a fourth order finite difference method for solving elliptic interface problems with the fft acceleration abstract in this paper a new cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary mib method and fast fourier transform fft the proposed augmented mib method consists of two major parts for solving elliptic interface problems in two dimensions for the interior part in treating a smoothly curved interface zeroth and first order jump conditions are enforced repeatedly by the mib scheme to generate fictitious values near the interface for the exterior part two layers of zero padding solutions are introduced beyond the original rectangular domain so that the fft inversion becomes feasible different types of boundary conditions including dirichlet neumann robin and their mix combinations can be imposed via the mib scheme to generate fictitious values near boundaries based on fictitious values at both interfaces and boundaries the augmented mib method reconstructs cartesian derivative jumps as auxiliary variables then by treating such variables as unknowns an enlarged linear system is obtained in the schur complement solution of such system the fft algorithm will not sense the solution discontinuities so that the discrete laplacian can be efficiently inverted therefore the fft based augmented mib not only achieves a fourth order of accuracy in dealing with interfaces and boundaries but also produces an overall complexity of o n 2 log n for a n n uniform grid moreover the augmented mib scheme can provide fourth order accurate approximations to solution gradients and fluxes

fast collocation methods for second kind integral equations in this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels for this purpose we construct multiscale interpolating functions and collocation functionals having vanishing moments moreover we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms an optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated the stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed

solving elliptic problems with discontinuities on irregular domains the voronoi interface method we introduce a simple method dubbed the voronoi interface method to solve elliptic problems with discontinuities across the interface of irregular domains this method produces a linear system that is symmetric positive definite with only its right hand side affected by the jump conditions the solution and the solution s gradients are second order accurate and first order accurate respectively in the l norm even in the case of large ratios in the diffusion coefficient this approach is also applicable to arbitrary meshes additional degrees of freedom are placed close to the interface and a voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach both the locations of the additional degrees of freedom and their voronoi discretizations are straightforward in two and three spatial dimensions

reproducing kernel method for solving fredholm integro differential equations with weakly singularity singular integral equations sies are often encountered in certain contact and fracture problems in solid mechanics numerical methods for solving sies have been the focus of much research including reproducing kernel methods however there are no reports on reproducing kernel methods for solving differential integral equations with weakly singular kernels we developed a reproducing kernel method for solving fredholm integro differential equations with weakly singular kernels in reproducing kernel hilbert space this involves changing a weakly singular kernel to a logarithm kernel to a kalman kernel weak singularity is removed by applying a smooth transformation to the kalman kernel solution representations are obtained in reproducing kernel hilbert space numerical experiments show that our reproducing kernel method is efficient

investigation of classical and fractional bose einstein condensation for harmonic potential abstract in this study classical and fractional gross pitaevskii gp equations were solved for harmonic potential and repulsive interactions between the boson particles using the homotopy perturbation method hpm to investigate the ground state dynamics of bose einstein condensation bec the purpose of writing fractional gp equations is to consider the system in a more realistic manner the memory effects of non markovian processes involving long range interactions between bosons with the restriction of the ergodic hypothesis and the effect of non gaussian distributions of bosons in the condensation can be taken into account with time fractional and space fractional gp equations respectively the obtained results of the fractional gp equations differ from the results of the classical one while the gauss distribution describing the homogeneous reversible and unitary system is obtained from the classical gp equation the probability density of the solution function of fractional gp equations is non conserved this situation describes the inhomogeneous irreversible and non unitary systems

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

xgfm recovering convergence of fluxes in the ghost fluid method abstract we introduce a simple but effective correction to the ghost fluid method gfm introduced by liu et al 2000 30 for capturing sharp interface conditions when solving poisson equations with discontinuities our method is shown to alleviate the gfm lack of convergence for the flux across the interfaces pointwise convergence of the gradient is recovered by ensuring consistent jump conditions with an iterative process compared to the gfm the discretization stencil and the symmetric positive definiteness of the resulting linear system are not modified only the right hand side of the linear system is affected we first show that the original gfm is consistent with the second order accurate symmetric scheme of gibou et al 2002 17 for dirichlet conditions in separate regions of the computational domain where the parameters and the solution are smooth but with matching conditions at the interface that are not fully consistent in general our proposed method recovers consistency and is shown to produce convergent gradients through several numerical examples in two and three spatial dimensions

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

an improvement on floating search algorithms for feature subset selection a new improved forward floating selection iffs algorithm for selecting a subset of features is presented our proposed algorithm improves the state of the art sequential forward floating selection algorithm the improvement is to add an additional search step called replacing the weak feature to check whether removing any feature in the currently selected feature subset and adding a new one at each sequential step can improve the current feature subset our method provides the optimal or quasi optimal close to optimal solutions for many selected subsets and requires significantly less computational load than optimal feature selection algorithms our experimental results for four different databases demonstrate that our algorithm consistently selects better subsets than other suboptimal feature selection algorithms do especially when the original number of features of the database is large

piecewise polynomial collocation for fredholm integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of initial or boundary value problems of linear fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

convergence of the ghost fluid method for elliptic equations with interfaces this paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps a weak formulation of the problem is first presented which then yields the existence and uniqueness of a solution to the problem by classical methods it is shown that the application of the ghost fluid method by fedkiw kang and liu to this problem can be obtained in a natural way through discretization of the weak formulation an abstract framework is given for proving the convergence of finite difference methods derived from a weak problem and as a consequence the ghost fluid method is proved to be convergent

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

travelling wave solutions of two nonlinear evolution equations by using the g g expansion method abstract by using the g g expansion method proposed recently we give the exact travelling wave solutions of two different types of nonlinear evolution equations in this paper the travelling wave solutions are expressed by three types of functions the computation for the method appears to be easier and faster by general mathematical software

an efficient algorithm for solving fredholm integro differential equations with weakly singular kernels in this paper an effective method based upon the piecewise homotopy perturbation method phpm is proposed for finding the numerical solutions of fredholm integro differential equations with weakly singular kernels for these equations the traditional homotopy perturbation method is divergent so we present a modified homotopy perturbation method for solving this kind of equation the advantages of the approach of this paper lie in the feature that this technique not only achieves an approximate solution with high accuracy but also improves the calculation speed two examples are given to show the effectiveness and convenience of this method

the travelling wave solutions for non linear initial value problems using the homotopy perturbation method in this article we have used the homotopy perturbation method hpm to find the travelling wave solutions for some non linear initial value problems in the mathematical physics these problems consist of the burgers fisher equation the kuramoto sivashinsky equation the coupled schordinger kdv equations and the long short wave resonance equations together with initial conditions the results of these problems reveal that the hpm is very powerful effective convenient and quite accurate to the systems of non linear equations it is predicted that this method can be found widely applicable in engineering and physics

solving poisson type equations with robin boundary conditions on piecewise smooth interfaces abstract we present two finite volume schemes to solve a class of poisson type equations subject to robin boundary conditions in irregular domains with piecewise smooth boundaries the first scheme results in a symmetric linear system and produces second order accurate numerical solutions with first order accurate gradients in the l norm for solutions with two bounded derivatives the second scheme is nonsymmetric but produces second order accurate numerical solutions as well as second order accurate gradients in the l norm for solutions with three bounded derivatives numerical examples are given in two and three spatial dimensions

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

solving elliptic interface problems with jump conditions on cartesian grids abstract we present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient the source term the solution and its flux are discontinuous across an irregular interface the algorithm produces second order accurate solutions and first order accurate gradients in the l norm on cartesian grids the condition number is bounded regardless of the ratio of the diffusion constant and scales like that of the standard 5 point stencil approximation on a rectangular grid with no interface numerical examples are given in two and three spatial dimensions

an evidential classifier based on feature selection and two step classification strategy in this paper we investigate ways to learn efficiently from uncertain data using belief functions in order to extract more knowledge from imperfect and insufficient information and to improve classification accuracy we propose a supervised learning method composed of a feature selection procedure and a two step classification strategy using training information the proposed feature selection procedure automatically determines the most informative feature subset by minimizing an objective function the proposed two step classification strategy further improves the decision making accuracy by using complementary information obtained during the classification process the performance of the proposed method was evaluated on various synthetic and real datasets a comparison with other classification methods is also presented highlightsa classifier is based on belief functions to tackle uncertain data the classifier composed by feature selection and a two step classification a new combination rule to better represent data uncertainty a new feature selection is based on minimizing uncertainty with sparse constraint two step classification improving accuracy of decision making

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

spline collocation method for integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions to initial or boundary value problems of fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms

solution of the epidemic model by homotopy perturbation method in this article the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered he s homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations governing on the problem the results are compared with the results obtained by adomian decomposition method some plots are presented to show the reliability and simplicity of the method

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

new general solutions to linear fredholm integro differential equations and their applications on solving the boundary value problems abstract this paper introduces the n general solution to linear fredholm integro differential equations and sets its properties the conditions for existence of classical general solution and solvability criteria for the equation are provided necessary and sufficient conditions for solvability of linear boundary value problems are established algorithms for construction of the n general solutions are obtained this paper also proposes approximate and numerical methods to solve the boundary value problems

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

the homotopy perturbation method and analytical solution of the problem of flow past a rotating disk analytical solutions have been obtained for the problem of steady laminar flow of a viscous incompressible fluid past a rotating disk using the homotopy perturbation method hpm the single parameter hpm is attempted first since it does not give adequately accurate results the extended hpm is invoked next in which the independent variable is scaled and the scale factor is expanded in a power series in p the homotopy parameter the coefficients in this power series are calculated by ensuring that the resulting solution is free of secular terms it is shown that the extended hpm solution converges to the true solution and that just eight terms in the perturbation expansion are sufficient to produce a highly accurate yet fully analytical solution

homotopy perturbation reproducing kernel method for nonlinear systems of second order boundary value problems in this paper based on homotopy perturbation method hpm and reproducing kernel method rkm a new method is presented for solving nonlinear systems of second order boundary value problems bvps hpm is based on the use of traditional perturbation method and homotopy technique the hpm can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases rkm is also an analytical technique which can solve powerfully linear bvps homotopy perturbation reproducing kernel method hp rkm combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear bvps three numerical examples are presented to illustrate the strength of the method

on the homotopy perturbation method for boussinesq like equations abstract we comment on some analytical solutions to a class of boussinesq like equations derived recently by means of the homotopy perturbation method hpm we show that one may obtain exactly the same result by means of the taylor series in the time variable we derive more general results by means of traveling waves and argue that a curious superposition principle proposed by the author may not be of any mathematical or physical significance

superconvergence of the iterated collocation methods for hammerstein equations in this paper we analyse the iterated collocation method for hammerstein equations with smooth and weakly singular kernels the paper expands the study which began in 16 concerning the superconvergence of the iterated galerkin method for hammerstein equations we obtain in this paper a similar superconvergence result for the iterated collocation method for hammerstein equations we also discuss the discrete collocation method for weakly singular hammerstein equations some discrete collocation methods for hammerstein equations with smooth kernels were given previously in 3 18

an improvement to homotopy perturbation method for solving system of linear equations in this paper we present an efficient numerical algorithm to find exact solutions for the system of linear equations based on homotopy perturbation method hpm a reliable modification is proposed and the modified method is employed to solve the system of linear equations the results are compared with those obtained by the original homotopy perturbation method two examples are given to illustrate the ability and reliability of the modified method the results reveal that the modified method is very simple and effective

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

bagging constraint score for feature selection with pairwise constraints constraint score is a recently proposed method for feature selection by using pairwise constraints which specify whether a pair of instances belongs to the same class or not it has been shown that the constraint score with only a small amount of pairwise constraints achieves comparable performance to those fully supervised feature selection methods such as fisher score however one major disadvantage of the constraint score is that its performance is dependent on a good selection on the composition and cardinality of constraint set which is very challenging in practice in this work we address the problem by importing bagging into constraint score and a new method called bagging constraint score bcs is proposed instead of seeking one appropriate constraint set for single constraint score in bcs we perform multiple constraint score each of which uses a bootstrapped subset of original given constraint set diversity analysis on individuals of ensemble shows that resampling pairwise constraints is helpful for simultaneously improving accuracy and diversity of individuals we conduct extensive experiments on a series of high dimensional datasets from uci repository and gene databases and the experimental results validate the effectiveness of the proposed method

product integration for weakly singular integral equations in this note we discuss the numerical solution of the second kind fredholm integral equation y t f t lambda int limits omega psi alpha t s g t s y s ds t in bar omega 1 where lambda in not subset backslash 0 the functions f g are given and continuous denotes the euclidean norm and 0 m is a given function on 0 which satisfies psi alpha j r leqq c j left begin array 20 c r a m j j geqq 0 m alpha 0 left begin array 20 c ln r j 0 r j j 0 end array right alpha m end array right with cj not depending on r here is the closure of a bounded domain m

on the existence of solutions for strongly nonlinear differential equations the objectives of this paper are twofold firstly to prove the existence of an approximate solution in the mean for some nonlinear differential equations we also investigate the behavior of the class of solutions which may be associated with the differential equation secondly we aim to implement the homotopy perturbation method hpm to find analytic solutions for strongly nonlinear differential equations

feature selection for monotonic classification monotonic classification is a kind of special task in machine learning and pattern recognition monotonicity constraints between features and decision should be taken into account in these tasks however most existing techniques are not able to discover and represent the ordinal structures in monotonic datasets thus they are inapplicable to monotonic classification feature selection has been proven effective in improving classification performance and avoiding overfitting to the best of our knowledge no technique has been specially designed to select features in monotonic classification until now in this paper we introduce a function which is called rank mutual information to evaluate monotonic consistency between features and decision in monotonic tasks this function combines the advantages of dominance rough sets in reflecting ordinal structures and mutual information in terms of robustness then rank mutual information is integrated with the search strategy of min redundancy and max relevance to compute optimal subsets of features a collection of numerical experiments are given to show the effectiveness of the proposed technique

a modified spectral method for solving operator equations in this paper we introduce a modified spectral method for solving the linear operator equation l u f l d l h 1 h 2 where h 1 and h 2 are normed vector spaces with norms 1 and respectively and d l is the domain of l also for each h h 2 h 2 h h where is an inner product on h 2 in this method we make a new set n n 0 for h 1 using l and two sets in h 1 and h 2 then using the new set n n 0 we solve this linear operator equation we show that this method does not have some shortcomings of spectral method also we prove the stability and convergence of the new method after introducing the method we give some conditions that under them the nonlinear operator equation l u n u f can be solved some examples are considered to show the efficiency of method

a boundary condition capturing method for incompressible flame discontinuities abstract in this paper we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other e g the vaporization of liquid water we consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow this method was designed as an extension of the ghost fluid method 1999 j comput phys 152 457 and relies heavily on the boundary condition capturing technology developed in liu et al 2000 j comput phys 154 15 for the variable coefficient poisson equation and in kang et al in press j comput phys for multiphase incompressible flow our new numerical method admits a sharp interface representation similar to the method proposed in helenbrook et al 1999 j comput phys 148 366 since the interface boundary conditions are handled in a simple and straightforward fashion the code is very robust e g no special treatment is required to treat the merging of flame fronts the method is presented in three spatial dimensions with numerical examples in one two and three spatial dimensions

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

homotopy perturbation method for solving hyperbolic partial differential equations this paper applies the homotopy perturbation method proposed by ji huan he to obtain approximate analytic solutions of hyperbolic partial differential equations the procedure of the method is systematically illustrated to give an extensive account of the method some examples are provided the results derived by this method will be compared with the results of characteristics method the results of homotopy perturbation method are of high accuracy verifying that the method is very effective and promising

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

invariant optimal feature selection a distance discriminant and feature ranking based solution the goal of feature selection is to find the optimal subset consisting of m features chosen from the total n features one critical problem for many feature selection methods is that an exhaustive search strategy has to be applied to seek the best subset among all the possible nm feature subsets which usually results in a considerably high computational complexity the alternative suboptimal feature selection methods provide more practical solutions in terms of computational complexity but they cannot promise that the finally selected feature subset is globally optimal we propose a new feature selection algorithm based on a distance discriminant fsdd which not only solves the problem of the high computational costs but also overcomes the drawbacks of the suboptimal methods the proposed method is able to find the optimal feature subset without exhaustive search or branch and bound algorithm the most difficult problem for optimal feature selection the search problem is converted into a feature ranking problem following rigorous theoretical proof such that the computational complexity can be greatly reduced the proposed method is invariant to the linear transformation of data when a diagonal transformation matrix is applied fsdd was compared with relieff and mrmrmid based on mutual information on 8 data sets the experiment results show that fsdd outperforms the other two methods and is highly efficient

the iterated galerkin method for linear integro differential equations abstract the iterated galerkin method of i h sloan which yields superconvergence for the approximate solution of integral equations is generalized to get similar results for the larger class of integro differential equations two approaches are investigated to obtain superconvergence in different norms moreover some additional aspects are discussed

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

application of he s semi inverse method to the nonlinear schrodinger equation a variational theory is established for the nonlinear schrodinger equation using he s semi inverse method on the basis of the variational principle obtained a solitary solution is obtained which is the same as debnath s result l debnath nonlinear partial differential equations for scientists and engineers birkhauser boston 1997 confirming the validity of our analysis the method is straightforward and concise

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

high order solution of poisson problems with piecewise constant coefficients and interface jumps we present a fast and accurate algorithm to solve poisson problems in complex geometries using regular cartesian grids we consider a variety of configurations including poisson problems with interfaces across which the solution is discontinuous of the type arising in multi fluid flows the algorithm is based on a combination of the correction function method cfm and boundary integral methods bim interface and boundary conditions can be treated in a fast and accurate manner using boundary integral equations and the associated bim unfortunately bim can be costly when the solution is needed everywhere in a grid e g fluid flow problems we use the cfm to circumvent this issue the solution from the bim is used to rewrite the problem as a series of poisson problems in rectangular domainswhich requires the bim solution at interfaces boundaries only these poisson problems involve discontinuities at interfaces of the type that the cfm can handle hence we use the cfm to solve them to high order of accuracy with finite differences and a fast fourier transform based fast poisson solver we present 2 d examples of the algorithm applied to poisson problems involving complex geometries including cases in which the solution is discontinuous we show that the algorithm produces solutions that converge with either 3rd or 4th order of accuracy depending on the type of boundary condition and solution discontinuity

determination of natural frequencies of fluid conveying pipes using homotopy perturbation method in the present paper the natural frequencies of fluid conveying pipes with pinned pinned boundary condition are derived explicitly in a systematical and straightforward way with the help of homotopy perturbation method numerical results are presented for two cases and the effect of fluid flow velocity on the natural frequencies is discussed good agreement with their experimental and fem counterparts is found numerically over ranges of practical interest

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

a partial differential equation approach to multidimensional extrapolation in this short note a general methodology for multidimensional extrapolation is presented the approach assumes a level set function exists which separates the region of known values from the region to be extrapolated it is shown that arbitrary orders of polynomial extrapolation can be formulated by simply solving a series of linear partial differential equations pdes examples of constant linear and quadratic extrapolation are given

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

a new modification of the homotopy perturbation method for linear and nonlinear operators in this paper we present an efficient modification of the homotopy perturbation method that will facilitate the calculations we then conduct a comparative study between the new modification and the homotopy perturbation method numerical illustrations are investigated to show the features of the technique the modified method accelerates the rapid convergence of the series solution and reduces the size of work the new modification introduces a promising tool for many linear and nonlinear problems

a stable projection method for the incompressible navier stokes equations on arbitrary geometries and adaptive quad octrees we present a numerical method for solving the incompressible navier stokes equations on non graded quadtree and octree meshes and arbitrary geometries the viscosity is treated implicitly through a finite volume approach based on voronoi partitions while the convective term is discretized with a semi lagrangian scheme thus relaxing the restrictions on the time step a novel stable implementation of the projection step is introduced making use of the marker and cell layout for the data the solver is validated numerically in two and three spatial dimensions

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

homotopy perturbation transform method for nonlinear equations using he s polynomials in this paper a combined form of the laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations this method is called the homotopy perturbation transform method hptm the nonlinear terms can be easily handled by the use of he s polynomials the proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round off errors the fact that the proposed technique solves nonlinear problems without using adomian s polynomials can be considered as a clear advantage of this algorithm over the decomposition method

a more efficient branch and bound algorithm for feature selection abstract an algorithm is presented to be able to select the globally optimal subset of d features from a larger d feature set this is a fundamental problem in statistical pattern recognition and combinatorial optimization exhaustive enumeration is computationally unfeasible in most applications this algorithm dynamically searches for the globally optimal solution on a minimum solution tree which is a subtree of the solution tree used in the traditional branch and bound algorithm the new algorithm is compared theoretically with the branch and bound algorithm the analysis and the experimental results show that it is more efficient than the traditional algorithm

a correction function method for poisson problems with interface jump conditions abstract in this paper we present a method to treat interface jump conditions for constant coefficients poisson problems that allows the use of standard black box solvers without compromising accuracy the basic idea of the new approach is similar to the ghost fluid method gfm the gfm relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface if the corrections are solution independent they can be moved to the right hand side rhs of the equations producing a problem with the same linear system as if there were no jumps only with a different rhs however achieving high accuracy is very hard if not impossible with the standard approaches used to compute the gfm correction terms in this paper we generalize the gfm correction terms to a correction function defined on a band around the interface this function is then shown to be characterized as the solution to a pde with appropriate boundary conditions this pde can in principle be solved to any desired order of accuracy as an example we apply this new method to devise a 4th order accurate scheme for the constant coefficients poisson equation with discontinuities in 2d this scheme is based on i the standard 9 point stencil discretization of the poisson equation ii a representation of the correction function in terms of bicubics and iii a solution of the correction function pde by a least squares minimization several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries its capability to capture sharp discontinuities and its high convergence rate

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

on one approach to solve the linear boundary value problems for fredholm integro differential equations necessary and sufficient conditions for the well posedness of linear boundary value problems for fredholm integro differential equations are established algorithms for finding their solutions are offered

numerical solutions of the nonlinear integro differential equations wavelet galerkin method and homotopy perturbation method this paper compares the homotopy perturbation method with the wavelet galerkin method for solving nonlinear integro differential equations from the computational viewpoint the homotopy perturbation method is more efficient and easy to use

fast branch bound algorithms for optimal feature selection a novel search principle for optimal feature subset selection using the branch bound method is introduced thanks to a simple mechanism for predicting criterion values a considerable amount of time can be saved by avoiding many slow criterion evaluations we propose two implementations of the proposed prediction mechanism that are suitable for use with nonrecursive and recursive criterion forms respectively both algorithms find the optimum usually several times faster than any other known branch bound algorithm as the algorithm computational efficiency is crucial due to the exponential nature of the search problem we also investigate other factors that affect the search performance of all branch bound algorithms using a set of synthetic criteria we show that the speed of the branch bound algorithms strongly depends on the diversity among features feature stability with respect to different subsets and criterion function dependence on feature set size we identify the scenarios where the search is accelerated the most dramatically finish in linear time as well as the worst conditions we verify our conclusions experimentally on three real data sets using traditional probabilistic distance criteria

a comparative study of suboptimal branch and bound algorithms abstract the branch and bound algorithm is widely known as an efficient approach for selecting optimal feature subsets if the optimality of the solution is allowed to be compromised it is possible to further improve the search speed of the branch and bound algorithm this paper studies the look ahead search strategy which can eliminate many solutions deemed to be suboptimal early in the branch and bound search we propose ways to incorporate the look ahead search scheme into four major branch and bound algorithms namely the basic branch and bound algorithm the ordered branch and bound algorithm the fast branch and bound algorithm and the adaptive branch and bound algorithm a comparative study of the look ahead scheme in terms of the computational cost and the solution quality on these suboptimal branch and bound algorithms is carried out on real data sets furthermore we test the feasible use of suboptimal branch and bound algorithms on a high dimensional data set and compare its performance with other well known suboptimal feature selection algorithms

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

numerical solution for the weakly singular fredholm integro differential equations using legendre multiwavelets an effective method based upon legendre multiwavelets is proposed for the solution of fredholm weakly singular integro differential equations the properties of legendre multiwavelets are first given and their operational matrices of integral are constructed these wavelets are utilized to reduce the solution of the given integro differential equation to the solution of a sparse linear system of algebraic equations in order to save memory requirement and computational time a threshold procedure is applied to obtain the solution to this system of algebraic equations through numerical examples performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation

variational iteration method for unsteady flow of gas through a porous medium using he s polynomials and pade approximants in this paper we apply the variational iteration method using he s polynomials for finding the analytical solution of unsteady flow of gas through a porous medium the proposed method is an elegant combination of he s variational iteration and the homotopy perturbation methods the suggested algorithm is quite efficient and is practically well suited for use in such problems the proposed iterative scheme finds the solution without any discretization linearization or restrictive assumptions the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution and to determine the initial slope a clear advantage of this technique over the decomposition method is that no calculation of adomian s polynomials is needed