gaussian quadrature for sums a rapidly convergent summation scheme gaussian quadrature is a well known technique for numerical integration recently gaussian quadrature with respect to discrete measures corresponding to finite sums has found some new interest in this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of gaussian formulas the abscissae of the gaussian summation have a very interesting asymptotic distribution function with a kink singularity we apply the gaussian summation technique to two problems which have been discussed in the literature we find that the gaussian summation has a very rapid convergence rate for the hardy littlewood sum for a large range of parameters

multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

end point corrected trapezoidal quadrature rules for singular functions abstract a group of quadrature formulae for end point singular functions is presented generalizing classical end point corrected trapezoidal quadrature rules the actual values of the end point corrections are obtained for each singularity as a solution of a system of linear algebraic equations the algorithm is applicable to a wide class of monotonic singularities and does not require that an analytical expression for the singularity be known only the knowledge of its first several moments and the ability to evaluate it on the interval of integration are needed keywords convergence numerical analysis numerical integration author

u p q u p 1 q is a generalized gelfand pair denote by g u p q the orthogonal group of the sesqui linear quadratic form x y x 1 overline y 1 cdots x p overline y p x p 1 overline y p 1 cdots x p q overline y p q on mathbb c p q and let h1 u p 1 q be the stabilizer of the first unit vector e1 let h0 u 1 and set h h0 h1 define the character l of h by chi l h chi l h 0h 1 h 0 l h 0 in h 0 h 1 in h 1 where l in mathbb z define the anti involution on g by sigma g overline g 1 in this note we show that any distribution t on g satisfying t h1gh2 l h1h2 t g g g h1 h2 h is invariant under the anti involution this result implies that g h1 is a generalized gelfand pair

complex gaussian quadrature of oscillatory integrals we construct and analyze gauss type quadrature rules with complex valued nodes and weights to approximate oscillatory integrals with stationary points of high order the method is based on substituting the original interval of integration by a set of contours in the complex plane corresponding to the paths of steepest descent each of these line integrals shows an exponentially decaying behaviour suitable for the application of gaussian rules with non standard weight functions the results differ from those in previous research in the sense that the constructed rules are asymptotically optimal i e among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence relative to the required number of evaluations of the integrand

fast fourier galerkin methods for solving singular boundary integral equations numerical integration and precondition we develop a fast fully discrete fourier galerkin method for solving a class of singular boundary integral equations we prove that the number of multiplications used in generating the compressed matrix is o nlog 3n and the solution of the proposed method preserves the optimal convergence order o n t where n is the order of the fourier basis functions used in the method and t denotes the degree of regularity of the exact solution moreover we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm

numerical solution of evolution equations by the haar wavelet method an efficient numerical method for solution of nonlinear evolution equations based on the haar wavelets approach is proposed the method is tested in the case of burgers and sine gordon equations numerical results obtained by computer simulation are compared with other available solutions these calculations demonstrate that the accuracy of the haar wavelet solutions is quite high even in the case of a small number of grid points

galerkinlevin method for highly oscillatory integrals in this paper a new method namely galerkinlevin method is proposed to investigate the highly oscillatory integrals indeed we construct an algorithm based on the levin and galerkin methods using the levin method the given integral is transformed into a certain ordinary differential equation then a galerkin method is applied for solving the obtained ode the efficiency of the approach will be shown by applying the procedure on some prototype examples a new algorithm for solving the highly oscillatory integrals is proposed using the levin method a given integral is transformed into a certain ode error and convergence analysis of the proposed algorithm is performed

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

an efficient method for evaluating the integral of a class of highly oscillatory functions highly oscillatory integrals require special techniques for their effective evaluation various studies have been conducted to find computational methods for evaluating such integrals in this paper we present an efficient numerical method to evaluate a class of generalised fourier integrals on a line or a square with integrands of the form f x e i k g x under the assumption that in the domain of integration both f and g are sufficiently smooth and that g does not have any stationary critical points numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised fourier integrals

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

a wavelet operational method for solving fractional partial differential equations numerically fractional calculus is an extension of derivatives and integrals to non integer orders and a partial differential equation involving the fractional calculus operators is called the fractional pde they have many applications in science and engineering however not only the analytical solution existed for a limited number of cases but also the numerical methods are very complicated and difficult in this paper we newly establish the simulation method based on the operational matrices of the orthogonal functions we formulate the operational matrix of integration in a unified framework by using the operational matrix of integration we propose a new numerical method for linear fractional partial differential equation solving in the method we 1 use the haar wavelet 2 establish a lyapunov type matrix equation and 3 obtain the algebraic equations suitable for computer programming two examples are given to demonstrate the simplicity clarity and powerfulness of the new method

using rationalized haar wavelet for solving linear integral equations the main purpose of this paper is to demonstrate that using rationalized haar wavelet for solving 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

symmetric gauss legendre quadrature formulas for composite numerical integration over a triangular surface abstract this paper first presents a gauss legendre quadrature method for numerical integration of i t f x y d x d y where f x y is an analytic function in x y and t is the standard triangular surface x y 0 x y 1 x y 1 in the cartesian two dimensional x y space we then use a transformation x x y y to change the integral i to an equivalent integral s f x y x y d d where s is now the 2 square in space 1 1 we then apply the one dimensional gauss legendre quadrature rules in and variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points we then propose the discretisation of the standard triangular surface t into n 2 right isosceles triangular surfaces t i i 1 1 n 2 each of which has an area equal to 1 2 n 2 units we have again shown that the use of affine transformation over each t i and the use of linearity property of integrals lead to the result i i 1 n n t i f x y d x d y 1 n 2 t h x y d x d y where h x y i 1 n n f x i x y y i x y and x x i x y and y y i x y refer to affine transformations which map each t i in x y space into a standard triangular surface t in x y space we can now apply gauss legendre quadrature formulas which are derived earlier for i to evaluate the integral i 1 n 2 t h x y d x d y we observe that the above procedure which clearly amounts to composite numerical integration over t and it converges to the exact value of the integral t f x y d x d y for sufficiently large value of n even for the lower order gauss legendre quadrature rules we have demonstrated this aspect by applying the above explained composite numerical integration method to some typical integrals

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

the inner structure of empirical mode decomposition abstract the empirical mode decomposition emd is a nonlinear method that is truly adaptive with good localization property in the time domain for analyzing non stationary complex data the emd has been proven useful in a wide range of applications however due to the nonlinear and complex nature of the sifting process the most essential step of the emd a firm mathematical foundation or a transparent physical description are still lacked for emd here we embark on constructing a mathematical theory of the sifting operator we first show that the sifting operator can be expressed as the data plus the sum of the responses to the impulses multiplied by the data value at the extrema such an expression of the sifting operator is then used to investigate the adaptive nature and the localizing effect of the emd alternatively the sifting operator can also be represented by a sifting matrix which depends nonlinearly on the extrema distribution based on the eigen decomposition of the sifting matrix the transfer function of the sifting process is analyzed finally we answer what an intrinsic mode function imf is from the wave perspective by exploring the physical basis of the imfs

delaminating quadrature method for multi dimensional highly oscillatory integrals a delaminating quadrature method for multi dimensional highly oscillatory integrals is put forward in comparison with the levin method the new method has the benefits of being numerically stable and having a high computational speed numerical experiments testified the advantages of this method

numerical solution of the controlled duffing oscillator by hybrid functions a numerical method for solving the controlled duffing oscillator is presented the method is based upon hybrid functions approximation the properties of hybrid functions which consists of block pulse functions plus legendre polynomials are given the method is based upon expanding various time functions in the system as their truncated hybrid functions the operational matrix of product is introduced this matrix together with the operational matrix of integration are utilized to reduce the solution of controlled duffing oscillator to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

numerical integration using wavelets in this paper an algorithm for obtaining approximate value of a definite integral as well as double integral using wavelets will be illustrated this approximation depends on the pure scaling functions expansion of the integrand function

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

haar wavelet method for nonlinear integro differential equations abstract a numerical method for solving nonlinear integral equations based on the haar wavelets is presented the method is applicable for volterra integral equations and integro differential equations it can be used also for solving boundary value problems of ordinary differential equations the efficiency of the proposed method is tested with the aid of four examples high accuracy even for a small number of collocation points is stated

numerical solution of fredholm integral equations of the first kind by using linear legendre multi wavelets in this paper we suggest an efficient method for solving fredholm integral equations of the first kind the continuous legendre multi wavelets constructed on 0 1 are utilized as a basis in galerkin method to reduce the solution of linear integral equations to a system of algebraic equations illustrative examples are included to demonstrate the efficiency and the application of the technique

on numerical solution of two dimensional nonlinear urysohn fuzzy integral equations based on fuzzy haar wavelets in the present paper using fuzzy haar wavelets we introduce an iterative method for numerical solving two dimensional nonlinear urysohn fuzzy integral equations 2dnufie we define a useful metric to prove the convergence and error estimate of the method moreover we investigate the numerical stability of the presented numerical successive approximations finally numerical experiments show the accuracy of the method

fast integration of highly oscillatory integrals with exotic oscillators in this paper we present an efficient filon type method for the integration of systems containing bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to airy transforms preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

a sparse fractional jacobi galerkin levin quadrature rule for highly oscillatory integrals abstract levin s quadrature rule is well known for its good performance on numerical treatment with highly oscillatory integrals however most existing work treats with sufficiently smooth integrands in this paper we study levin s quadrature rule for a class of singular and oscillatory integrals based on fractional jacobi polynomials a class of differential equation solvers for levin s equation are developed which leads to the fractional jacobi galerkin levin method the discretized equation is turned into a sparse linear system by properly choosing jacobi polynomials and the inner product furthermore convergence analysis with respect to the oscillation is presented by studying coefficients of the fractional jacobi expansion of the error function numerical experiments indicate that in contrast to existing quadrature rules the new method is efficient for computing oscillatory integrals with stationary points and unknown singular parameters

numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets in this work we present a computational method for solving nonlinear fredholm integral equations of the second kind which is based on the use of haar wavelets error analysis is worked out that shows efficiency of the method finally we also give some numerical examples

a new approach to sparse decomposition of nonstationary signals with multiple scale structures using self consistent nonlinear waves abstract a new method for signal decomposition is proposed and tested based on self consistent nonlinear wave equations with self sustaining physical mechanisms in mind the new method is adaptive and particularly effective for dealing with synthetic signals consisting of components of multiple time scales by formulating the method into an optimization problem and developing the corresponding algorithm and tool we have proved its usefulness not only for analyzing simulated signals but more importantly also for real clinical data

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

numerical solution of differential equations by using chebyshev wavelet operational matrix of integration in this paper we present operational matrix of integration omi of chebyshev wavelets basis and the product operation matrix pom of it some comparative examples are included to demonstrate the superiority of operational matrix of chebyshev wavelets to those of legendre wavelets

efficient numerical methods for bessel type of oscillatory integrals in this paper new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of bessel functions j x meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by bessel and bessel trigonometric functions in addition multi resolution quadrature rules based on hybrid functions and haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems

on the computational complexity of the empirical mode decomposition algorithm abstract it has been claimed that the empirical mode decomposition emd and its improved version the ensemble emd eemd are computation intensive in this study we will prove that the time complexity of the emd eemd which has never been analyzed before is actually equivalent to that of the fourier transform numerical examples are presented to verify that emd eemd is in fact a computationally efficient method

numerical integrations over an arbitrary quadrilateral region in this paper double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method the physical region is transformed into a standard quadrilateral finite element using the basis functions in local space then the standard quadrilateral is subdivided into two triangles and each triangle is further discretized into 4xn 2 right isosceles triangles with area 12n 2 and thus composite numerical integration is employed in addition the affine transformation over each discretized triangle and the use of linearity property of integrals are applied finally each isosceles triangle is transformed into a 2 square finite element to compute new n 2 extended symmetric gauss points and corresponding weight coefficients where n is the lower order conventional gauss legendre quadratures these new gauss points and weights are used to compute the double integral examples are considered over an arbitrary domain and rational and irrational integrals which can not be evaluated analytically

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

efficient quadrature for highly oscillatory integrals involving critical points this paper based on the levin collocation method and levin type method together with composite two point gauss legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points the effectiveness and accuracy of the quadrature are tested

hybrid function method for solving fredholm and volterra integral equations of the second kind numerical solutions of fredholm and volterra integral equations of the second kind via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions integration of the cross product a special product matrix and a related coefficient matrix with optimal order are applied to solve these integral equations the main characteristic of this technique is to convert an integral equation into an algebraic hence the solution procedures are either reduced or simplified accordingly the advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function for the solutions of integral equations we propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions the high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples the hybrid function method is superior to other piecewise constant orthogonal functions w f blyth r l may p widyaningsih volterra integral equations solved in fredholm form using walsh functions anziam j 45 e 2004 c269 c282 m h reihani z abadi rationalized haar functions method for solving fredholm and volterra integral equations j comp appl math 200 2007 12 20 for these problems

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

new quadrature rules for highly oscillatory integrals with stationary points in this paper new algorithms are being proposed for evaluation of highly oscillatory integrals hois with stationary point s the algorithms are based on modified levin quadrature mlq with multiquadric radial basis functions rbfs coupled with quadrature rules based on hybrid functions of order 8 hfq8 and haar wavelets quadrature hwq aziz et al 2011 part of the new procedure presented in this paper is comprised of transplanting monomials which are used in the conventional levin method by the rbfs the linear and hermite polynomials based quadratures xiang 2007 are being replaced by the new methods based on hwq and hfq8 respectively both the methods are merged with mlq to obtain the numerical solution of highly oscillatory integrals having stationary points the accuracy of the new methods is neither dampened by presence of the stationary point s nor by the large value of frequency parameter theoretical facts about the error analysis of the new methods are analyzed and proved numerical examples are included to show efficiency and accuracy of the new methods a new framework for the numerical solution of highly oscillatory integrals is proposed the integrals are bifurcated in the neighborhood of stationary point the integral on the smaller subinterval is solved by hybrid functions and haar wavelets the integral on the longer subinterval is solved by the meshless method with mq radial basis functions convergence analysis of the proposed methods is performed

efficient filon type methods for int_a bf x rm e rm i omega g x rm d x based on the transformation y g x some new efficient filon type methods for integration of highly oscillatory function int a bf x rm e rm i omega g x rm d x with an irregular oscillator are presented one is a moment free filon type method for the case that g x has no stationary points in a b the others are based on the filon type method or the asymptotic method together with filon type method for the case that g x has stationary points the effectiveness and accuracy are tested by numerical examples

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

on application of fast and adaptive periodic battle lemarie wavelets to modeling of multiple lossy transmission lines in this paper the continuous operator is discretized into matrix forms by galerkin s procedure using periodic battle lemarie wavelets as basis testing functions the polynomial decomposition of wavelets is applied to the evaluation of matrix elements which makes the computational effort of the matrix elements no more expensive than that of method of moments mom with conventional piecewise basis testing functions a new algorithm is developed employing the fast wavelet transform fwt owing to localization cancellation and orthogonal properties of wavelets very sparse matrices have been obtained which are then solved by the lsqr iterative method this algorithm is also adaptive in that one can add at will finer wavelet bases in the regions where fields vary rapidly without any damage to the system orthogonality of the wavelet basis functions to demonstrate the effectiveness of the new algorithm we applied it to the evaluation of frequency dependent resistance and inductance matrices of multiple lossy transmission lines numerical results agree with previously published data and laboratory measurements the valid frequency range of the boundary integral equation results has been extended two to three decades in comparison with the traditional mom approach the new algorithm has been integrated into the computer aided design tool magicad which is used for the design and simulation of high speed digital systems and multichip modules panet al ieee trans hyb manuf technol 15 4 465 1992

numerical solution of first kind fredholm integral equations with wavelets galerkin method wgm and wavelets precondition in this paper we use orthogonal wavelets as basis in projection methods for obtaining numerical solution of the fredholm integral equations of the first kind fk1 if this basis has the mra property then we show that the projection methods specially wgm must be convergent at the end for showing efficiency of method we use some examples

haar wavelet approximation for the solution of cubic nonlinear schrodinger equations abstract in this study haar wavelet collocation method is used for the numerical solution of 1d and 2d cubic nonlinear schrodinger equations with initial and dirichlet boundary conditions the space derivatives are estimated through haar wavelet collocation method whereas for time derivative we have used crank nicolson scheme the proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate

approximation of highly oscillatory integrals containing special functions abstract a modified form of levin s method based on multi quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory bessel function of the first kind of order and oscillatory airy function the method converts the oscillatory integrals into a system of coupled ordinary differential equations odes and subsequently numerical solution of the coupled odes is obtained by meshless collocation procedure a multi resolution quadrature is used to tackle singularity of the proposed method theoretical error analysis of the proposed methods is performed in asymptotic sense numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory airy and bessel integrals