on particular solution of ordinary differential equations with constant coefficients in this paper a new method is presented for obtaining the particular solution of ordinary differential equations with constant coefficients an explicit formula of the particular solution is derived from the use of an upper triangular toeplitz matrix the validity of the approach is illustrated by some examples

an algebraic method to solve the radial schrodinger equation we propose a method of numerical integration of differential equations of the type x 2y f x y 0 by approximating its solution with solutions of equations of the type x 2y ax 2 bx c y 0 this approximation is performed by segmentary approximation on an interval we apply the method to obtain approximate solutions of the radial schrodinger equation on a given interval and test it for two different potentials we conclude that our method gives a similar accuracy than the taylor method of higher order

a new spectral method for a class of linear boundary value problems in this paper we use bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems the proposed method has been applied to several bvps with dirichlet neumann and mixed types boundary conditions numerical examples provide favorable comparisons with other existing methods

performance of the taylor series method for odes daes this paper revisits the use of the taylor series method for the numerical integration of odes and daes the numerical method is implemented using an efficient variable step variable order scheme several numerical tests comparing with well established numerical codes are presented

moment balancing templates for d k constrained codes and run length limited sequences the first order moment of d k constrained codes is investigated in this paper a generalized moment balancing template is proposed to encode a d k sequence into a single insertion or deletion correcting codeword without losing the constraint property by relocating 0 s in moment balancing runs which appear in a pairwise manner of a d k sequence the first order moment of this sequence can be modified to satisfy the varshamov tenengolts construction with a reasonably large base in the modulo system introduced by the varshamov tenengolts construction this generalized moment balancing template can be applied to run length limited sequences the asymptotic bound of the redundancy introduced by the template for d k sequences is of the same order as the universal template for random sequences and therefore the redundancy is small and suitable for long sequences of practical interest

nonlinear scattering theory at low energy sequel abstract we extend the nonlinear scattering theory and apply it to schrodinger and klein gordon equations with various kinds of nonlinear terms including some nonlocal ones

an approximation to solutions of linear ode by cubic interpolation we present a method of integration for non autonomous non homogeneous systems of linear ordinary differential equation ode which is based in both the cubic polynomial segmentary interpolation and the minimal square method this method is valid for nonhomogeneous ordinary linear second order differential equations in the neighborhood of regular and singular regular points we illustrate the method with the mathieu and bessel equations and two other equations that arise in the study of quantum systems with axial symmetry which are versions of the spheroidal wave equation

higher order numeric wazwaz el sayed modified adomian decomposition algorithms in this paper we develop new numeric modified adomian decomposition algorithms by using the wazwaz el sayed modified decomposition recursion scheme and investigate their practicality and efficiency for several nonlinear examples we show how we can conveniently generate higher order numeric algorithms at will by this new approach including by using examples 12th order and 20th order numeric algorithms furthermore we show how we can achieve a much larger effective region of convergence using these new discrete solutions we also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th order numeric modified decomposition algorithm over the classic 4th order runge kutta algorithm by example the efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by duan for generation of the adomian polynomials to high orders

on a semilinear wave equation the cauchy problem and the asymptotic behavior of solutions abstract the semilinear wave equation u m 2 u u p 2 u v u p 0 in r 3 is studied where denotes the d alembertian operator and means spatial convolution under mild assumptions on the real valued function v and 2 p 3 the well posedness of the cauchy problem is proved furthermore some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as t in the case of zero mass our results extend that of perla menzala and strauss where case p 2 was studied

a mechanical algorithm for solving ordinary differential equation in this paper by using the theories and methods of mathematics analysis ordinary differential equation and computer algebra a reliable algorithm for solving ordinary differential equation ode was established and a new maple procedure mechodeproc was established too the results of the examples indicated that the procedure mechodeproc based on taylor polynomial method was simple and effective and could provide an accuracy approximate solution or exact solution of the ordinary differential equation this would be useful for solving differential equation

a numerical method for solving linear non autonomous systems based on linear programming in this article we propose an approximate solution method for solving linear non autonomous systems with an initial condition in this method the initial value problem is transformed into a linear programming problem to obtain an approximate solution the existence and uniqueness criterion for solution is also proved in a linear programming manner finally we propose a method to find the fundamental matrix of these systems numerical examples are considered

a finite element approach for solution of burgers equation in this study for the burgers equation which is used a model problem in turbulence and shock wave theory was given a finite element solution considering the numerical results obtained for different values of viscosity compared with cole s analytical solution it was seen that they were in good agreement

a new algorithm for symbolic integral with application in this paper by using the theories and methods of mathematical analysis and computer algebra a new algorithm of symbolic integral was established a new maple procedure myint for it was established too and it was applied to the automatic proving the mean value theorems for derivatives the results indicated that the algorithm of symbolic integral had been established in this paper had advantage of simple idea excellent property for operation and powerful competence this would be useful for the problem of solving differential equations automatic proving some mathematical theorems and so on

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

spline algorithms for the hartree fock equation for the helium ground state spline algorithms are evaluated for the non linear integro differential equation describing the hartree fock approximation for the he 1s21s ground state the error in the energy decreases as h2k 2 where h is a grid parameter and k is the order of the spline it is shown that for higher order splines the method is fast and accurate and contrary to the conclusion reached by altenberger siczek and gilbert that spline methods are suitable for scf atomic structure calculations accuracy and timing studies are presented as well as comparisons with other accurate procedures

a numerical solution of burgers equation based on least squares approximation abstract burgers equation which is one dimensional non linear partial differential equation was converted to p non linear ordinary differential equations by using the method of discretization in time each of them was solved by the least squares method for various values of viscosity at different time steps the numerical solutions obtained were compared with the exact solutions it was seen that both of them were in excellent agreement while the exact solution was not available for viscosity smaller than 0 01 it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay

numerical solution of functional differential integral and integro differential equations this paper describes a numerical method based on lagrange interpolation and chebyshev interpolation to treat functional integral equations of volterra type and fredholm type also the method can be extended to functional differential and integro differential equations various numerical examples are treated

numerical solution of kdv equation using modified bernstein polynomials abstract here we present an algorithm for approximating numerical solution of korteweg de vries kdv equation in a modified b polynomial basis a set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions for the time variable the system of equations is solved using fourth order runge kutta method the accuracy of the solutions is dependent on the size of the b polynomial basis set we have presented our numerical result with an exact analytical result excellent agreement is found between exact and approximate solutions this procedure has a potential to be used in more complex system of differential equations where no exact solution is available

numerical solutions of eighth order bvp by the galerkin residual technique with bernstein and legendre polynomials in this paper galerkin method with bernstein and legendre polynomials as basis functions is used for the numerical solutions of eighth order linear and nonlinear differential equations for two different cases of boundary conditions in this method the basis functions are transformed into a new set of basis functions which vanish at the boundary where the essential types of boundary conditions are defined and a matrix formulation is derived for solving the eighth order boundary value problems bvps the numerical results obtained by the proposed method reflect that the performance of the present method is reliable more efficient and converges monotonically to the exact solutions

an algorithm for solving a nonlinear integro differential equation in this paper an algorithm based on adomian s decomposition method is developed to approximate the solution of the nonlinear integro differential equationu t x t email protected t 0a email protected email protected email protected u x x t email protected f x t 0

regular article integration by cell algorithm for slater integrals in a spline basis an algorithm for evaluating slater integrals in a b spline basis is introduced based on the piecewise property of the b splines the algorithm divides the two dimensional r 1 r 2 region into a number of rectangular cells according to our chosen grid and implements the two dimensional integration over each individual cell using gaussian quadrature over the off diagonal cells the integrands are separable so that each two dimensional cell integral is reduced to a product of two one dimensional integrals furthermore the scaling invariance of the b splines in the logarithmic region of the chosen grid is fully exploited such that only some of the cell integrations need to be implemented the values of given slater integrals are obtained by assembling the cell integrals this algorithm significantly improves the efficiency and accuracy of the traditional method that relies on the solution of differential equations and renders the b spline method more effective when applied to multi electron atomic systems

the best linear approximation for nonlinear systems in this paper we propose a method for finding the best linearization of nonlinear ode problems by l 2 norm for this means we try to obtain the best approximation of a nonlinear system as a linear time varying system our method for obtaining the above approximation is on the basis of analytic and numerical approaches finally we examine our methods to some numerical examples

a numerical solution of burgers equation in this paper the burgers equation which is one dimensional quasi linear parabolic partial differential equation was solved by a variational method constructed on the method of discretization in time the numerical results obtained by these ways for various values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

quadratic hamiltonians in phase space quantum mechanics the dynamical evolution is described within the phase space formalism by means of the moyal propagator which is the symbol of the evolution operator quadratic hamiltonians on the phase space are distinguished in that their moyal bracket with any function equals their poisson bracket it is shown that for general time independent quadratic hamiltonians the moyal propagators transform covariantly under linear canonical transformations they are then derived and classified in a fully explicit manner using the theory of hamiltonian normal forms the authors present several tables of propagators it is proved that these propagators belong to the moyal algebra of distributions and that the spectrum of the hamiltonian may be obtained directly as the support of the fourier transform of the moyal propagator with respect to time from that the quantum mechanical problem for these hamiltonians is in principle completely solved the appropriate path integral formalism for phase space quantum mechanics leading back to the same results is outlined in an appendix

an algorithm for solving daes with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving differential algebraic equation systems was established and a new maple procedures mainproc was established too ten examples are presented to illustrate the implementation of the program

on a wave equation with a cubic convolution abstract we study the well posedness of the cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation u tt u m 2 u v u 2 0 in euclidean space

classical and quantum three dimensional integrable systems with axial symmetry we study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection we assume that the three constants of motion are the hamiltonian h with the standard form of a kinetic part plus a potential depending on the position only the z component of the angular momentum l and a hamiltonian like constant h for which the kinetic part is quadratic in the momenta we find the explicit form of these potentials compatible with complete integrability the classical equations of motion written in terms of two arbitrary potential functions are separated in oblate spheroidal coordinates the quantization of such systems leads to a set of two differential equations that can be presented in the form of spheroidal wave equations

a multiple insertion deletion correcting code for run length limited sequences a code construction is proposed to add a multiple insertion deletion error correcting capability to a run length limited sequence the codewords of this code are themselves run length limited the insertion deletion correcting capability is achieved by requiring several weighted sums of run lengths in the codewords to satisfy certain congruences modulo primes the construction is similar to the number theoretic code proposed by dolecek and anantharam which can correct multiple repetition errors or equivalently multiple insertions of zeros it is shown that if the codewords in this code are run length limited then the code is capable of correcting both insertions and deletions of zeros and ones an algorithm is proposed for decoding over a multiple insertion deletion channel following the work of dolecek and anantharam a systematic encoding method is also proposed for the codes furthermore it is shown that the proposed construction has a higher rate asymptotically than the helberg code which is unconstrained in terms of run lengths even though our construction has the additional run length constraints the need for run length limited codes that can correct insertion deletion errors is motivated by bit patterned media for magnetic recording

an algorithm for solving nonlinear singular perturbation problems with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving nonlinear singular perturbation problems was established a new maple procedure spp was established too the procedure spp give not only the approximate analytic solutions but also the numerical solutions of spp six examples are presented to illustrate the implementation of the program the method is very simple and effective for most of singular perturbation problems this would be useful for solving differential equations

solutions of differential equations in a bernstein polynomial basis an algorithm for approximating solutions to differential equations in a modified new bernstein polynomial basis is introduced the algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the galerkin method to determine the expansion coefficients to construct a solution matrix formulation is used throughout the entire procedure however accuracy and efficiency are dependent on the size of the set of bernstein polynomials and the procedure is much simpler compared to the piecewise b spline method for solving differential equations a recursive definition of the bernstein polynomials and their derivatives are also presented the current procedure is implemented to solve three linear equations and one nonlinear equation and excellent agreement is found between the exact and approximate solutions in addition the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques this procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations

a simple taylor series expansion method for a class of second kind integral equations a simple yet effective taylor series expansion method is presented for a class of fredholm integral equations of the second kind with smooth and weakly singular kernels the equations studied in this paper arise in a number of applications e g potential theory radiative equilibrium radiative heat transfer and electrostatics the approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple closed form the approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics

numerical solution of some classes of integral equations using bernstein polynomials this paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using bernstein polynomials as basis the integral equations considered are fredholm integral equations of second kind a simple hypersingular integral equation and a hypersingular integral equation of second kind the method is explained with illustrative examples also the convergence of the method is established rigorously for each class of integral equations considered here

a new computational method for laplace transforms by decomposition method in this paper adomian decomposition method is applied to develop a new computational method for laplace transforms some examples are provided to illustrate the simplicity and reliability of the new method

integrable systems in ellipsoidal coordinates the most general potential for complete integrability of three dimensional classical and quantum problems is obtained starting from a certain class of two second order integrals of motion we also show that these potentials lead to separation of variables in both classical and quantum cases

a simple approach to the particular solution of constant coefficient ordinary differential equations an eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is presented it is shown that the exponentials are the eigenfunctions of such equations solutions corresponding to products of powers and exponentials are obtained the singular case is studied and a fast algorithm for its implementation is presented

moment balancing templates for spectral null codes the generalized moment balancing templates for spectral null codes are investigated in this paper a new approach based on the insertion of a determined number of vectors at determinate indices of a spectral null code word is found to balance the moment of the code word moment balanced code words preserve the spectral null properties and can guarantee the correction of one insertion or one deletion error as compared to the universal template for an uncoded sequence the redundancy is determined by the structure of the original spectral null code however it can be kept small the new approach involves the applications of the optimized moment balancing template and nonbinary enumeration of the balancing moment values

codes correcting a single insertion deletion of a zero or a single peak shift codes of d k sequences of constant hamming weight are considered perfect codes correcting a single insertion deletion or peak shift are defined and shown to exist a systematic method to construct large classes of perfect codes is given a number of related problems are considered

solving the problem of biological species living together by adomian decomposition method in this paper we use adomian decomposition method for solving the system of nonlinear integro differential equations derived from considering biological species living together

legendre spectral collocation method for the solution of the model describing biological species living together in this article numerical techniques have been developed to solve a system of two nonlinear integro differential equations which arises in biological model this model has been solved by legendre spectral collocation method and bernstein polynomials the present methods reduce this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by newton s method also the results obtained by present methods have been compared with that of by some existing methods available in literature some illustrative examples have been discussed to show the applicability and accuracy of the present methods

bernstein series solution of a class of linear integro differential equations with weakly singular kernel abstract in this study a new collocation method based on the bernstein polynomials is introduced for the approximate solution of a class of linear volterra integro differential equations with weakly singular kernel if the exact solution is polynomial then the exact solution can be obtained if the exact solution is not a polynomial then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver in addition the method is presented with error and stability analysis

on global solution to the klein gordon hartree equation below energy space in this paper we consider the cauchy problem for klein gordon equation with a cubic convolution nonlinearity in r3 by making use of bourgain s method in conjunction with a precise strichartz estimate of s klainerman and d tataru we establish the hs s 1 global well posedness of the cauchy problem for the cubic convolution defocusing klein gordon hartree equation before arriving at the previously discussed conclusion we obtain global solution for this non scaling equation with small initial data in hs0 hs0 1 where s0 6 but not 2 1 for this equation that we consider is a subconformal equation in some sense in doing so a number of nonlinear prior estimates are already established by using bony s decomposition flexibility of klein gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate we establish this commutator estimate by exploiting cancellation property and utilizing coifman and meyer multilinear multiplier theorem as far as we know it seems that this is the first result on low regularity for this klein gordon hartree equation

use of modified bernstein polynomials to solve kdv burgers equation numerically abstract numerical solution of korteweg de veries burgers kdvb equation is presented using modified bernstein polynomials b polynomials over the spatial domain b polynomials are used to expand the desired solution requiring discretization with only the time variable galerkin method is used to determine the expansion coefficients to construct initial trial functions we use fourth order runge kutta method to solve the system of equations for the time variable the accuracy of the solutions is dependent on the size of the b polynomial basis set numerical results obtained using this method are compared with existing analytical results excellent agreement is found between the exact solution and approximate solution obtained by this method

an algorithm for adomian decomposition method in this paper a reliable technique for calculating adomian polynomials for nonlinear operators is developed the algorithm is fulfilled with symbolic computation so that the adomian decomposition methods for solving differential equations can be mechanized in maple ten examples are presented to illustrate the implementation of the program

solving system of volterra fredholm integral equations with bernstein polynomials and hybrid bernstein block pulse functions this work approximates the unknown functions based on the bernstein polynomials and hybrid bernstein block pulse functions in conjunction with the collocation method for the numerical solution of system of fredholm volterra integral equations in both methods the system of integral equations is approximated by the gauss quadrature formula with respect to the legendre weight function the proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods moreover the convergence analysis of these algorithms will be shown by preparing some theorems numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods

solving concrete examples by adomian method in this note we extend the adomian decomposition method to solve two concrete problems the population problem and a problem from polymer rheology

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

the quantum square well with moving boundaries a numerical analysis we present some numerical discussions concerning the infinite square well in one dimension with moving boundaries our results show that if the speed of displacement is small objects of physical relevance like probability density averaged position or mean value of the energy have a smooth behavior on the contrary if this speed becomes large many irregularities arise which has a difficult qualitative explanation these irregularities manifest themselves as sharp bumps on the probability distribution or a chaotic shape on the averaged values of position and energy none of these patterns is the result of numerical errors and therefore we conclude that an unknown and very nontrivial effect is produced at high speeds of the moving wall

nonasymptotic upper bounds for deletion correcting codes explicit nonasymptotic upper bounds on the sizes of multiple deletion correcting codes are presented in particular the largest single deletion correcting code for q ary alphabet and string length is shown to be of size at most qn q q 1 n 1 an improved bound on the asymptotic rate function is obtained as a corollary upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length and this idea is demonstrated by application to sets of run length limited strings the problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph this problem is formulated as an integer linear program the upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program the nonasymptotic bounds derived imply the known asymptotic bounds of levenshtein and tenengolts and improve on known nonasymptotic bounds numerical results support the conjecture that in the binary case the varshamov tenengolts codes are the largest single deletion correcting codes

mechanization for solving spp by reducing order method in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of reduction of order approximation method for solving singular perturbation problems was established a new maple procedure redordproc was established too the procedure redordproc give not only the approximate analytic solutions but also the numerical solutions of the kinetics problems some examples are presented to illustrate the implementation of the program

moment balancing templates constructions to add insertion deletion correction capability to error correcting or constrained codes templates are constructed to extend arbitrary additive error correcting or constrained codes i e additional redundant bits are added in selected positions to balance the moment of the codeword the original codes may have error correcting capabilities or constrained output symbols as predetermined by the usual communication system considerations which are retained after extending the code using some number theoretic constructions in the literature insertion deletion correction can then be achieved if the template is carefully designed the number of additional redundant bits for the insertion deletion correction can be kept small in some cases of the same order as the number of parity bits in a hamming code of comparable length

a new algorithm for integral of trigonometric functions with mechanization in this paper by using the theories of mathematical analysis the authors systematically studied the problem of integrating of trigonometric functions utilizing omnipotence expressions tan x2 t a new integral algorithm for the integral of trigonometric functions was established and a new maple program trigint for the calculation mechanizing for the integral of trigonometric functions was established trigint could evaluate not only the integral of trigonometric functions utilizing omnipotence expresssions correctly but also all the other kinds of functions the results indicated that the algorithm established in this paper had advantages of simple idea excellent property for operation and powerful competence this would be useful for some calculating problems of engineering mathematics

one step recursive method for solving systems of differential equations in this work an iteration one step method to integrate systems of nonlinear ordinary differential equations with initial values is presented

numerical solution of 2d fractional optimal control problems by the spectral method along with bernstein operational matrix abstractthis paper presents an approximate method to solve a class of two dimensional fractional optimal control problems with nonlinear dynamical system to implement the new method by considering the initial boundary conditions the unknown state and control functions are approximated by the bernstein polynomials b polynomials basis using spectral ritz method then the problem is reduced to an unconstrained nonlinear optimisation problem meanwhile to reduce computational complexity a new fractional operational bernstein matrix generalised on an arbitrary interval is constructed and applied the choice of polynomial basis functions along with the ritz method provides good flexibility in which all the given initial and boundary conditions are imposed at last we extensively argue the convergence of the new method and several illustrative test problems are added to demonstrate the applicability and effectiveness of the new procedure moreover our achievements are compared with the previous results

solving generalized pantograph equations by shifted orthonormal bernstein polynomials abstract in this paper we introduce shifted orthonormal bernstein polynomials sobps and derive the operational matrices of integration and delays for these polynomials then we apply them to convert the pantograph equations to a system of linear equations an important property of this method is that the condition number of the coefficient matrix of the system is small which confirms that our method is stable error analysis and comparison with other methods are given to confirm the validity efficiency and applicability of the proposed method

on the modified taylor s approximation for the solution of linear and nonlinear equations fourier series and taylor s expansions are commonly used in the fields of science and engineering a whole range of interesting physical problems can be solved using one or the other of these standard techniques yet there is a class of problems whose solutions exhibit near sinusoidal or repetitive behavior that cannot be solved using the above expansions it is for these types of problems that the modified taylor expansion has been developed it is a method that combines the advantages of the repetitive behavior of sinusoidal functions and polynomial series this technique has not been employed for tackling equations whose solutions are near sinusoidal we discuss the method and apply it to a number of interesting problems that show its utility examples include the solution of differential integral integro differential and functional equations in addition we propose an algorithm for the numerical integration of a function which exhibits near periodic behavior namely the bessel function most of the symbolic and numerical computations have been performed using the computer algebra system maple

numerical solution of linear fredholm integral equation by using hybrid taylor and block pulse functions in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate the solution of a linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

numerical solution of a singular integro differential equation this paper presents a method based on polynomial approximation using bernstein polynomial basis to obtain approximate numerical solution of a singular integro differential equation with cauchy kernel the numerical results obtained by the present method compares favorably with those obtained by various galerkin methods earlier in the literature also the convergence of the method is established rigorously for the studied integro differential equation

numerical solution of volterra integral equations of the second kind with convolution kernel by using taylor series expansion method in this paper we present a taylor series expansion method for a class of volterra integral equations of second kind with smooth or weakly singular kernels this method use taylor series approximation method for integral equation and transform the integral equation to an nth order linear differential equation boundary conditions for differential equation produce in easy way this method gives an approximate simple and closed form solution for integral equation some numerical example use to illustrate the accuracy of method