fast calculation of laurent expansions for matrix inverses previously described algorithms for calculating the laurent expansion of the inverse of a matrix valued analytic function become impractical already for singularity orders as low as around p 6 since they require over o 28p matrix multiplications and correspondingly large amounts of memory in place of using mathematically exact recursions we show that for floating point calculations a rational approximation approach can avoid this cost barrier without any significant loss in accuracy

fractional high order methods for the nonlinear fractional ordinary differential equation abstract in this paper we consider the nonlinear fractional order ordinary differential equation 0 d t y t f y t t 0 n 1 n y i 0 y 0 i i 0 1 2 n 1 where f y t satisfies the l condition i e f y 1 t f y 2 t l y 1 y 2 in t 0 t fractional order linear multiple step methods are introduced the high order 2 6 approximations of the fractional order ordinary differential equation with an initial value are proposed the consistence convergence and stability of the fractional high order methods are proved finally some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods

the convergence of fully discrete galerkin approximations for the benjamin bona mahony bbm equation abstract in the present paper we analyze a second order in time fully discrete finite element method for the bbm equation the discretization in space is based on the standard galerkin method for the time discretization the crank nicolson scheme is used we also prove the convergence of a linearized galerkin modification scheme

linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

development of the kansa method for solving seepage problems using a new algorithm for the shape parameter optimization abstract in this research the kansa or multiquadric method mq has been developed for solving the seepage problems in 2d and 3d arbitrary domains this research is the first application of this method for seepage analysis in both confined and unconfined porous media the domain decomposition approach has been employed for applying mq method easily in inhomogeneous and irregular complex geometries and decreasing the computational costs for determining the optimum shape parameter that affects strongly the accuracy of mq and other rfb methods a new scheme that decreases drastically the computational time is introduced the efficiency of the proposed algorithm has been examined under various radial basis functions variations of number of interpolating points and points distribution through a numerical example with analytical solution eventually three examples including different boundary conditions are presented comparing results of the examples with other numerical methods indicates that the present approach has high capability and accuracy in solving seepage problems

a meshless local collocation method for time fractional diffusion wave equation abstract in this manuscript we present a radial basis function based local collocation method for solving time fractional diffusion wave equation the advantage of the local collocation method is only the discretization points located in each sub domain surrounding the collocation point need to be considered it also avoids the ill conditioning problem arises due to the use of global collocation method the theoretical stability and convergence of the proposed time semi discrete scheme are also discussed numerical experiments for one dimension and two dimension problems are carried out it is shown that the present method is efficient and numerical results have nice agreements with theoretical result

a three level average implicit finite difference scheme to solve equation obtained by coupling the rosenau kdv equation and the rosenau rlw equation in the present work a mathematical model to obtain the solution of the nonlinear wave by coupling the rosenau kdv equation and the rosenau rlw equation is proposed the solution properties are also derived a numerical tool is applied to the model by using a three level average implicit finite difference technique the fundamental conservative properties of the equation are preserved by the presented numerical scheme and the existence and uniqueness of the numerical solution are proved moreover the convergence and stability of the numerical solution are also shown the new method give second order accurate in time and space thus the presented results can be constructed to demonstrate the viability of the new model

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

some new solitonary solutions of the modified benjamin bona mahony equation abstract in this paper we use the exp function method to construct some new soliton solutions of the benjamin bona mahony and modified benjamin bona mahony equations these equations have important and fundamental applications in mathematical physics and engineering sciences the exp function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation this method provides the concise and straightforward solution in a very easier way the results obtained in this paper can be viewed as a refinement and improvement of the previously known results

theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions multivariate interpolation of smooth data using smooth radial basis functions is considered the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically explicit criteria for different types of limits are given using the results for the limits the dependence of the error on the shape parameter of the radial basis function is investigated the mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

optimal decay rates of solutions for a multidimensional generalized benjamin bona mahony equation abstract in this paper we study the optimal decay rates of solutions for the generalized benjamin bona mahony equation in multi dimensional space n 3 by using fourier transform and the energy method we obtain the l q 2 q convergence rates of the solutions under the condition that the initial data is small the optimal decay rates obtained in this paper are found to be the same as the decay rate for the heat equation

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

global attractors for doubly nonlinear evolution equations with non monotone perturbations abstract this paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non monotone perturbations in a reflexive banach space setting in order to construct global attractors an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach since solutions of the cauchy problem for the equation might not be unique moreover the preceding abstract theory is applied to a generalized allen cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients

galerkin finite element solution for benjamin bona mahony burgers equation with cubic b splines abstract in this article we study solitary wave solutions of the nonlinear benjamin bona mahony burgers bbm burgers equation based on a lumped galerkin technique using cubic b spline finite elements for the spatial approximation the existence and uniqueness of solutions of the galerkin version of the solutions have been established an accuracy analysis of the galerkin finite element scheme for the spatial approximation has been well studied the proposed scheme is carried out for four test problems including dispersion of single solitary wave interaction of two three solitary waves and development of an undular bore then we propose a full discrete scheme for the resulting ivp von neumann theory is used to establish stability analysis of the full discrete numerical algorithm to display applicability and durableness of the new scheme error norms l 2 l and three invariants i 1 i 2 and i 3 are computed and the acquired results are demonstrated both numerically and graphically the obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

maximal monotonicity for the sum of two subdifferential operators in lp spaces abstract this paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive banach spaces and proving the maximal monotonicity in l p l p of the nonlinear elliptic operator u m u u with a maximal monotone graph

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

applying multiquadric quasi interpolation to solve burgers equation abstract in this paper we develop a kind of univariate multiquadric mq quasi interpolation and use it to solve burgers equation with viscosity at first we construct the mq quasi interpolation which possesses the properties of linear reproducing and preserving monotonicity next we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable then we verify our method for two examples with distinguishing initial value condition one example is tested for three reynolds number that is r 10 r 100 and r 10 000 from the numerical experiments we see that the presented method in this paper is valid although the accuracy of the technique is not higher than hon and mao s one another example is used to examine the travelling of the shock we can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate mq quasi interpolation the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

the propagation of polarization in double refraction abstract we define systems of uniaxial type which generalize maxwell s system of equations for uniaxial crystals we compute the geometric invariants of these systems using energy estimates we prove results on the propagation of sobolev polarization sets the polarization sets refine the wave front sets of vector valued distributions

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

a linearly implicit conservative difference scheme for the generalized rosenau kawahara rlw equation this paper concerns the numerical study for the generalized rosenau kawahara rlw equation obtained by coupling the generalized rosenau rlw equation and the generalized rosenau kawahara equation we first derive the energy conservation law of the equation and then develop a three level linearly implicit difference scheme for solving the equation we prove that the proposed scheme is energy conserved unconditionally stable and second order accurate both in time and space variables finally numerical experiments are carried out to confirm the energy conservation the convergence rates of the scheme and effectiveness for long time simulation

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

numerical scheme for a model of shallow water waves in 2 1 dimensions abstract a nonlinear difference scheme is considered for the two dimensional rosenau burgers equation some priori estimates existence and uniqueness of the difference solution have been shown a second order convergence in the uniform norm and stability are proved also a convergent iterative algorithm is presented all results are obtained without any restrictions on the meshsizes at last numerical experiments are carried out to support the theoretical claims

the method of variably scaled radial kernels for solving two dimensional magnetohydrodynamic mhd equations using two discretizations the crank nicolson scheme and the method of lines mol abstract mhd equations have many applications in physics and engineering the model is coupled equations in velocity and magnetic field and has a parameter namely hartmann the value of hartmann number plays an important role in the equations when this parameter increases using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem in the present paper a numerical meshless method based on radial basis functions rbfs is provided to solve mhd equations for approximating the spatial variable a new approach which is introduced by bozzini et al 2015 is applied the method will be used here is based on the interpolation with variably scaled kernels the methodology of the new technique is defining the scale function c on the domain r d then the interpolation problem from the data locations x j r d transforms to the new interpolation problem in the data locations x j c x j r d 1 bozzini et al 2015 the radial kernels used in the current work are multiquadrics mq inverse quadric iq and wendland s function of course the latter one is based on compactly supported functions to discretize the time variable two techniques are applied one of them is the crank nicolson scheme and another one is based on mol the numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem also in appendix we provide a computational algorithm for implementing the new technique in matlab software

stable computations with flat radial basis functions using vector valued rational approximations one commonly finds in applications of smooth radial basis functions rbfs that scaling the kernels so they are flat leads to smaller discretization errors however the direct numerical approach for computing with flat rbfs rbf direct is severely ill conditioned we present an algorithm for bypassing this ill conditioning that is based on a new method for rational approximation ra of vector valued analytic functions with the property that all components of the vector share the same singularities this new algorithm rbf ra is more accurate robust and easier to implement than the contour pade method which is similarly based on vector valued rational approximation in contrast to the stable rbf qr and rbf ga algorithms which are based on finding a better conditioned base in the same rbf space the new algorithm can be used with any type of smooth radial kernel and it is also applicable to a wider range of tasks including calculating hermite type implicit rbf fd stencils we present a series of numerical experiments demonstrating the effectiveness of this new method for computing rbf interpolants in the flat regime we also demonstrate the flexibility of the method by using it to compute implicit rbf fd formulas in the flat regime and then using these for solving poisson s equation in a 3 d spherical shell

exponential decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations abstract in this paper we investigate the exponential time decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations e u t u t x x u x x u x f u x 0 t 0 x r with prescribed initial data i u 0 x u 0 x u as x here 0 r are constants u are two given constants satisfying u u and the nonlinear function f u c 2 r is assumed to be either convex or concave based on the existence of traveling waves the local stability and the algebraic decay rate to traveling waves of solutions to the cauchy problem e and i established in yin et al 2007 13 we show an exponential decay rate of the solutions to the cauchy problem e and i toward the traveling waves mentioned above by employing the space time weighted energy method which was initiated by kawashima and matsumura in 1985 14 and later elaborated by matsumura and nishihara 1994 15 and nishikawa 1998 16

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

the numerical simulation of the phase field crystal pfc and modified phase field crystal mpfc models via global and local meshless methods abstract in the current work we consider the two dimensional time dependent phase field crystal pfc and the modified phase field crystal models mpfc to obtain their numerical solutions for this purpose we apply two numerical meshless methods based on radial basis functions rbfs and also two meshless methods which are based on moving least squares method mls four techniques developed in this paper are globally radial basis functions grbfs radial basis functions pseudo spectral rbfs ps moving least squares mls and generalized moving least squares gmls approximations two methods based on rbfs are global and the other methods based on moving least squares are local procedures as is well known the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non regular domains with different choices of grids in high dimensions applying the new methods on spatial domain and also using the semi implicit scheme for time variable yield a linear system of algebraic equations to solve this linear system the lu decomposition algorithm and command backslash in matlab software for gmls method are applied the implementation of boundary conditions in the rbfs ps is applied directly because the boundary conditions of the mentioned problems are periodic some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper moreover in the appendix of the paper a matlab code for gmls method is written

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

a new class of oscillatory radial basis functions radial basis functions rbfs form a primary tool for multivariate interpolation and they are also receiving increased attention for solving pdes on irregular domains traditionally only nonoscillatory radial functions have been considered we find here that a certain class of oscillatory radial functions including gaussians as a special case leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat this flat limit is important in that it generalizes traditional spectral methods to completely general node layouts interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions such as multiquadrics and inverse multiquadratics

lebesgue function for multivariate interpolation by radial basis functions multivariate interpolation and approximation is a powerful tool for intuition of real world in this paper we study lebesgue function and lebesgue constant for multivariate interpolation by conditionally positive definite rbfs also we give a conjecture about similar bernstein and erdos problem to multivariate case

on the propagation of polarization in conical refraction on etudie la propagation et la distribution des ensembles de polarisation pour des systemes d equations ayant une singularite conique involutive de l ensemble caracteristique

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

the lie group method based on radial basis functions for solving nonlinear high dimensional generalized benjamin bona mahony burgers equation in arbitrary domains abstract the aim of this paper is to introduce a new numerical method for solving the nonlinear generalized benjamin bona mahony burgers gbbmb equation this method is combination of group preserving scheme gps with radial basis functions rbfs which takes advantage of two powerful methods one as geometric numerical integration method and the other meshless method thus we introduce this method as the lie group method based on radial basis functions lg rbfs in this method we use kansas approach to approximate the spatial derivatives and then we apply gps method to approximate first order time derivative one of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions to demonstrate this point we solve nonlinear gbbmb equation on various geometric domains in one two and three dimension spaces the results of numerical experiments are compared with analytical solutions and the method presented in dehghan et al 2014 to confirm the accuracy and efficiency of the presented method

a new algorithm based on lucas polynomials for approximate solution of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equation in this paper a new method based on hybridization of lucas and fibonacci polynomials is developed for approximate solutions of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equations firstly time discretization is made by using finite difference approaches after that unknown function and its derivatives are expanded to lucas series based on these series expansion differentiation matrices are derived by utilizing fibonacci polynomials by doing so the solution of the mentioned equations is reduced to the solution of an algebraic system of equations by solving this system of equations the lucas series coefficients are obtained then substituting these coefficients into lucas series expansion approximate solutions can be constructed successively the main goal of this paper is to indicate that lucas polynomial based method is appropriate for 1d and 2d nonlinear problems efficiency and performance of the proposed method are judged on six test problems which consists of the 1d and 2d version of mentioned equation by calculating l2 and l error norms feasibility of the method is verified by obtained accurate results

legendre wavelets based numerical algorithm for simulation of multidimensional benjamin bona mahony burgers and sobolev equations abstract in this work a numerical algorithm is developed with the help of legendre wavelets and quasilinearization method for simulation of multidimensional benjamin bona mahony burgers bbmb and sobolev equations initially non linear equation is linearized by quasilinearization method and then legendre wavelets are used for both space and temporal discretization the algorithm is developed without the aid of finite difference method to preserve the accuracy of the legendre wavelets in numerical experiments time dependent 1d 2d and 3d problems are considered to check the accuracy and efficiency of the algorithm a comparison is made with the analytical and numerical solutions available in the literature it is concluded that the present algorithm without finite difference based legendre method gives better results than finite difference based haar wavelet methods for time dependent 2d bbmb and 2d sobolev equations also numerical experiments reveal that the proposed algorithm produces more accurate results even with comparatively coarser grids

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

vector field approximation using radial basis functions in this article we investigate the performance of rbf pde methods for approximating solenoidal fields it is well known that global rbf collocation methods present a trade off principle which means that smoothness implies high convergence order plus ill conditioning on the other hand local methods for solving this problem have recently appeared in the literature in this paper we perform a numerical investigation of the differences between rbf global and local methods in order to investigate the possible advantage of using local methods for the approximation of vector fields more precisely we compare the local hermite interpolation technique using inverse multiquadrics against the non symmetric collocation method of kansa

the use of compact boundary value method for the solution of two dimensional schrodinger equation in this paper a high order and accurate method is proposed for solving the unsteady two dimensional schrodinger equation we apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulting linear system of ordinary differential equations the proposed method has fourth order accuracy in both space and time variables moreover this method is unconditionally stable due to the favorable stability property of boundary value methods the results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature these results show that the combination of a compact finite difference approximation of fourth order and a fourth order boundary value method gives an efficient algorithm for solving the two dimensional schrodinger equation

numerical solutions of two dimensional sobolev and generalized benjamin bona mahony burgers equations via haar wavelets abstract this work addresses numerical method based on haar wavelets and finite differences to solve two dimensional linear nonlinear sobolev and non linear generalized benjamin bona mahony burgers ngbbmb equations the temporal part is discretized using finite differences while spatial part is approximated by two dimensional haar wavelets with this strategy computing solution of two dimensional pdes reduces to computing solution of linear system of algebraic equations collocation approach is then applied to determine the wavelet coefficients from linear system this paper shows that two dimensional haar wavelets are suitable and effective for two dimensional linear and non linear pdes for validation of the proposed scheme different problems have been solved and error norms l l 2 are computed computation verifies that current scheme has good outcome

on the convergence of the modified rosenau and the modified benjamin bona mahony equations abstract we consider the modified rosenau and the modified benjamin bona mahony equations which contain nonlinear dispersive effects we prove that as the diffusion parameter tends to zero the solutions of the dispersive equations converge to entropy solutions of a scalar conservation law the proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the l p setting

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

a unified theory of radial basis functions native hilbert spaces for radial basis functions ii abstract this contribution provides a new formulation of the theory of radial basis functions in the context of integral operators instead of fourier transforms the most important tools now are expansions into eigenfunctions this unifies the theory of radial basis functions in r d with the theory of zonal functions on the sphere sd 1 and the theory of kernel functions on riemannian manifolds new characterizations of native spaces and positive definite functions are included the paper is a self contained continuation of an earlier survey r schaback international series of numerical mathematics vol 132 birkhauser basel 1999 pp 255 282 over the native spaces associated to not necessarily radial basis functions

error estimate in fractional differential equations using multiquadratic radial basis functions the purpose of this paper is to investigate the collocation method based on multiquadric mq radial basis functions rbfs for fractional differential equations fdes in the process of doing this a scattered data rbf interpolation is considered a posterior error bound for the smooth solutions of fdes in the proposed method is established in the l 2 norm using an alternate characterization of the native space numerical results are presented which confirm the theoretical prediction of the convergence behavior of the proposed method

on the convergence of difference schemes for the benjamin bona mahony bbm equation abstract in this paper we analyze a crank nicolson type finite difference scheme for the bbm equation we prove the second order convergence in discrete h 1 norm of the difference approximate solutions the existence stability and uniqueness are also discussed in detail at last a linearized difference scheme is given and its convergence is also proved

a new class of polynomial functions for approximate solution of generalized benjamin bona mahony burgers gbbmb equations abstract in this paper a new class of polynomial functions equipped with a parameter is used which is applied to approximate the gbbmb in this method the first time discretization of the gbbmb equations is made by using finite difference approaches then at any spatial point a nonlinear equations system is generated using the collocation points and solving a nonlinear system in the least squares method the coefficients of functions are obtained in some examples a linearization is performed first and then the collocation method is applied

galerkin finite element method for the rosenau rlw equation finite element galerkin solutions for the rosenau rlw rrlw equation are considered the existence and uniqueness of the weak solution are shown and the semidiscrete galerkin scheme is studied using appropriate projections for a second order accuracy in time we propose the galerkin crank nicolson fully discrete method we also prove the convergence of the extrapolated crank nicolson scheme finally some numerical experiments are given to demonstrate the validity and accuracy of our method

convergence analysis for a second order elliptic equation by a collocation method using scattered points a collocation method using scattered points applied to a second order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials we show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

numerical simulation of kadomtsev petviashvili benjamin bona mahony equations using finite difference method in this paper the finite difference method is employed to solve kadomtsev petviashvili benjamin bona mahony ii kp bbm ii partial differential equations the time and space variable are discretized by the crank nicholson method and the central difference scheme respectively the consistence and stability are also proved some examples are investigated to verify the efficiency of the present method

a conservative linear difference scheme for the 2d regularized long wave equation abstract in the present work a three level in time linear and conservative implicit finite difference scheme for solving the 2d regularized long wave equation is proposed the existence uniqueness and conservations for mass and energy of the numerical solution are proved by the discrete energy method the new scheme is shown to be second order convergent and unconditionally stable numerical examples are provided to show the present scheme to be efficient and reliable

asymptotically periodic solutions of fractional differential equations abstract we study the existence of pseudo s asymptotically periodic mild solutions for a class of abstract fractional differential equation

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

l p error estimates for shifted surface spline interpolation on sobolev space the accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant f is reasonably smooth however for functions which have smoothness below a certain order associated with the basis function no approximation power has yet been established hence the purpose of this study is to discuss the lp approximation order 1 p of interpolation to functions in the sobolev space wpk with k max 0 d 2 d p we are particularly interested in using the shifted surface spline which actually includes the cases of the multiquadric and the surface spline moreover we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met

a stabilized radial basis finite difference rbf fd method with hybrid kernels abstract recent developments have made it possible to overcome grid based limitations of finite difference fd methods by adopting the kernel based meshless framework using radial basis functions rbfs such an approach provides a meshless implementation and is referred to as the radial basis generated finite difference rbf fd method in this paper we propose a stabilized rbf fd approach with a hybrid kernel generated through a hybridization of the gaussian and cubic rbf this hybrid kernel was found to improve the condition of the system matrix consequently the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard rbf fd methods coupled with present stable algorithms unlike other rbf fd approaches the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity and the size of the stencils as an application we solve the frequency domain acoustic wave equation in a 2d half space in order to suppress spurious reflections from truncated computational boundaries absorbing boundary conditions have been effectively implemented

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

on the propagation of polarization sets for systems of real principal type we define and study the polarization sets for vector valued distributions systems of real principal type are defined and we prove that for such systems of pseudodifferential operators we have a unique propagation of the polarization sets for the solutions

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

radial basis functions method for solving the fractional diffusion equations abstract fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes the paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions rbfs for the numerical solution of a class of initial boundary value fractional diffusion equations with variable coefficients on a finite domain the space fractional derivatives are defined by using riemann liouville fractional derivative we first provide riemann liouville fractional derivatives for the five kinds of rbfs including the powers gaussian multiquadric matern and thin plate splines in one dimension the time dependent fractional diffusion equation is discretized in space with the rbf collocation method and the remaining system of ordinary differential equations odes is advanced in time with an ode method using a method of lines approach some numerical results are given in order to demonstrate the efficiency and accuracy of the method additionally some physical properties of this fractional diffusion system are simulated which further confirm the effectiveness of our method the stability of the linear systems arising from discretizing riemann liouville fractional differential operator with rbfs is also analysed

new periodic and soliton solutions for the generalized bbm and burgers bbm equations abstract in this paper we consider the benjamin bona mahony equation bbm and we obtain new exact solutions for it by using a generalization of the well known tanh coth method new periodic and soliton solutions for the generalized bbm and burgers bbm equations are formally derived

enhancing finite differences with radial basis functions polynomials are used together with polyharmonic spline phs radial basis functions rbfs to create local rbf finite difference rbf fd weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences fd the presented method replicates the convergence properties of fd but for arbitrary node layouts it is tested on the 2d compressible navier stokes equations at low mach number relevant to atmospheric flows test cases are taken from the numerical weather prediction community and solved on bounded domains thus attention is given on how to handle boundaries with the rbf fd method as well as a novel implementation for hyperviscosity comparisons are done on cartesian hexagonal and quasi uniform node layouts consideration and guidelines are given on phs order polynomial degree and stencil size the main advantages of the present method are 1 capturing the basic physics of the problem surprisingly well even at very coarse resolutions 2 high order accuracy without the need of tuning a shape parameter and 3 the inclusion of polynomials eliminates stagnation saturation errors a matlab code is given to calculate the differentiation weights for this novel approach

nonlinear stability of generalized benjamin bona mahony burgers shock profiles in several dimensions abstract this paper is concerned with nonlinear stability of planar viscous shock profiles of the generalized benjamin bona mahony burgers equations in two dimensions our analysis is motivated by j goodman s work 1989 4 on the nonlinear stability of scalar viscous shock profiles in two dimensions and is based on some new decay estimates on the planar viscous shock profiles of the generalized benjamin bona mahony burgers equations

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

evaluation of mixed crank nicolson scheme and tau method for the solution of klein gordon equation abstract numerical method based on the crank nicolson scheme and the tau method is proposed for solving nonlinear klein gordon equation nonlinear klein gordon equation is reduced by crank nicolson scheme to the system of ordinary differential equations then tau method is used to solve this system by using interpolating scaling functions and operational matrix of derivative the order of convergence is proposed and some numerical examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

numerical analysis for fourth order compact conservative difference scheme to solve the 3d rosenau rlw equation in this paper a fourth order compact and energy conservative difference scheme for three dimensional rosenau rlw equation is proposed the scheme is a two level and nonlinear implicit scheme it is proved by the discrete energy method that the compact scheme is solvable the convergence and stability of the difference scheme is obtained and its numerical convergence order is o 2 h4 in the l norm we discuss an iterative algorithm for solving the nonlinear algebraical system generated by the nonlinear compact scheme and prove its convergence numerical experiment results show that the theory is accurate and the method is efficient and reliable

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

error analysis of interpolating element free galerkin method to solve non linear extended fisher kolmogorov equation abstract nonlinear partial differential equations pdes play an important role in the modeling of the natural phenomena as they have great significance in real world applications this investigation proposes a new algorithm to find the numerical solution of the non linear extended fisher kolmogorov equation firstly the time variable is discretized by a second order finite difference scheme the rate of convergence and stability of the semi discrete formulation are studied by the energy method the existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail furthermore the interpolating element free galerkin approach based on the interpolation moving least squares approximation is employed to derive a fully discrete scheme finally the error estimate of the full discrete plan is proposed and its convergence order is o 2 m 1 in which and m denote the time step the radius of the weight function and smoothness of the exact solution of the main problem respectively

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme