stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions stable computation of differentiation matrices and scattered node stencils based on gaussian radial basis functions

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

a quasi radial basis functions method for american options pricing abstract based on the idea of quasi interpolation and radial basis functions approximation a fast and accurate numerical method is developed for solving the black scholes equation for valuation of american options prices since the method does not require solving a resultant full matrix the ill conditioning problem resulting from using the radial basis functions as a global interpolant can be avoided the method has been shown to be effective in solving problems with free boundary condition as indicated in the numerical computation for the american option pricing an excellent approximation of the solution as well as the free optimal exercise boundary can be obtained

regularity criteria for the 3d mhd equations via one directional derivative of the pressure abstract in this paper we consider the cauchy problem for the 3d viscous mhd equations and provide some regularity criteria involving only one directional derivative of the pressure say 3 p in particular if 3 p l 0 t l r 3 with 2 3 2 3 2 3 then the solution remains smooth on 0 t

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

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

bmo and the regularity criterion for weak solutions to the magnetohydrodynamic equations in this paper we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations we derive the regularity of weak solutions in the marginal class moreover our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

coherent structures in strongly interacting many body systems ii classical solutions and quantum fluctuations for pt i see ibid vol 22 p 4877 1989 previously i the author considered a second quantised hamiltonian which can be used to model a large number of important strongly interacting many body systems he has shown that the dynamics of such systems can be described exactly using quantum fields the equations of motion derived in this way are highly nonlinear partial differential equations pde he now uses a standard field theoretical approach where initially the fields are treated as classical functions he studies the solutions of these equations which remarkably may be obtained making full use of very recent mathematical discoveries the symmetry reduction method he finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper these cases may correspond to different physical situations and the last the second order case corresponds to the most general physical situation subsequently he outlines possible quantisation procedures for these fields he then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

a high order multivariate approximation scheme for scattered data sets we present a high order multivariate approximation scheme for scattered data sets each data point is represented as a taylor series and the high order derivatives in the taylor series are treated as random variables the approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares the approximation is an interpolation when the data points are given as exact or a nonlinear regression function when nonzero measurement errors are associated with the data points using this formulation the gradient information on each data point can be used to significantly reduce the approximation error all parameters of the approximation scheme can be computed automatically from the data points an uncertainty bound of the approximation function is also produced by the scheme numerical experiments demonstrate that although this method is more computationally intensive than traditional methods it produces more accurate approximation functions

a regularity criterion for the three dimensional mhd equations in terms of one directional derivative of the pressure in this work we will establish a sufficient condition for the regularity criterion to the 3d mhd equations in terms of one directional derivative of the pressure more precisely we prove that if x 3 x t l 2 1 r 0 t m 2 3 r r 3 for 0 r 1 then the solution u b is regular we obtain some improvement of the related results in berselli and galdi 2002 cao and wu 2010 jia and zhou 2012 lin and du 2013 and yamazaki 2013

deformations of the riccati equation by using miura type transformations using some different miura type transformations a c integrable ordinary differential equation the riccati equation is deformed to some different s integrable models such as the 1 1 dimensional and 2 1 dimensional sinh gordon equations and mikhailov dodd bullough equations

regularity criteria for the 3d mhd equations involving one current density and the gradient of one velocity component abstract in this paper we consider the cauchy problem for the 3 d mhd equations and provide a regularity criterion involving u 3 and j 3 where u 3 and j 3 are the third component of the velocity and the current density respectively as by products we show that assumptions on two vorticity and one current density component are enough to ensure the smoothness of the solution and improve some previous results skalak 2014 zhou and pokorn 2009 2010 for the navier stokes equations

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

space time adaptive finite difference method for european multi asset options the multi dimensional black scholes equation is solved numerically for a european call basket option using a priori a posteriori error estimates the equation is discretized by a finite difference method on a cartesian grid the grid is adjusted dynamically in space and time to satisfy a bound on the global error the discretization errors in each time step are estimated and weighted by the solution of the adjoint problem bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations comparisons are made with monte carlo and quasi monte carlo methods in one dimension and the performance of the method is illustrated by examples in one two and three dimensions

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

similarity solutions of the kadomtsev petviashvili equation using the direct method introduced by clarkson and kruskal ck 1989 recently the author obtains the similarity reductions of the kadomtsev petviashvili equation kpe some of them possess the same painleve properties as that of the boussinesq equation be obtained by ck however the similarity solutions of kpe may include some more than three arbitrary functions of time there exist some other types of similarity reductions of the kpe different from the similarity reductions of the be obtained by ck

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

on a regularity criterion for the navier stokes equations involving gradient of one velocity component we improve the regularity criterion for the incompressible navier stokes equations in the full three dimensional space involving the gradient of one velocity component the method is based on recent results of cao and titi see regularity criteria for the three dimensional navier stokes equations indiana univ math j 57 2643 2008 and kukavica and ziane see navier stokes equations with regularity in one direction j math phys 48 065203 2007 in particular for s 2 3 we get that the solution is regular if u3 lt 0 t ls r3 2 t 3 s 2312

self similar solutions for a coupled system of nonlinear schrodinger equations this work is devoted to the study of self similar solutions of the 2 1 dimensional coupled nonlinear schrodinger equations which occur in nonlinear optics 1 psi 1 z psi 1 xx psi 1 yy eta mod psi 1 to 2 1 h mod psi 2 mod 2 psi 1 0 and square root 1 psi 2 psi 2 xx psi 2 yy eta mod psi 2 mod 2 1 h mod psi 1 psi 2 0 where psi 1 and psi 2 are complex functions h is a non vanishing real parameter eta or 1 and in or 1 the authors give the point symmetry properties of the model and calculate generic 2 1 dimensional symmetry reductions some exact and approximate solutions are obtained in particular they use a variational approach to determine and classify a set of physically relevant localized nonsingular self similar solutions

the interaction representation in the self focusing theory abstract the general scheme of dividing the solution of the nonlinear schrodinger equation 2d case into the focused singular part of the beam called the peak and the rest nonsingular part the background is presented the peak is compared with a manifold of self similar modes whose dynamics is analysed it is shown that if the beam power is less than critical the self similar beam is strictly localized if the beam power is more than critical the self similar solution is a quasi localized mode with tunnel emission of power from the paraxial region and the width of the peak tends to zero the parametric peak background interaction analysis proves that the limiting self focusing law a t t 0 t ln ln t 0 t 1 2 is determined only by the peak dynamics

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

stability of isotropic singularities for the nonlinear schro dinger equation abstract we describe a method by which the fully two and three dimensional cubic schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities in both cases anisotropic initial data collapse very rapidly towards isotropic singularities in three dimensions the solutions become self similar with a blowup rate t t 1 2 in two dimensions the self similarity is weakly broken and the blowup rate is t t ln ln 1 t t 1 2 the stability of the singular isotropic solutions is very firmly backed by the numerical results

lie symmetries of a generalised non linear schrodinger equation ii exact solutions for pt i see ibid vol 21 p 1493 1988 the authors obtain group invariant solutions of the non linear equation i psi t delta psi a0 psi a1 psi mod psi mod 2 a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended galilei group for a1a2 not 0 and the extended galilei similitude group for a1 0 or a2 0 they use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved whenever possible with the help of a singularity analysis solutions are obtained in terms of elementary functions jacobi elliptic functions and painleve transcendents

lie symmetries of a generalised non linear schrodinger equation iii reductions to third order ordinary differential equations for pt ii see ibid vol 22 p 469 1989 the study of group invariant solutions of the generalised non linear schrodinger equation gnlse is continued it is shown that eight types of subgroups of the symmetry group lead via symmetry reduction to third order real ordinary differential equations giving both the phase and the absolute value of the solution only two of the reductions provide to a painleve type equation and both of them only for the cubic gnlse this equation is solved in terms of the fourth painleve transcendent

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

some new regularity criteria for the 3d mhd equations abstract in this paper we consider the regularity criteria for solutions to the 3d incompressible magnetohydrodynamics mhd equations the first one involves h u 1 u 2 u and 3 b then the criterion concerning h u and h b 1 b 2 b is presented finally we show that the weak solution u b is actually regular provided that u 3 3 u b and 3 b are in suitable serrin type integrability classes respectively

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

similarity reductions of the kp equation by a direct method basing on a direct method developed by clarkson and kruskal 1989 the kadomtsev petviashvili kp equation has been reduced to three types of 1 1 dimensional partial differential equations which are equivalent to the three types of the similarity reduction equations obtained by the classical lie approach but with different independent variables more arbitrary functions which have been missed by the classical lie approach have been included in the solutions of the kp equation for instance the third type of reduction obtained by the direct method can be divided into three subcases and the third type of solution of the kp equation obtained by the classical lie approach is only a special case of one subcase of the authors results

improved radial basis function methods for multi dimensional option pricing in this paper we have derived a radial basis function rbf based method for the pricing of financial contracts by solving the black scholes partial differential equation as an example of a financial contract that can be priced with this method we have chosen the multi dimensional european basket call option we have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter for other non optimal choices of shape parameter values the resulting convergence rate is algebraic we propose an adapted node point placement that improves the accuracy compared with a uniform distribution compared with an adaptive finite difference method the rbf method is 20 40 times faster in one and two space dimensions and has approximately the same memory requirements

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

coherent structures in strongly interacting many body systems i derivation of dynamics numerous examples in condensed matter physics possess an effective hamiltonian structure involving two body anharmonic interactions between elementary excitations the effective second quantised hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting the author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking he introduces quantum field operators and calculates their equations of motion the interaction coefficients are expanded in a taylor series about the critical mode to second order to illustrate the method but he gives the expansion to infinite order for completeness it is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations pde for the field operator he shows that the form of the second order equation is exact due to renormalisation theory in the case where the hamiltonian coefficients represent operators in real space which are isotropic and incorporates symmetries and topologies which would appear in an expansion to infinite order this special situation arises in a very wide range of physical examples these equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non linear analysis in particular the method of symmetry reduction for pde provides a complete set of symmetry variables for the type of equations used in the study

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

a new regularity criterion for the 3d incompressible mhd equations in terms of one component of the gradient of pressure abstract in this paper we study the global regularity issue for the 3d incompressible mhd equations it is proved that if the pressure p satisfies 3 p l 0 t l r 3 with 2 3 2 3 and 1 then the corresponding weak solution u b is regular on 0 t

some serrin type regularity criteria for weak solutions to the navier stokes equations we consider the regularity criteria for a weak solution u u1 u2 u3 to the navier stokes equations in r3 denoting by 1 2 3 curlu the vorticity we then prove that u is smooth provided u3 3 or 3u3 3 are in some serrin type integrability classes

new similarity solutions for the modified boussinesq equation in this paper the author presents some new similarity solutions of the modified boussinesq equation which is a completely integrable soliton equation these new similarity solutions include reductions to the second and fourth painleve equations which are not obtainable using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques

focusing and multi focusing solutions of the nonlinear schro dinger equation abstract a method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic schrodinger equations in two and three dimensions and describe their universal properties the same method is applied to simulate the multi focusing phenomena produced by simple models of saturating nonlinearities

new symmetry reductions and exact solutions of the davey stewartson system i reductions to ordinary differential equations in this article symmetry reductions and exact solutions are presented for the 2 1 dimensional davey stewartson ds system which has the completely integrable dsi and dsii systems as special cases these symmetry reductions are obtained using the direct method originally developed by clarkson and kruskal to study symmetry reductions of the boussinesq equation which involves no group theoretic techniques the ds system is reduced directly to ordinary differential equations with no intermediate step using these reductions exact solutions of the ds system including some expressible in terms of the second and fourth painleve equations and elementary functions are obtained

on the regularity criteria for weak solutions to the magnetohydrodynamic equations abstract we study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations which generalize the results in c he z xin on the regularity of solutions to the magneto hydrodynamic equations j differential equations 213 2 2005 235 254 our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

on the 3d navier stokes equations with regularity in pressure abstract in this paper we investigate the global regularity of leray hopf weak solutions to the 3d navier stokes equations in terms of pressure in particular we establish three serrin type regularity criteria in framework of the anisotropic lebesgue spaces our result can be understood as a generalization of the notable works of y zhou 2006 23 c cao and e titi 2008 5 and y zhou and m pokorn 2010 24

enhancing credit default swap valuation with meshfree methods in this paper we apply the meshfree radial basis function rbf interpolation to numerically approximate zero coupon bond prices and survival probabilities in order to price credit default swap cds contracts we assume that the interest rate follows a cox ingersoll ross process while the default intensity is described by the exponential vasicek model several numerical experiments are conducted to evaluate the approximations by the rbf interpolation for one and two factor models the results are compared with those estimated by the finite difference method fdm we find that the rbf interpolation achieves more accurate and computationally efficient results than the fdm our results also suggest that the correlation between factors does not have a significant impact on cds spreads

local rbf method for multi dimensional partial differential equations abstract in this paper a local meshless differential quadrature collocation method is utilized to solve multi dimensional reaction convection diffusion pdes numerically in some cases global version of the meshless method is considered as well the meshless methods approximate solution on scattered and uniform nodes in both local and global sense in the case of convection dominated pdes the local meshless method is coupled with an upwind technique to avoid spurious oscillations for this purpose a physically motivated local domain is utilized in the flow direction both regular and irregular geometries are taken into consideration numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains

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

remarks on the regularity criteria for generalized mhd equations abstract we study the cauchy problem for the generalized mhd equations and prove some regularity criteria involving the integrability of u in the morrey multiplier spaces

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

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

solving inverse stokes problems by modified collocation trefftz method in this paper the two dimensional inverse stokes problems governed by bi harmonic equations are stably solved by the modified collocation trefftz method mctm in some practical applications of the stokes problems part of the boundary conditions cannot be measured in advance so the mathematical descriptions of such problems are known as the inverse stokes problems when numerical simulation is adopted for solutions of the inverse stokes problems the solutions will become extremely unstable which means that small perturbations in the boundary conditions will result in large errors of the final results hence we adopted the mctm for stably and efficiently analyzing the inverse stokes problems the mctm is one kind of boundary type meshless methods so the mesh generation and the numerical quadrature can be avoided besides the numerical solution is expressed as a linear combination of t complete functions modified by a characteristic length by enforcing the satisfactions of the boundary conditions at every boundary node a system of linear algebraic equations will be yielded the unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix the numerical solutions and their derivatives can be easily obtained by linear summation three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two dimensional inverse stokes problems

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

navier stokes equations with regularity in one direction we consider sufficient conditions for the regularity of leray hopf solutions of the navier stokes equations we prove that if the third derivative of the velocity u x3 belongs to the space lts0lxr0 where 2 s0 3 r0 2 and 9 4 r0 3 then the solution is regular this extends a result of beirao da veiga chin ann math ser b 16 407 412 1995 c r acad sci ser i math 321 405 408 1995 by making a requirement only on one direction of the velocity instead of on the full gradient the derivative u x3 can be substituted with any directional derivative of u

similarity structure of wave collapse abstract similarity transformations of the cubic schrodinger equation cse are investigated the transformations are used to remove the explicit time variation in the cse and reduce it to differential equations in the spatial variables only two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time and that some similarity solutions must be classified as improper solutions the nature of the collapsing singularity is reexamined

multi dimensional option pricing using radial basis functions and the generalized fourier transform we show that the generalized fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi dimensional option pricing we derive a general algorithm including a transformation of the black scholes equation into the heat equation that can be used in any number of dimensions numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes furthermore the gain increases with the number of dimensions