fibonacci and lucas p polynomials

integral equation method for the continuous spectrum radial schrodinger equation a new approach to the numerical solution of boundary value problems for differential equations which originated in recent papers by greengard and rokhlin is improved and adapted to the numerical solution of the radial schrodinger equation the approach is based on the conversion of the differential equation into an integral equation together with the application of a spectral type clenshaw curtis quadrature method through numerical examples the integral equation method is shown to be superior to finite difference methods