a hybrid 2 d 3 d multilevel green s function interpolation method for electrically large multilayered problems the multilevel green s function interpolation method has been implemented to solve the general electrically large multilayered problems a new hybrid 2 d and 3 d h2 d 3 d multilevel interpolation method and a cross level translation scheme are devised for the specific problems from which a quad tree is constructed the interpolation of the multilayered medium green s function using radial basis function is then studied to further accelerate the computation a subdomain fast fourier transform scheme and a sparse approximated inverse preconditioner with a new discarding strategy are also incorporated into the algorithm as numerical examples typical microstrip antenna arrays a metasurface antenna and electromagnetic bandgap integrated antenna arrays are analyzed with the proposed method and excellent performance is observed in both accuracy and efficiency

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

corrections to comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method aug 10 2691 2699 in the above titled paper ibid vol 58 no 8 pp 2691 2699 aug 10 table ii contained errors errors also appeared in equations 2 and 3 corrections are presented here

an improved multilevel green s function interpolation method with adaptive phase compensation an improved multilevel green s function interpolation method mlgfim with adaptive phase compensation apc is proposed the difficulty in applying interpolation approaches to the fast varying phase term in the integral equation kernel for full wave electromagnetic em simulations is eradicated by using the phase compensation and adaptive direction separation ads the multilevel tree structure in mlgfim keeps the number of direction separation invariant at all levels attributing to the recursive interpolation with multilevel phase compensation the proposed mlgfim apc in conjunction with the lagrange chebyshev interpolation yield an o n log n cpu time and o n computer memory requirement for surface scattering problems by introducing a transition level the mlgfim apc can adaptively incorporate interpolation techniques of conventional interpolation ci transition interpolation ti and phase compensation interpolation pi corresponding to electromagnetic simulation of problems of small moderate and large electrical sizes respectively large scale microstrip antenna arrays are simulated to illustrate the accuracy and efficiency of the proposed method it is found that the cpu time scales better than o n log n for these co planar problems

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

an efficient solution for cauchy likesystems of linear equations using the displacement approach and transformation of vectors the complexity ofsolving strongly nonsingular cauchy like systems of linear equations email protected email protected will be improved off a factor of log n from the known algorithms of the recursive triangular factorization where f is a field and email protected f n x 1 is a vector and c f n x n is a strongly nonsingular cauchy like matrix the efficient algorithms were presented by morf bitmead and anderson to solve strongly nonsingular toeplitz like equations of linear systems by using the recursive triangular factorization in 1980 recently this approach of the recursive triangular factorization was extended to solve cauchy like systems with the complexity of o n log 3n operations by pan et al this is the best known complexity bound by using the direct approach of recursive triangular factorization in cauchy like cases however these algorithms are still slower than the well known algorithms with the asymptotic bound of o n log 2n operations which have been proposed by the means of reducing cauchy like matrices into toeplitz like matrices in our present paper the algorithms of the recursive triangular factorization are modified so that the complexity bound of the direct recursive approach can be decreased to o n log 2n operations this matches the asymptotic bound without transforming to toeplitz like matrices our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors expressed in the given cauchy like matrix into the special vectors where the entries are unit roots the applications of structured matrices include nevanlinna pick tangential interpolation problems

time dependent solutions for a fractional schrodinger equation with delta potentials we investigate for an arbitrary initial condition the time dependent solutions for a fractional schrodinger equation in the presence of delta potentials by using the green function approach the solutions obtained show an anomalous spreading asymptotically characterized by a power law behavior which is governed by the order of the fractional spatial operator present in the schrodinger equation

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

some solutions to the space fractional schrodinger equation using momentum representation method the space fractional schrodinger equation with linear potential delta function potential and coulomb potential is studied under momentum representation using fourier transformation by use of mellin transform and its inverse transform we obtain the energy levels and wave functions expressed in h function for a particle in linear potential field the wave function expressed also by the h function and the unique energy level of the bound state for the particle of even parity state in delta function potential well which is proved to have no action on the particle of odd parity state is also obtained the integral form of the wave functions for a particle in coulomb potential field is shown and the corresponding energy levels which have been discussed in laskin s paper phys rev e 66 056108 2002 are proved to satisfy an equality of infinite limit of the h function all of these results contain the ones of the standard quantum mechanics as their special cases

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

fractional schrodinger equation with noninteger dimensions the spatial and time dependent solutions of the schrodinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated they are obtained by using the green function approach in two situations the free case and in the presence of a harmonic potential the results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process

toy model of fractional transport of cancer cells due to self entrapping a simple mathematical model is proposed to study the influence of cell fission on transport the model describes fractional in time tumor development which is a one dimensional continuous time random walk the model is relevant for consideration of both solid and diffusive cancers

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

exact solutions of fractional schrodinger like equation with a nonlocal term we study the time space fractional schrodinger equation with a nonlocal potential by the method of fourier transform and laplace transform the green function and hence the wave function is expressed in terms of h functions graphical analysis demonstrates that the influence of both the space fractal parameter and the nonlocal parameter on the fractional quantum system is strong indeed the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative

parameter estimation for the fractional schrodinger equation using bayesian method in this paper the fractional schrodinger equation is studied the bayesian method is put forward to estimate some relevant parameters of the equation results show that the estimated values can fit well with the exact solution the varying initial values and maximum iterations have little effect on the estimated results it indicates that the bayesian method is efficient for the multi parameter estimation for the fractional schrodinger equation this method can also be used to estimate parameters for the fractional schrodinger equation in other potential field

subdiffusion in random compressible flows in this work we study the diffusion of admixture particles in a one dimensional velocity field given by a gradient of a random potential this refers us to the case of random compressible flows where previously only scaling estimates were available we develop a general approach which allows to solve this problem analytically with its help we derive the macroscopic transport equation and rigorously show in which cases transport can be subdiffusive we find the fourier laplace transform of the green s function of this equation and prove that for some potential distributions it satisfies the subdiffusive equation with fractional derivative with respect to time

n dimensional fractional diffusion equation and green function approach spatially dependent diffusion coefficient and external force we investigate an n dimensional fractional diffusion equation with radial symmetry by using the green function approach we consider in our analysis the spatial dependence on the diffusion coefficient and the presence of an external force in particular we employ boundary conditions in a finite interval and after we extend it to a semi infinite interval we also show that a rich class of diffusive processes including normal and anomalous ones can be obtained from the solutions found here

the implementation of multilevel green s function interpolation method for full wave electromagnetic problems we extend the multilevel green s function interpolation method mlgfim developed for quasi electrostatic problems to full wave simulations the difficulty in applying the interpolation approach lies in the additional rapidly changing phase term associated with the full wave green s functions to enhance the efficiency and accuracy of the full wave green s function interpolation a scattered point set consisting of two staggered tartan grids in conjunction with radial basis function interpolation is employed to further reduce the computational complexity the qr factorization technique is applied to compress the low rank green s function matrices for electromagnetic scattering from pec spheres up to a diameter of eight wavelengths the proposed method compares well with mie s scattering in accuracy and shows the o nlogn efficiency as the method is kernel independent its extension to structures in layered medium is straightforward in the numerical simulations of finite microstrip patch arrays up to 11 by 11 elements the proposed method demonstrates very favorable dependencies of cpu time and memory storage requirement versus the number of unknowns

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

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

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

toric varieties and modular forms let n r be a lattice and let deg n be a piecewise linear function that is linear on the cones of a complete rational polyhedral fan under certain conditions on deg the data n deg determines a function f that is a holomorphic modular form of weight r for the congruence subgroup 1 l moreover by considering all possible pairs n deg we obtain a natural subring l of modular forms with respect to 1 l we construct an explicit set of generators for l and show that l is stable under the action of the hecke operators finally we relate l to the hirzebruch elliptic genera that are modular with respect to 1 l

periods of modular forms and identities between eisenstein series borisov and gunnells observed in 2001 that certain linear relations between products of two holomorphic weight 1 eisenstein series had the same structure as the relations between periods of modular forms a similar phenomenon exists in higher weights we give a conceptual reason for this observation in arbitrary weight this involves an unconventional way of expanding the rankin selberg convolution of a cusp form with an eisenstein series we also prove a partial result towards understanding the action of a hecke operator on a product of two eisenstein series

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

solution of a modified fractional diffusion equation abstract recently a modified fractional diffusion equation has been proposed i sokolov j klafter from diffusion to anomalous diffusion a century after einstein s brownian motion chaos 15 2005 026103 a v chechkin r gorenflo i m sokolov v yu gonchar distributed order time fractional diffusion equation frac calc appl anal 6 3 2003 259279 i m sokolov a v checkin j klafter distributed order fractional kinetics acta phys pol b 35 2004 1323 for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term in this letter we give the solution of the modified equation on an infinite domain in contrast to the solution of the traditional fractional diffusion equation the solution of the modified equation requires an infinite series of fox functions instead of a single fox function

solutions for a schrodinger equation with a nonlocal term we obtain time dependent solutions for a schrondiger equation in the presence of a nonlocal term by using the green function approach these solutions are compared with recent results obtained for the fractional schrodinger equation as well as for the usual one the nonlocal term incorporated in the schrodinger equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution

a fast volume surface integral equation solver for scattering from composite conducting dielectric objects this paper presents a fast hybrid volume surface integral equation approach for the computation of electromagnetic scattering from objects comprising both conductors and dielectric materials the volume electric field integral equation is applied to the material region and the surface electric field integral equation is applied on the conducting surface the method of moments mom is used to convert the integral equation into a matrix equation and the precorrected fft p fft method is employed to reduce the memory requirement and cpu time for the matrix solution the present approach is sufficiently versatile in handling problems with either open or closed conductors and dielectric materials of arbitrary inhomogeneity due to the combination of the surface and volume electric field integral equations the application of the precorrected fft method facilitates the solving of much larger problems than can be handled by the conventional mom

comparison of interpolating functions and interpolating points in full wave multilevel green s function interpolation method the difficulty for the solution of full wave electromagnetic problems using multilevel green s function interpolation method mlgfim lies in applying interpolating approaches to efficiently and accurately approximate green s function with rapidly changing phase term we compare various interpolating schemes when radial basis function rbf is employed for the interpolation of scattered data of green s function we show that the infinitely smooth gaussian ga rbf has the best interpolation accuracy in order to improve the interpolation efficiency a new kind of staggered tartan grid is proposed a good calculation method for the shape parameter in ga rbf is given to solve its sensitivity to the group size and the number of interpolation points based on the analysis of variation of the number of interpolation points with electric length of the group adaptive choice of the types of interpolation functions and interpolation points are employed numerical examples show that the computational efficiency of this new interpolation scheme is much improved over the previously reported ones

behavior of fractional diffusion at the origin the present work discusses the fractional diffusion equation based on the riemann liouville fractional time derivatives it was shown that the normalization conservation constraint leads to the divergency of diffusive agent concentration at the origin this divergency implies an external source of the diffusive agent at r 0 thus the riemann liouville fractional time derivative implies a loss of diffusive agent mass which is compensated for by the source of this agent at the origin in contrast the absence of the normalization conservation constraint does not lead to any divergences in the limit r 0 and at the same time provides the decay of normalization

toric modular forms and nonvanishing of l functions in a previous paper 1 we defined the space of toric forms t l and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group 1 l in this article we prove the following theorem modulo eisenstein series the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that l f 1 6 0 the proof uses work of merel and involves an explicit computation of the intersection pairing on manin symbols

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

generalized fractional schrodinger equation with space time fractional derivatives in this paper the generalized fractional schrodinger equation with space and time fractional derivatives is constructed the equation is solved for free particle and for a square potential well by the method of integral transforms fourier transform and laplace transform and the solution can be expressed in terms of mittag leffler function the green function for free particle is also presented in this paper finally we discuss the relationship between the cases of the generalized fractional schrodinger equation and the ones in standard quantum

toric modular forms of higher weight in the papers 1 2 we used the geometry of complete polyhedral fans to construct a subring t l of the modular forms on 1 l and showed that for weight two the cuspidal part of t l coincides with the space of cusp forms of analytic rank zero in this paper we show that in weights greater than two the cuspidal part of t l coincides with the space of all cusp forms

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

the role of the fox wright functions in fractional sub diffusion of distributed order the fundamental solution of the fractional diffusion equation of distributed order in time usually adopted for modelling sub diffusion processes is obtained based on its mellin barnes integral representation such solution is proved to be related via a laplace type integral to the fox wright functions a series expansion is also provided in order to point out the distribution of time scales related to the distribution of the fractional orders the results of the time fractional diffusion equation of a single order are also recalled and then re obtained from the general theory

an openmp parallelized multilevel green s function interpolation method accelerated by fast fourier transform technique a parallelized multilevel green s function interpolation method mlgfim accelerated by fast fourier transform fft technique is proposed the difficulties in applying various improved interpolation schemes to effectively reduce the number of interpolation points are overcome by using the fft technique in order to accelerate the convergence property of the iterative solution using the proposed algorithm a recently proposed preconditioning scheme i e multiplicative calderon preconditioner is adopted to transform the first kind integral operator to the second kind albeit an increase of computer memory storage requirement an openmp parallel implementation of the mlgfim fft algorithm on a share memory computer system is developed to analyze various electrically large electromagnetic scattering problems including a nasa almond a 20 wavelength cylinder capped with two half spheres and a 37 wavelength cylinder array numerical results illustrate good computational performance of the proposed algorithm