on the numerical evaluation of electrostatic fields in dense random dispersions of cylinders we consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials consisting of dense random dispersions of cylinders in a uniform background this is a well studied problem from the viewpoint of homogenization and effective medium theory but one for which accurate numerical simulations have been difficult to obtain size effects in particular have been neglected due to the expense of solving the field equation in the presence of large numbers of close to touching inclusions such features require very fine discretizations even with the use of adaptive gridding and cause the linear systems which arise to be highly ill conditioned in this paper we present a new integral equation method for the solution of the interface problem which uses a recently developed method of images to resolve the close to touching interactions and the fast multipole method to compute far field interactions only minutes of workstation time are needed to solve the field equation with thousands of inclusions allowing us to carry out large scale statistical studies of the effective conductivity of random two phase materials at a variety of volume fractions and contrast ratios

a cartesian grid embedded boundary method for solving the poisson and heat equations with discontinuous coefficients in three dimensions we present a method for solving poisson and heat equations with discontinuous coefficients in two and three dimensions it uses a cartesian cut cell embedded boundary method to represent the interface between materials as described in johansen and colella 1998 matching conditions across the interface are enforced using an approximation to fluxes at the boundary overall second order accuracy is achieved as indicated by an array of tests using non trivial interface geometries both the elliptic and heat solvers are shown to remain stable and efficient for material coefficient contrasts up to 10 6 thanks in part to the use of geometric multigrid a test of accuracy when adaptive mesh refinement capabilities are utilized is also performed an example problem relevant to nuclear reactor core simulation is presented demonstrating the ability of the method to solve problems with realistic physical parameters

variational iteration method for the time fractional fornberg whitham equation this paper presents the approximate analytical solutions to solve the nonlinear fornberg whitham equation with fractional time derivative by using initial values explicit solutions of the equations are solved by using a reliable algorithm like the variational iteration method the fractional derivatives are taken in the caputo sense the present method performs extremely well in terms of efficiency and simplicity numerical results for different particular cases of a are presented graphically

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

computing eigenelements of boundary value problems with fractional derivatives variational iteration method has been successfully implemented to handle linear and nonlinear differential equations the main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper first a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in caputo sense second the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives numerical examples are tested to show the pertinent features of this method this approach provides a new way to investigate eigenvalue problems with fractional order derivatives

electrostatic fields without singularities theory algorithms and error analysis the following problems that arise in the computation of electrostatic forces and in the boundary element method are considered given two convex interior disjoint polyhedra in 3 space endowed with a volume charge density which is a polynomial in the cartesian coordinates of r 3 compute the coulomb force acting on them given two interior disjoint polygons in 3 space endowed with a surface charge density which is polynomial in the cartesian coordinates of r 3 compute the normal component of the coulomb force acting on them for both problems adaptive gaussian approximation algorithms are given which for n gaussian points in time o n achieve absolute error o c n for a constant c 1 such a result improves upon previously known best asymptotic bounds this result is achieved by blending techniques from integral geometry computational geometry and numerical analysis in particular integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities

fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation in this paper we use the fractional variational homotopy perturbation iteration method fvhpim with modified riemann liouville derivative to solve a time fractional diffusion equation using this method a rapid convergent sequence tending to the exact solution of the equation can be obtained to show the efficiency of the considered method some numerical examples are presented

fast finite element calculation of effective conductivity of random continuum microstructures the recursive poincare steklov operator method abstract the effective conductivity of a block of composite can be extracted from the dirichlet to neumann poincare steklov operator pso for that block in this paper a domain decomposition method for computing the pso over a finite element mesh is discussed a new numerical strategy is introduced to accelerate the computation of this operator using the schur complement to calculate the pso for the smallest subdomains then recursively merging subdomain psos up to the full domain at each step of the algorithm information extraneous to the pso is discarded the effective conductivity values computed by this method are identical to those obtained from a basic finite element method an order of magnitude faster and with much less computer memory consumed as proof of concept effective conductivity measurements are presented for a percolating random fractal like microstructure across a range of phase fractions

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

short note accelerating a fft based solver for numerical homogenization of periodic media by conjugate gradients in this short note we present a new technique to accelerate the convergence of a fft based solver for numerical homogenization of complex periodic media proposed by moulinec and suquet 1 the approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to vainikko 2 to give a linear system which can be efficiently solved by conjugate gradient methods computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high contrast coefficients at a low overhead per iteration

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

approximate solutions of fractional zakharov kuznetsov equations by vim this paper presents the approximate analytical solution of a fractional zakharov kuznetsov equation with the help of the powerful variational iteration method the fractional derivatives are described in the caputo sense several examples are given and the results are compared to exact solutions the results show that the variational iteration method is very effective convenient and simple to use

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

symmetry analysis of the time fractional gaudrey dodd gibbon equation abstract in this paper based on lie symmetry analysis method we study the invariance properties of the time fractional gaudrey dodd gibbon equation using two kinds of different similarity variables this equation can be reduced to two kinds of different nonlinear ordinary differential equations of fractional order the fractional derivatives corresponding to reduction equations are usually known as the erdelyi kober fractional derivative and riemann liouville fractional derivative respectively

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

local fourier analysis of multigrid for the curl curl equation we present a local fourier analysis of multigrid methods for the two dimensional curl curl formulation of maxwell s equations both the hybrid smoother proposed by hiptmair and the overlapping block smoother proposed by arnold falk and winther are considered the key to our approach is the identification of two dimensional eigenspaces of the discrete curl curl problem by decoupling the fourier modes for edges with different orientations this procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods additionally we identify the helmholtz splitting in fourier space this allows several well known properties to be recovered in fourier space such as the commutation properties of the classical nedelec prolongator and the equivalence of the curl curl operator and the vector laplacian for divergence free vectors we show how the approach used in this paper can be generalized to two and three dimensional problems in h curl and h div and to other types of regular meshes

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

hilfer fractional advection diffusion equations with power law initial condition a numerical study using variational iteration method abstract we propose a hilfer advection diffusion equation of order 0 1 and type 0 1 and find the power series solution by using variational iteration method power series solutions are expressed in a form that is easy to implement numerically and in some particular cases solutions are expressed in terms of mittag leffler function absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters for power law initial conditions it is shown that the hilfer advection diffusion pde gives the same solutions as the caputo and riemann liouville advection diffusion pde to leading order the fractional solution compared to the non fractional solution increases rapidly with for 0 7 at a given time t but for 0 7 this factor is weakly sensitive to we also show that the truncation errors arising when using the partial sum as approximate solutions decay exponentially fast with the number of terms n used we find that for 0 7 the number of terms needed is weakly sensitive to the accuracy level and to the fractional order n 20 but for 0 7 the required number of terms increases rapidly with the accuracy level and also with the fractional order

second order poisson nernst planck solver for ion transport the poisson nernst planck pnp theory is a simplified continuum model for a wide variety of chemical physical and biological applications its ability of providing quantitative explanation and increasingly qualitative predictions of experimental measurements has earned itself much recognition in the research community numerous computational algorithms have been constructed for the solution of the pnp equations however in the realistic ion channel context no second order convergent pnp algorithm has ever been reported in the literature due to many numerical obstacles including discontinuous coefficients singular charges geometric singularities and nonlinear couplings the present work introduces a number of numerical algorithms to overcome the abovementioned numerical challenges and constructs the first second order convergent pnp solver in the ion channel context first a dirichlet to neumann mapping dnm algorithm is designed to alleviate the charge singularity due to the protein structure additionally the matched interface and boundary mib method is reformulated for solving the pnp equations the mib method systematically enforces the interface jump conditions and achieves the second order accuracy in the presence of complex geometry and geometric singularities of molecular surfaces moreover two iterative schemes are utilized to deal with the coupled nonlinear equations furthermore extensive and rigorous numerical validations are carried out over a number of geometries including a sphere two proteins and an ion channel to examine the numerical accuracy and convergence order of the present numerical algorithms finally application is considered to a real transmembrane protein the gramicidin a channel protein the performance of the proposed numerical techniques is tested against a number of factors including mesh sizes diffusion coefficient profiles iterative schemes ion concentrations and applied voltages numerical predictions are compared with experimental measurements

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

guaranteed upper lower bounds on homogenized properties by fft based galerkin method abstract guaranteed upper lower bounds on homogenized coefficients arising from the periodic cell problem are calculated in a scalar elliptic setting our approach builds on the recent variational reformulation of the moulinec suquet 1994 fast fourier transform fft homogenization scheme by vond ejc et al 2014 which is based on the conforming galerkin approximation with trigonometric polynomials upper lower bounds are obtained by adjusting the primal dual finite element framework developed independently by dvo ak 1993 and wi ckowski 1995 to the fft based galerkin setting we show that the discretization procedure differs for odd and non odd number of grid points thanks to the helmholtz decomposition inherited from the continuous formulation the duality structure is fully preserved for the odd discretizations in the latter case a more complex primal dual structure is observed due to presence of the trigonometric polynomials associated with the nyquist frequencies these theoretical findings are confirmed with numerical examples to conclude the main advantage of the fft based approach over conventional finite element schemes is that the primal and the dual problems are treated on the same basis and this property can be extended beyond the scalar elliptic setting

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

analytical solution of a fractional diffusion equation by adomian decomposition method this paper presents an analytical solution of a fractional diffusion equation by adomian decomposition method by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

an efficient higher order fast multipole boundary element solution for poisson boltzmann based molecular electrostatics in order to compute polarization energy of biomolecules we describe a boundary element approach to solving the linearized poisson boltzmann equation our approach combines several important features including the derivative boundary formulation of the problem and a smooth approximation of the molecular surface based on the algebraic spline molecular surface state of the art software for numerical linear algebra and the kernel independent fast multipole method is used for both simplicity and efficiency of our implementation we perform a variety of computational experiments testing our method on a number of actual proteins involved in molecular docking and demonstrating the effectiveness of our solver for computing molecular polarization energy

new version fast multipole method accelerated electrostatic calculations in biomolecular systems in this paper we present an efficient and accurate numerical algorithm for calculating the electrostatic interactions in biomolecular systems in our scheme a boundary integral equation bie approach is applied to discretize the linearized poisson boltzmann pb equation the resulting integral formulas are well conditioned for single molecule cases as well as for systems with more than one macromolecule and are solved efficiently using krylov subspace based iterative methods such as generalized minimal residual gmres or biconjugate gradient stabilized bicgstab methods in each iteration the convolution type matrix vector multiplications are accelerated by a new version of the fast multipole method fmm the implemented algorithm is asymptotically optimal o n both in cpu time and memory usage with optimized prefactors our approach enhances the present computational ability to treat electrostatics of large scale systems in protein protein interactions and nano particle assembly processes applications including calculating the electrostatics of the nicotinic acetylcholine receptor nachr and interactions between protein sso7d and dna are presented

norm preserving discretization of integral equations for elliptic pdes with internal layers i the one dimensional case we investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers in one dimension the equations that arise can be solved analytically and the condition numbers estimated in various l p norms we show that high order accurate nystrom discretization leads to well conditioned finite dimensional linear systems if and only if the discretization is both norm preserving in a correctly chosen l p space and adaptively refined in the internal layer

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

fast analytical methods for macroscopic electrostatic models in biomolecular simulations we review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications including the poisson boltzmann pb and the generalized born models for electrostatic solvation energy the focus is on analytical approaches for hybrid solvation models especially the image charge method for a spherical cavity and also the generalized born theory as an approximation to the pb model this review places much emphasis on the mathematical details behind these methods

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

two dimensional modelling of the river rhine the system for operational water level forecast and prediction of fortunately not daily pollutant transport for the river rhine is in daily use this model is based on the saint venant or one dimensional shallow water equations the model is augmented by additional terms and equations to model the effect of dead zones and the transport of soluble components the next step is to move towards two dimensional models an important problem that arises is that the domain of the fluid is not fixed by the given data but depends on the water level and is therefore part of the solution of the model even worse depending on the topography of the river bed even the topology of the fluid domain may change as islands may appear at low water and get flooded at high water situations

wavelets method for solving fractional optimal control problems in this paper an efficient and accurate computational method based on the legendre wavelets lws is proposed for solving a class of fractional optimal control problems focps in the proposed method the focp under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved to this end the fractional derivative of the state variable and the control variable are expanded by the lws with unknown coefficients then the operational matrix of the riemann liouville fractional integration with some properties of the lws are employed to achieve a nonlinear algebraic equation in place of the performance index and a linear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients finally the method of constrained extrema which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined lagrange multipliers is applied as a result the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable control variable and lagrange multipliers furthermore the efficiency of the proposed method is shown for some concrete examples the results reveal that the proposed method is very accurate and efficient

discretization of the induced charge boundary integral equation boundary element methods bems for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries in this paper we consider a more accurate implementation of bem in the context of ions in aqueous solution near proteins but our results are applicable more generally the ions that modulate protein function are often within a few angstroms of the protein which leads to the significant accumulation of polarization charge at the protein solvent interface computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using bem in particular the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the bem matrix are calculated we demonstrate that when the dielectric interface is discretized into flat tiles the qualocation method of tausch et al ieee trans comput comput aided des 20 1398 2001 to compute the bem matrix elements is always more accurate than the traditional centroid collocation method qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

local fourier analysis for cell centered multigrid methods on triangular grids this work is focused on the design of efficient multigrid methods for discretizations on voronoi meshes associated with regular refinements of acute triangles for this purpose a special local fourier analysis for this type of discretizations is developed this analysis helps us to obtain appropriate novel smoothers depending on the geometry of the grid a two and three grid analysis are performed to analyze the behavior of the multigrid methods numerical test calculations validate the theoretical predictions furthermore the application of this analysis for the solution of more real problems by using semi structured grids is shown

a new approach of superconvergence analysis for two dimensional time fractional diffusion equation abstract in this paper a new approach of superconvergent estimate of bilinear finite element is established for two dimensional time fractional diffusion equation under fully discrete scheme the novelty of this approach is the combination technique of the interpolation and ritz projection as well as the superclose estimate in h 1 norm between them which avoids the difficulty of constructing a postprocessing operator for ritz projection operator and reduces the regularity requirement of the exact solution at the same time three numerical examples are carried out to verify the theoretical analysis

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

the effective conductivity of arrays of squares large random unit cells and extreme contrast ratios an integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two component checkerboard like composites with complicated unit cells at very high contrast ratios the scheme extends recent work on multi component checkerboards at medium contrast ratios general improvement include the simplification of a long range preconditioner the use of a banded solver and a more efficient placement of quadrature points this together with a reduction in the number of unknowns allows for a substantial increase in achievable accuracy as well as in tractable system size results accurate to at least nine digits are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106 furthermore the scheme is flexible enough to handle complex valued conductivities and using a homotopy method purely negative contrast ratios examples of the accurate computation of resonant spectra are given

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

combined equivalent charge formulations and fast wavelet galerkin bem for 3 d electrostatic analysis we describe a new equivalent charge formulation ecf i e the combined ecf cecf for electrostatic analysis of structures consisting of conductors and dielectrics the cecf uses a weighted combination of the single and the adjoint double layer operators to account for the potential on the conductor dielectric surface and is found to have better conditioning than the standard ecf a perturbation approach is presented to insure that the capacitances are computed accurately even when the permittivity ratios of the dielectrics are vary large unlike the original perturbation approach this new approach uses only one system matrix with different right hand sides a wavelet galerkin boundary element method wgbem for solving the ecf and cecf is developed based on the new variable order wgbem introduced in j numer math 2004 12 3 233 254 the wavelets are directly constructed on the usual boundary element triangulation this enables the proposed wgbem to solve electrostatic problems in complicated geometries unstructured meshes and comparatively coarse discretizations the quasi vanishing moment wavelets introduced in comput methods appl mech engng 2008 197 4000 4006 are used to further reduce the memory and cpu time requirements several numerical examples show that the proposed cecf converges faster than the ecf and that the wgbem using both the ecf and cecf has almost linear complexity in solving large scale 3 d electrostatic problems moreover since the truncated non standard form is computed once and then stored the wgbem is very suitable for solving problems with multiple right hand sides like the perturbation approach in this paper copyright 2009 john wiley sons ltd

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

numerical quadratures for layer potentials over curved domains in r 3 numerical quadratures for calculating simple and double layer potentials over curved domains in mathbb r 3 are given more generally these numerical quadratures are shown to be useful for evaluating surface integral involving kernels which are pseudohomogeneous of degree 1 in addition to potential flow problems such kernels are encountered when dealing with helmholtz problem stokes flow problems or linear elasticity problems

quadrature methods for 2d and 3d problems in this paper we give an overview on well known stability and convergence results for simple quadrature methods based on low order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds first we explain the methods for the case of second kind equations then we discuss what is known for the analysis of pseudodifferential equations we explain why these simple methods are not recommended for integral equations over domains with dimension higher than one finally for the solution of a two dimensional singular integral equation we prove a new result on a quadrature method based on product rules

high order numerical quadratures for layer potentials over curved domains in ℝ3 abstract high order numerical quadratures for approximating layer potentials over curved domains in 3 are given more generally these numerical quadratures are shown to be useful for evaluating surface integral involving kernels which are pseudo homogeneous of degree 1 a numerical application is presented

a novel high order approximate scheme for two dimensional time fractional diffusion equations with variable coefficient abstract based on the spatial quasi wilson nonconforming finite element method and temporal l 2 1 formula a fully discrete approximate scheme is proposed for a two dimensional time fractional diffusion equations with variable coefficient on anisotropic meshes in order to demonstrate the stable analysis and error estimates several lemmas are provided which focus on high accuracy about projection and superclose estimate between the interpolation and projection unconditionally stable analysis are derived in l 2 norm and broken h 1 norm moreover convergence result of accuracy o h 2 2 and superclose property of accuracy o h 2 2 are deduced by combining interpolation with projection where h and are the step sizes in space and time respectively and then the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

fast adaptive summation of point forces in the two dimensional poisson equation direct summation of the velocity field introduced by point vortices tends to be time consuming since the velocity of each vortex is found as a sum over all other vortices the resulting number of numerical operations is proportional to the square of the number of vortices here a relatively simple procedure is outlined which significantly reduces the number of operations by replacing selected partial sums by asymptotic series the resulting number of operations appears to vary roughly in proportion to the number of unknowns corresponding to a fast solver

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

fractional variational iteration method for fractional initial boundary value problems arising in the application of nonlinear science in this paper we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified riemann liouville fractional derivative proposed by g jumarie fractional order lagrange multiplier has been considered solution has been plotted for different values of a

the variational iteration method for solving a neutral functional differential equation with proportional delays in this paper the variational iteration method is applied to neutral functional differential equations with proportional delays illustrative examples are given to show the efficiency of the method we also compare the performance of the method with that of a particular runge kutta method and a one leg q method

an analysis of quadrature errors in second kind boundary integral methods this paper considers the effect of quadrature errors on galerkin methods for approximating the solution of second kind fredholm integral equations with weakly singular kernels defined on a smooth closed surface in three dimensions a quadrature strategy for galerkin methods that use piecewise poly nomials both continuous and discontinuous on a quadrilateral based mesh is proposed the error resulting from this further discretization is proved to be no larger than the original discretization error and moreover the amount of work needed to compute the resulting matrix for the corresponding linear system is shown to be proportional to the number of matrix entries

analytical solution of a fractional diffusion equation by variational iteration method in the present paper the analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful variational iteration method by using an initial value the explicit solutions of the equation for different cases have been derived which accelerate the rapid convergence of the series solution the present method performs extremely well in terms of efficiency and simplicity numerical results for different particular cases of the problem are presented graphically

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

numerical solution of fractional differential equations using the generalized block pulse operational matrix the riemann liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration also the generalized block pulse operational matrices of differentiation are derived based on the above results we propose a way to solve the fractional differential equations the method is computationally attractive and applications are demonstrated through illustrative examples

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions

a boundary element formulation of protein electrostatics with explicit ions a novel formulation based upon continuum electrostatics to compute the electrostatic potential in and around two biomolecules in a solvent with ionic strength is presented many if not all current methods rely on the non linear poisson boltzmann equation to include ionic strength the present formulation however describes ionic strength through the inclusion of explicit ions which considerably extends its applicability and validity range the method relies on the boundary element method and results in two very similar coupled integral equations valid on the dielectric boundaries of two molecules respectively the method can be employed to estimate the total electrostatic energy of two protein molecules at a given distance and orientation in an electrolyte solution with zero to moderately high ionic strength the formulation is equally applicable to the case of a single solute in an electrolyte a number of monte carlo simulations of protein like solutes in nacl at various concentrations have been performed and demonstrates the method s usefulness to estimate the electrostatic contribution to the potential of mean force between two biomolecules in an electrolyte