stabilized multiscale finite element method for the stationary navier stokes equations abstract in the paper a stabilized multiscale finite element method for the stationary incompressible navier stokes equations is considered the method is a petrov galerkin approach based on the multiscale enrichment of the standard polynomial space enriched with the unusual bubble functions which no longer vanish on every element boundary for the velocity space the stability of the p 1 p 0 triangular element or the q 1 p 0 quadrilateral element is established and the optimal error estimates of the stabilized multiscale finite element method for the stationary navier stokes equations are obtained

multi quadric quasi interpolation method coupled with fdm for the degasperis procesi equation in this paper the degasperis procesi dp equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form u p based on the equivalent formulation of dp equation a new numerical method is presented the multi quadric mq quasi interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order tvd method is used to approximate the time derivative several examples are presented to demonstrate the effectiveness of the proposed method

on the link between umbilic geodesics and soliton solutions of nonlinear pdes in this paper we describe a new class of soliton solutions called umbilic solitons for certain nonlinear integrable pdes these umbilic solitons have the property that as the space variable x tends to infinity the solution tends to a periodic wave and as x tends to minus infinity it tends to a phase shifted wave of the same shape the equations admitting solutions in this new class include the dym equation and equations in its hierarchy the methods used to find and analyse these solutions are those of algebraic and complex geometry we look for classes of solutions by constructing associated finite dimensional integrable hamiltonian systems on riemann surfaces in particular in this setting we use geodesics on n dimensional quadrics to find the spatial or x flow which together with the commuting t flow given by the equation itself defines new classes of solutions amongst these geodesics particularly interesting ones are the umbilic geodesics which then generate the class of umbilic soliton solutions this same setting also enables us to introduce another class of solutions of dym like equations which are related to elliptic and umbilic billiards

an efficient chebyshev tau method for solving the space fractional diffusion equations fractional diffusion equations fdes have recently been paid much attention finding accurate and efficient methods for solving fdes has become an active research undertaking in this paper an efficient method based on the shifted chebyshev tau idea is presented to solve an initial boundary value problem for the fdes the method is derived by expanding the required approximate solution as the elements of shifted chebyshev polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from a computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and only a small number of shifted chebyshev polynomials is needed

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

a simple empirical formula of origin intensity factor in singular boundary method for two dimensional hausdorff derivative laplace equations with dirichlet boundary abstract this paper presents a simple empirical formula of origin intensity factor in singular boundary method sbm solution of hausdorff derivative laplace equations the sbm with the empirical formula is mathematically more simple and computationally more efficient than using the other techniques for origin intensity factor numerical experiments simulate the steady heat conduction through fractal media governed by the hausdorff laplace equation and show the efficiency and reliability benefits of the present sbm empirical formula

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three

the maps with polynomial basis functions for solving axisymmetric time fractional equations abstract this study presents a parallel meshless approach for time fractional equations in the axisymmetric geometry in the present parallel meshless approach the method of approximate particular solutions maps in conjunction with laplace transform extended precision arithmetic epa and multiple scale technique mst is implemented in this application the laplace transform is employed to eliminate the dependence on time the maps is used to obtain the solution of time independent equation in laplace space furthermore the solution of time dependent equation is restored by stehfest s algorithm here the multiple scale technique mst and extended precision arithmetic epa are applied to alleviate the effect of an ill conditioned matrix on a numerical inverse laplace transform then multi cores parallel computing can improve effectively computing speed of the proposed meshless approach in the numerical implementation both the 2d axisymmetric equations and the original 3d equations are considered and compared

high accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems we investigate the mechanical quadrature methods mqms for solving boundary integral equations bies of the three dimensional axisymmetric elasticity equations with dirichlet s problems the convergence of the mqms is proved based on anselone s collective compact theory moreover an asymptotic error expansion of the mqms shows that the approximation order is of o h 3 numerical examples are tested and the results verify the theoretical analysis

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

boundary particle method for laplace transformed time fractional diffusion equations highlights error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations our method avoids costly convolution integral calculation and dilemmatic selection of time step size a truly boundary only meshless method is applied to laplace transformed inhomogeneous problem our method effectively simulates 3d long time history fractional diffusion systems this paper develops a novel boundary meshless approach laplace transformed boundary particle method ltbpm for numerical modeling of time fractional diffusion equations it implements laplace transform technique to obtain the corresponding time independent inhomogeneous equation in laplace space and then employs a truly boundary only meshless boundary particle method bpm to solve this laplace transformed problem unlike the other boundary discretization methods the bpm does not require any inner nodes since the recursive composite multiple reciprocity technique rc mrm is used to convert the inhomogeneous problem into the higher order homogeneous problem finally the stehfest numerical inverse laplace transform nilt is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding bpm solutions in comparison with finite difference discretization the ltbpm introduces laplace transform and stehfest nilt algorithm to deal with time fractional derivative term which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability consequently it can effectively simulate long time history fractional diffusion systems error analysis and numerical experiments demonstrate that the present ltbpm is highly accurate and computationally efficient for 2d and 3d time fractional diffusion equations

singular boundary method for wave propagation analysis in periodic structures abstract a strong form boundary collocation method the singular boundary method sbm is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures the sbm is of several advantages including mathematically simple easy to program meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method due to the periodic behaviors of the structures the sbm coefficient matrix can be represented as a block toeplitz matrix by employing three different fast toeplitz matrix solvers the computational time and storage requirements are significantly reduced in the proposed sbm analysis to demonstrate the effectiveness of the proposed sbm formulation for wave propagation analysis in periodic structures several benchmark examples are presented and discussed the proposed sbm results are compared with the analytical solutions the reference results and the comsol software

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

the method of fundamental solutions for axisymmetric potential problems in this paper we investigate the application of the method of fundamental solutions mfs to two classes of axisymmetric potential problems in the first the boundary conditions as well as the domain of the problem are axisymmetric and in the second the boundary conditions are arbitrary in both cases the fundamental solutions of the governing equations and their normal derivatives which are required in the formulation of the mfs can be expressed in terms of complete elliptic integrals the method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem copyright 1999 john wiley sons ltd

fundamental kernel based method for backward space time fractional diffusion problem based on kernel based approximation technique we devise in this paper an efficient and accurate numerical scheme for solving a backward space time fractional diffusion problem bstfdp the kernels used in the approximation are the fundamental solutions of the space time fractional diffusion equation expressed in terms of inverse fourier transform of mittag leffler functions the use of inverse fast fourier transform ifft technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the bstfdp since the bstfdp is intrinsic ill posed we apply the standard tikhonov regularization technique to obtain a stable solution to the highly ill conditioned resultant system of linear equations for choosing optimal regularization parameter we combine the regularization technique with the generalized cross validation gcv method for an optimal placement of the source points in the use of fundamental solutions meanwhile the proposed algorithm also speeds up the previous method given in dou and hon 2014 several numerical examples are constructed to verify the accuracy and efficiency of the proposed method

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

a meshless singular boundary method for transient heat conduction problems in layered materials abstract the transient heat conduction problems in layered materials are common encountered in practical application but bring great challenge to most numerical algorithms in this study the singular boundary method sbm a meshless boundary collocation technique is first employed to numerically simulate such problems based on the time dependent fundamental solution of diffusion equation taking the boundary conditions and interface conditions into account the computing system of the sbm for layered materials is established and then is solved the proposed method fully inherits the merits of conventional boundary type methods while possessing its distinctive advantages furthermore it is simple straightforward computationally efficient and stable since it does not need to discretize temporal derivative term three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme numerical results clearly indicate the efficiency accuracy and stability of the presented sbm for solving transient heat conduction problems in layered materials

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

on a nonlinear hyperbolic variational equation ii the zero viscosity and dispersion limits we consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 with the simplest initial data such that u x blows up in finite time we prove that the zero viscosity limit selects a unique global weak solution of the partial differential equation without viscosity we also present numerical experiments which indicate that the zero dispersion limit selects a different global weak solution of the same initial value problem

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

analyticity of the cauchy problem for two component hunter saxton systems abstract this paper is mainly concerned with the periodic cauchy problem for a generalized two component hunter saxton system with analytic initial data the analyticity of its solutions is proved in both variables globally in space and locally in time the obtained result can be also applied to its special cases the classical integrable two component hunter saxton system the generalized hunter saxton equation and the classical hunter saxton equation

a boundary type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations abstract this study presents a parallel meshless solver for transient heat conduction analysis of slender functionally graded materials fgms with exponential variations in the present parallel meshless solver a strong form boundary collocation method the boundary knot method bkm in conjunction with laplace transform is implemented to solve the heat conduction equations of slender fgms with exponential variations this method is mathematically simple easy to parallel meshless and without domain discretization however two ill posed issues the ill conditioning dense bkm matrix and numerical inverse laplace transform process may lead to incorrect numerical results here the extended precision arithmetic epa and the domain decomposition method ddm have been adopted to alleviate the effect of these two ill posed issues on numerical efficiency of the present method then the parallel algorithm has been employed to significantly reduce the computational cost and enhance the computational capacity for the fgm structures with larger length width ratio to demonstrate the effectiveness of the present parallel meshless solver for transient heat conduction analysis several benchmark examples are considered under slender fgms with exponential variations the present results are compared with the analytical solutions the conventional boundary knot method and comsol simulation

an efficient two grid method for miscible displacement problem approximated by mixed finite element methods abstract in the paper we present an efficient two grid method for the miscible displacement problem which discretized by mixed finite element methods for the pressure equation and concentration equation at the same time and then analyzed the error estimate of the two gird algorithm at last the numerical experiment presented confirmed the theoretical results compared with the standard mixed finite element methods this two grid scheme based on the mixed methods can keep the same convergence order and cost much less work

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

polynomial particular solutions for solving elliptic partial differential equations abstract in the past polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms in this paper a closed form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions the newly derived particular solution is further coupled with the method of particular solutions mps for numerically solving a large class of elliptic partial differential equations in contrast to the use of chebyshev polynomial basis functions the proposed approach is more flexible in selecting the collocation points inside the domain the polynomial basis functions are well known for yielding ill conditioned systems when their order becomes large the multiple scale technique is applied to circumvent the difficulty of ill conditioning problem five numerical examples are presented to demonstrate the effectiveness of the proposed algorithm

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd

existence results for the einstein scalar field lichnerowicz equations on compact riemannian manifolds in the null case this is the second in our series of papers concerning positive solutions of the einstein scalar field lichnerowicz equations let m g be a smooth compact riemannian manifold without boundary of dimension n geqslant 3 f and a geqslant 0 are two smooth functions on m with int m f dv g 0 and int m a dv g 0 in this article we prove two results involving the following equation arising from the hamiltonian constraint equation for the einstein scalar field equation in general relativity delta g u f u 2 star 1 frac a u 2 star 1 where delta g rm div g nabla g and 2 star 2n n 2 first we prove that if either supmf and int madvg or supma is sufficiently small the equation admits one positive smooth solution second we show that the equation always admits one and only one positive smooth solution provided rm sup m f leqslant 0 we should emphasize that we allow a to vanish somewhere along with these two results existence and non existence for related equations are also considered