variable order fractional differential operators in anomalous diffusion modeling abstract the purpose of this paper is to offer a unified discussion of variable order differential operators in anomalous diffusion modeling the characteristics of the new models in contrast to constant order fractional diffusion models change with time space concentration or other independent quantities we introduced a classification of variable order fractional diffusion models based on the possible physical origins which prompt the variable order some potential applications of the variable order fractional diffusion models are also discussed

a rbf based differential quadrature method for solving two dimensional variable order time fractional advection diffusion equation abstract numerical simulation technique of two dimensional variable order time fractional advection diffusion equation is developed in this paper using radial basis function based differential quadrature method rbf dq to the best of the authors knowledge this is the first application of this method to variable order time fractional advection diffusion equations for the general case of irregular geometries the meshless local form of rbf dq is used and the multiquadric type of radial basis functions is selected for the computations this approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the dirichlet and neumann boundary conditions on the irregular boundaries the method is validated by the well documented test examples involving variable order fractional modeling of air pollution the numerical results demonstrate that the proposed method provides accurate solutions for two dimensional variable order time fractional advection diffusion equations

harnack inequality for quasilinear elliptic equations on riemannian manifolds abstract we study viscosity solutions to degenerate and singular elliptic equations div f u u u h of p laplacian type on riemannian manifolds where an even function f c 1 r c 2 0 is supposed to be strictly convex on 0 under the assumption that either f c 2 r or its convex conjugate f c 2 r with some structural condition we establish a locally uniform abp type estimate and the krylov safonov type harnack inequality on riemannian manifolds with the use of an intrinsic geometric quantity to the operator here the c 2 regularities of f and f account for degenerate and singular operators respectively

second order approximations for variable order fractional derivatives fractional calculus allows variable order of fractional operators which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and or time in this paper we derive two second order approximation formulas for the variable order fractional time derivatives involved in anomalous diffusion and wave propagation we then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub diffusion and super diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable order function finally we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super diffusion and also simulations of the burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and or time taken together our results demonstrate that variable order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long standing issues of outflow boundary conditions and monotonicity of integer order pdes

harmonic functions and volume growth of fingerless ends let m be a noncompact complete riemannian manifold with finitely many ends in this paper we study the existence of green s function for the p laplace equation on m in terms of a certain volume growth we also show that the dimension of the linear space of polynomial growth harmonic functions is finite if a volume comparison condition and a mean value inequality for nonnegative subharmonic functions hold in sufficiently large parts of each end

numerical techniques for the variable order time fractional diffusion equation abstract in this paper we consider the variable order time fractional diffusion equation we adopt the coimbra variable order vo time fractional operator which defines a consistent method for vo differentiation of physical variables the coimbra variable order fractional operator also can be viewed as a caputo type definition although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature here an approximate scheme is first proposed the stability convergence and solvability of this numerical scheme are discussed via the technique of fourier analysis numerical examples are provided to show that the numerical method is computationally efficient

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

a space time discontinuous galerkin method for the elastic wave equation abstract in this work we present a new high order space time discretization method based on a discontinuous galerkin paradigm for the second order visco elastodynamics equation after introducing the method we show that the resulting space time discontinuous galerkin formulation is well posed stable and retains optimal rate of convergence with respect to the discretization parameters namely the mesh size and the polynomial approximation degree a set of two and three dimensional numerical experiments confirms the theoretical bounds

an analysis of a finite difference and a galerkin technique applied to the simulation of advection and diffusion of air pollutants from a line source abstract a finite difference scheme and a galerkin scheme are compared with respect to a very accurate solution describing time dependent advection and diffusion of air pollutants from a line source in an atmosphere vertically stratified and limited by an inversion layer the accurate solution was achieved by applying the finite difference scheme on a very refined grid with a very small time step the grid size and time step were defined according to stability and accuracy criteria discussed in the text it is found that for the problem considered the two methods can be considered equally accurate however the galerkin method gives a better approximation in the vicinity of the source this was assumed to be partly due to the different way the source term is taken into account in the two methods improvement of the accuracy of the finite difference scheme was achieved by approximating at every step the contribution of the source term by a gaussian puff moving and diffusing with the velocity and diffusivity of the source location instead of utilizing a stepwise function for the numerical approximation of the function representing the source term

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

the galerkin finite element method for a multi term time fractional diffusion equation we consider the initial boundary value problem for a diffusion equation involving multiple time fractional derivatives on a bounded convex polyhedral domain we analyze a space semidiscrete scheme based on the standard galerkin finite element method using continuous piecewise linear functions nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived which cover both smooth and nonsmooth data further we develop a fully discrete scheme based on a finite difference discretization of the time fractional derivatives and discuss its stability and error estimate extensive numerical experiments for one and two dimensional problems confirm the theoretical convergence rates

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

combined dg cg time stepping for wave equations the continuous and discontinuous galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order this eliminates the doubling of the number of variables that results when a second order problem is written as a first order system stability convergence and accuracy of these schemes is established in the context of the wave equation it is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy and that this problem can be circumvented using interpolants matched with the time stepping scheme

harnack inequality for degenerate and singular operators of p laplacian type on riemannian manifolds we study viscosity solutions to degenerate and singular elliptic equations of p laplacian type on riemannian manifolds the krylov safonov type harnack inequality for the p laplacian operators with 1 p infty is established on the manifolds with ricci curvature bounded from below based on abp type estimates we also prove the harnack inequality for nonlinear p laplacian type operators assuming that a nonlinear perturbation of ricci curvature is bounded below

on the variable order dynamics of the nonlinear wake caused by a sedimenting particle abstract in this work we develop a variable order vo differential equation of motion for a spherical particle sedimenting in a quiescent viscous liquid in particular we examine the various force terms in the equation of motion and propose a new form for the history drag acting on the particle we show that the variable order formulation allows for an effective way to express the dynamic transition of the dominant forces over the entire time of the motion of the particle from rest to terminal velocity the use of vo operators also allows us to examine the evolving dynamics of the wake during sedimentation using numerical data from a finite element simulation of a sedimenting particle we first solve for the order of the derivative that returns the correct decay of the history force we then propose a relatively simple expression for the history force that is a function of the reynolds number and particle to fluid density ratio the new history drag expression correlates very well r 2 0 99 with the numerical data for terminal reynolds numbers ranging from 2 5 to 20 and for particle to fluid density ratios of interest in practice 1 10

radial point interpolation collocation method rpicm for partial differential equations this paper presents a truly meshfree method referred to as radial point interpolation collocation method rpicm for solving partial differential equations this method is different from the existing point interpolation method pim that is based on the galerkin weak form because it is based on the collocation scheme no background cells are required for numerical integration radial basis functions are used in the work to create shape functions a series of test examples were numerically analysed using the present method including 1 d and 2 d partial differential equations in order to test the accuracy and efficiency of the proposed schemes several aspects have been numerically investigated including the choice of shape parameter c with can greatly affect the accuracy of the approximation the enforcement of additional polynomial terms and the application of the hermite type interpolation which makes use of the normal gradient on neumann boundary for the solution of pdes with neumann boundary conditions particular emphasis was on an efficient scheme namely hermite type interpolation for dealing with neumann boundary conditions the numerical results demonstrate that good improvement on accuracy can be obtained after using hermite type interpolation the h convergence rates are also studied for rpicm with different forms of basis functions and different additional terms

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous l 2 dirichlet boundary data this paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with l 2 boundary terms and smooth right hand side for the case of l 2 boundary data the rates of convergence in negative norms are derived in the case of smooth forcing term and zero boundary data optimal rates of convergence in positive norms are provided

fractional spectral collocation methods for linear and nonlinear variable order fpdes while several high order methods have been developed for fractional pdes fpdes with fixed order there are no such methods for fpdes with field variable order these equations allow multiphysics simulations seamlessly e g from diffusion to sub diffusion or from wave dynamics transitioning to diffusion by simply varying the fractional order as a function of space or time we develop an exponentially accurate fractional spectral collocation method for solving linear nonlinear fpdes with field variable order following the spectral theory developed in 1 for fractional sturm liouville eigenproblems we introduce a new family of interpolants called left right sided and central fractional lagrange interpolants we employ the fractional derivatives of left right sided riemann liouville and riesz type and obtain the corresponding fractional differentiation matrices by collocating the field variable fractional orders we solve several fpdes including time and space fractional advection equation time and space fractional advection diffusion equation and finally the space fractional burgers equation to demonstrate the performance of the method in addition we develop a spectral penalty method for enforcing inhomogeneous initial conditions our numerical results confirm the exponential like convergence of the proposed fractional collocation methods

weighted finite difference techniques for the one dimensional advection diffusion equation various numerical techniques will be developed and compared for solving the one dimensional advection diffusion equation with constant coefficient these techniques are based on the two level finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes the new methods are more accurate and are more efficient than the conventional techniques these schemes are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

continuous finite elements in space and time for the heat equation on analyse une nouvelle methode variationnelle pour l approximation de l equation de la chaleur a l aide des elements finis continus dans l espace et le temps on presente les estimations d erreurs et les resultats d experiences numeriques

a discontinuous galerkin method for higher order ordinary differential equations in this paper we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal o tp 1 convergence rate in the l2 norm we further show that the p degree discontinuous solution of differential equation of order m and its first m 1 derivatives are o t2p 2 m superconvergent at the end of each step we also establish that the p degree discontinuous solution is o tp 2 superconvergent at the roots of p 1 m degree jacobi polynomial on each step finally we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme