a global optimization procedure for the location of a median line in the three dimensional space abstract a global optimization procedure is proposed to find a line in the euclidean three dimensional space which minimizes the sum of distances to a given finite set of three dimensional data points although we are using similar techniques as for location problems in two dimensions it is shown that the problem becomes much harder to solve however a problem parameterization as well as lower bounds are suggested whereby we succeeded in solving medium size instances in a reasonable amount of computing time

explicit methods for fractional differential equations and their stability properties the use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated in this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed some numerical experiments are presented in order to validate theoretical results

the high order compact numerical algorithms for the two dimensional fractional sub diffusion equation in this paper performing the average operators on the space variables a numerical scheme with third order temporal convergence for the two dimensional fractional sub diffusion equation is considered for which the unconditional stability and convergence in l1 l norm are strictly analyzed for 0 0 9569347 by using the discrete energy method therewith adding small perturbation terms we construct a compact alternating direction implicit difference scheme for the two dimensional case finally some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all 0 1

coercive inequalities for gibbs measures abstract we prove the generalized nash and logarithmic nash inequalities for gibbs measures with dirichlet form associated to the kawasaki dynamics

towards a statistical theory of texture evolution in polycrystals most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries the energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a material across a wide range of scales a central problem in materials science is to develop technologies capable of producing an arrangement of grains a texture that provides for a desired set of material properties one of the most challenging aspects of this problem is to understand the role of topological reconfigurations during coarsening here we propose an upscaling procedure suitable for large complex systems the procedure is based on numerical experimentation combined with stochastic tools and consists of large scale numerical simulations of a system at a microscopic level statistical analysis of the microscopic data and formulation of the model based on stochastic characteristics predicted by the statistical analysis this approach promises to be valuable in establishing the effective model of microstructural evolution in realistic two and three dimensional systems to test ideas we use our upscaling procedure to study the mesoscopic behavior of a reduced one dimensional network of grain boundaries despite the simplicity of its formulation this model exhibits highly nontrivial behavior characterized by growth and disappearance of grain boundaries and develops probability distributions similar to those observed in higher dimensional simulations here we focus on the grain deletion events which are common to all coarsening systems

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

a solution algorithm for non convex mixed integer optimization problems with only few continuous variables abstract geometric branch and bound techniques are well known solution algorithms for non convex continuous global optimization problems with box constraints several approaches can be found in the literature differing mainly in the bounds used the aim of this paper is to extend geometric branch and bound methods to mixed integer optimization problems i e to objective functions with some continuous and some integer variables mixed integer non linear and non convex optimization problems are extremely hard containing several classes of np hard problems as special cases we identify for which type of mixed integer non linear problems our method can be applied efficiently derive several bounding operations and analyze their rates of convergence theoretically moreover we show that the accuracy of any algorithm for solving the problem with fixed integer variables can be transferred to the mixed integer case our results are demonstrated theoretically and experimentally using the truncated weber problem and the p median problem for both problems we succeed in finding exact optimal solutions

a computational method for solving variable order fractional nonlinear diffusion wave equation abstract in this paper we generalize a one dimensional fractional diffusion wave equation to a one dimensional variable order space time fractional nonlinear diffusion wave equation v os tfnd we where the variable order fractional derivatives are considered in the caputo type to solve this introduced equation an easy to follow method is proposed which is based on the chebyshev cardinal functions coupling with the tau and collocation methods to carry out the method an operational matrix of variable order fractional derivative omv ofd is derived for the chebyshev cardinal functions to be employed for expanding the unknown function the proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling the experimental results confirm the applicability and effectiveness of the method

discrete time orthogonal spline collocation method with application to two dimensional fractional cable equation discrete time orthogonal spline collocation osc methods are presented for the two dimensional fractional cable equation which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons the proposed scheme is based on the osc method for space discretization and finite difference method for time which is proved to be unconditionally stable and convergent with the order o min 2 1 2 2 h r 1 in l 2 norm where h and r are the time step size space step size and polynomial degree respectively and 1 and 2 are two different exponents of fractional derivatives with 0 1 2 1 numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein and super convergence phenomena at the partition nodes is also exhibited which is a characteristic of the osc methods namely the rates of convergence in the maximum norm at the partition nodes in u x and u y are approximately h r 1 in our numerical experiment

approximating the basset force by optimizing the method of van hinsberg et al in this work we put the method proposed by van hinsberg et al 29 to the test highlighting its accuracy and efficiency in a sequence of benchmarks of increasing complexity furthermore we explore the possibility of systematizing the way in which the method s free parameters are determined by generalizing the optimization problem that was considered originally finally we provide a list of worked out values ready for implementation in large scale particle laden flow simulations

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

a short note on quenching phenomena for semilinear parabolic equations abstract in this paper we investigate initial boundary value problems for semilinear parabolic differential equations with singular term a criterion for the appearance of quencing phenomena of classical solution to the above problems on a bounded domain is given and a global existence and nonexistence results of the above problems on unbounded domains are obtained

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

convergence and stability of compact finite difference method for nonlinear time fractional reaction diffusion equations with delay abstract this paper is concerned with numerical solutions of nonlinear time fractional reaction diffusion equations with time delay a linearized compact finite difference scheme is proposed to solve the equations in terms of a new developed fractional gronwall type inequality convergence and stability of the proposed scheme are obtained numerical experiments are given to illustrate the theoretical results

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

convergence and restart in branch and bound algorithms for global optimization application to concave minimization and d c optimization problems a general branch and bound conceptual scheme for global optimization is presented that includes along with previous branch and bound approaches also grid search techniques the corresponding convergence theory as well as the question of restart capability for branch and bound algorithms used in decomposition or outer approximation schemes are discussed as an illustration of this conceptual scheme a finite branch and bound algorithm for concave minimization is described and a convergent branch and bound algorithm based on the previous one is developed for the minimization of a difference of two convex functions

a computational wavelet method for variable order fractional model of dual phase lag bioheat equation abstract in this study we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue several models with different levels of simplifications have been proposed one of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time is the 2d transient dual phase lag dpl variable order fractional energy equation for finding the numerical solution of this general case we propose an efficient semi discrete method based on the two dimensional legendre wavelets 2d lws precisely the variable order fractional derivatives of the model are discretized in the first stage and then the response of the model is expanded by the 2d lws consequently the main problem is transformed into an equivalent system of algebraic equations which can be simply tackled the stability of the proposed method is examined theoretically and experimentally also the procedure is described for one example to examine the computational efficiency of method the experimental results show the stability and spectral accuracy of the proposed method according to the achieved results increasing the fractional order from 0 1 to 1 0 leads to increment of maximum tissue temperatures by about 29 near the center of the targeted region

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

a discontinuous petrov galerkin method for time fractional diffusion equations we propose and analyze a time stepping discontinuous petrov galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time fractional subdiffusion problems we prove the existence uniqueness and stability of approximate solutions and derive error estimates to achieve high order convergence rates from the time discretizations the time mesh is graded appropriately near t 0 to compensate for the singular temporal behavior of the exact solution near t 0 caused by the weakly singular kernel but the spatial mesh is quasi uniform in the l infty 0 t l 2 omega norm 0 t is the time domain and omega is the spatial domain for sufficiently graded time meshes a global convergence of order k m alpha 2 h r 1 is shown where 0 alpha 1 is the fractional exponent k is the maximum time step h is the maximum diameter of the elements of the spatial mesh and m and r are the degrees of approximate solutions in time an

a galerkin finite element method for the modified distributed order anomalous sub diffusion equation abstract in this paper we consider a modified time distributed order anomalous sub diffusion equation for the description of processes becoming less anomalous over the course of time first we discretize the integral term using a mid point quadrature rule and transform the time distributed order fractional diffusion equation into a multi term equation then applying the backward difference method in time and galerkin finite element method in space we present a fully discrete scheme to solve the time distributed order fractional diffusion equation moreover the stability and convergence of the proposed method are also discussed numerical examples are given to verify the accuracy and stability of our scheme

quenching for a parabolic equation with variable coefficient modeling mems technology abstract this paper deals with a parabolic equation with variable coefficient modeling mems technology and we will prove the solution of the problem quenches in finite time in the case without stationary solutions numerical simulations are presented to illustrate the theoretical results

three point combined compact difference schemes for time fractional advection diffusion equations with smooth solutions a ccd method with the l1 formula is proposed for solving time fractional advection diffusion equations the method is globally at least fifth order accurate in space and 2 th order accurate in time the unconditional stability is proved by means of fourier analysis numerical experiments are conducted to demonstrate the efficiency the present method preserves the higher order accuracy for advection dominated problems with the smooth solutions in this paper we study the numerical solution for the time fractional advection diffusion equation tfade a high order accurate three point combined compact difference scheme with the l1 formula is proposed to solve a class of tfades the method is globally 2 th order accurate in time and at least fifth order accurate in space for the constant coefficient tfades subject to periodic boundary conditions where is the order of time fractional derivative in the governing equation the unconditional stability and the high order convergence are proved using the fourier analysis method the proposed method can be extended to treat the variable coefficient tfades subject to other local boundary conditions several numerical examples are computed to validate the numerical accuracy effectiveness and robustness of the present method for tfades with smooth solutions and different boundary conditions moreover the proposed algorithm preserves the higher order accuracy for advection dominated problems with the smooth solutions

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

random fractional generalized airy differential equations a probabilistic analysis using mean square calculus abstract the aim of this paper is to study a generalization of fractional airy differential equations whose input data coefficient and initial conditions are random variables under appropriate hypotheses assumed upon the input data we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense afterwards we provide reliable explicit approximations for the main statistical information of the solution process mean variance and covariance further we show a set of numerical examples where our obtained theory is illustrated more precisely we show that our results for the random fractional airy equation are in full agreement with the corresponding to classical random airy differential equation available in the extant literature finally we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional airy differential equation by combining the knowledge of the mean and the variance and the principle of maximum entropy

numerical solution of degenerate stochastic kawarada equations via a semi discretized approach abstract the numerical solution of a highly nonlinear two dimensional degenerate stochastic kawarada equation is investigated a semi discretized approximation in space is comprised on arbitrary nonuniform grids exponential splitting strategies are then applied to advance solutions of the semi discretized scheme over adaptive grids in time it is shown that key quenching solution features including the positivity and monotonicity are well preserved under modest restrictions the numerical stability of the underlying splitting method is also maintained without any additional restriction computational experiments are provided to not only illustrate our results but also provide further insights into the global nonlinear convergence of the numerical solution

on the quenching of a nonlocal parabolic problem arising in electrostatic mems control abstract we consider a nonlocal parabolic model for a micro electro mechanical system specifically for a radially symmetric problem with monotonic initial data it is shown that the solution quenches so that touchdown occurs in the device in a situation where there is no steady state it is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained numerical simulations illustrating the results are given

general pade approximation method for time space fractional diffusion equation in recent years fractional differential equations have attracted much attention due to their wide application in this paper we present a novel numerical method for the space time riesz caputo fractional diffusion equation which discrete the riesz derivative by a fourth order fractional compact difference scheme then the above space time riesz caputo fractional diffusion equation change into a fractional ordinary differential equation fode system again using laplace and inverse laplace transforms one can get the analytical solution of the fode system furthermore we approximate the mittag leffler function by the global pade approximation and obtain the numerical method for the space time riesz caputo fractional diffusion equation finally two numerical examples are presented to show that the numerical results are in good line with our theoretical analysis

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

a note on the stochastic lattice gas model it is proved without imposing translation invariance that the only stationary states for the infinite two dimensional stochastic lattice gas or binary alloy model are the canonical gibbs states and that these dynamics describe a strong return to equilibrium

numerical solution of time fractional nonlinear kleingordon equation using sincchebyshev collocation method in this paper we proposed a new numerical scheme to solve the time fractional nonlinear kleingordon equation the fractional derivative is described in the caputo sense the method consists of expanding the required approximate solution as the elements of sinc functions along the space direction and shifted chebyshev polynomials of the second kind for the time variable the proposed scheme reduces the solution of the main problem to the solution of a system of nonlinear algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

asymptotic equivalence of fluctuation fields for reversible exclusion processes with speed change on considere des processus d exclusion reversibles stationnaires avec changement de vitesse et on etudie la forme asymptotique des champs de fluctuation

quadratic spline collocation method for the time fractional subdiffusion equation in this paper exploiting the quadratic spline collocation qsc method we numerically solve the time fractional subdiffusion equation with dirichelt boundary value conditions the coefficient matrix of the discretized linear system is investigated in detail theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with o 3 h 3 under the l norm provided that the solution v x t has four order continual derivative with respects to x and t and it can achieve the accuracy of o 4 h 4 at collocation points where h are the step sizes in time and space respectively

solution of the multisource weber and conditional weberproblems by d c programming d c programming is a recent technique of global optimization that allows the solution of problems whose objective function and constraints can be expressed as differences of convex i e d c functions many such problems arise in continuous location theory the problem first considered is to locate a known number of source facilities to minimize the sum of weighted euclidean distances between a user s fixed location and the source facility closest to the location of each user we also apply d c programming to the solution of the conditional weber problem an extension of the multisource weber problem in which some facilities are assumed to be already established in addition we consider a generalization of weber s problem the facility location problem with limited distances where the effective service distance becomes a constant when the actual distance attains a given value computational results are reported for problems with up to 10 000 users and two new facilities 50 users and three new facilities 1 000 users 20 existing facilities and one new facility or 200 users 10 existing and two new facilities

several effective algorithms for nonlinear time fractional models abstract in this paper several effective algorithms are proposed to simulate the time fractional models a complete analysis of the algorithms for nonlinear problems is presented while the previous investigations mainly focus on construction and analysis of l1 type method for the linear time fractional problems after that by using the sum of exponentials approximation we develop the corresponding accelerated methods we obtain that computational cost is reduced from o n 2 to o log n or o log 2 n where n denotes total number of time steps finally several numerical experiments on several real world models are proposed to confirm the effectiveness of the algorithms

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

gbsss the generalized big square small square method for planar single facility location abstract big square small square is a geometrical branch and bound algorithm devised by p hansen et al or 33 1251 1265 for the solution of constrained planar minisum single facility location problems with l p norms and continuous non decreasing costfunctions the method basically works by splitting the studied planar region into squares and either rejecting or further processing these squares by the evaluation of a lower bound we present a modified version of this algorithm aimed at correcting a small failure to converge accelerating the calculations minimising the information to be stored and most importantly determining a region of near optimality furthermore the method is applicable to any planar single facility problem with distances measured by mixed norms and as an objective any continuous function of the distances this includes nearly all the models which have been proposed in the literature

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

stability and convergence of finite difference schemes for a class of time fractional sub diffusion equations based on certain superconvergence this paper is devoted to the construction and analysis of finite difference methods for solving a class of time fractional subdiffusion equations based on the certain superconvergence at some particular points of the fractional derivative by the traditional first order grunwald letnikov formula some effective finite difference schemes are derived the obtained schemes can achieve the global second order numerical accuracy in time which is independent of the values of anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme respectively are established for the one dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method furthermore the extension to the two dimensional case is also considered numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes the superconvergent points of the traditional first order grunwald letnikov formula to approximate the riemann liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time fractional sub diffusion equations the obtained schemes can achieve the global second order accuracy in time which is independent of the anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme are presented respectively the strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method the extension to the two dimensional case is also discussed the compact difference scheme with the convergence of second order in time and fourth order in space is derived along with the strict stability and convergence analysis several numerical experiments are implemented to illustrate the computational efficiency numerical accuracy and robustness of the proposed schemes

implicit numerical approximation scheme for the fractional fokker planck equation in this paper we consider an anomalous subdiffusion process governed by fractional fokker planck equation an effective numerical method for approximating fokker planck equation in a bounded domain is presented the stability and convergence of the numerical method are analyzed some numerical examples are presented to show the application of the present technique the numerical results exhibit the good performance of our theoretical analysis

mixed spline function method for reaction subdiffusion equations in this paper we develop two classes of finite difference schemes for the reaction subdiffusion equations by using a mixed spline function in space direction forward and backward difference in time direction respectively it has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters by fourier method we prove that one class of difference scheme is unconditionally stable and convergent the other is conditionally stable and convergent finally some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

on the positivity monotonicity and stability of a semi adaptive lod method for solving three dimensional degenerate kawarada equations abstract this paper concerns the numerical solution of three dimensional degenerate kawarada equations these partial differential equations possess highly nonlinear source terms and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes arbitrary fixed nonuniform spatial grids which are not necessarily symmetric are considered throughout this study the numerical solution is advanced through a semi adaptive local one dimensional lod integrator the temporal adaptation is achieved via a suitable arc length monitoring mechanism criteria for preserving the positivity and monotonicity are investigated and acquired the numerical stability of the splitting method is proven in the von neumann sense under the spectral norm extended stability expectations are proposed and investigated

the big triangle small triangle method for the solution of nonconvex facility location problems in this paper we propose to modify the big square small square global optimization search in the plane with a big triangle small triangle approach the triangulation of the feasible region is obtained by using voronoi diagrams the resulting algorithm was tested on the obnoxious facility location and the attraction repulsion weber problems with excellent results

a compact adi method and its extrapolation for time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions a compact alternating direction implicit adi finite difference method is proposed for two dimensional time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions the unconditional stability and convergence of the method is proved the error estimates in the weighted l 2 l 2 and l l norms are obtained the proposed method has the fourth order spatial accuracy and the temporal accuracy of order min 2 1 min 2 1 where 0 1 0 1 is the order of the fractional derivative in order to further improve the temporal accuracy two richardson extrapolation algorithms are presented numerical results demonstrate the accuracy of the compact adi method and the high efficiency of the extrapolation algorithms

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

solving time fractional advection dispersion equation by variable weights particle tracking method particle tracking method is an efficient and reliable method to solve time fractional advection dispersion equation which can describe anomalous transport in heterogeneous porous media however this method will lead to severe fluctuation or disappearance of solutions if the concentration value is small a variable weights method is developed to conquer the shortcoming of particle tracking method then one dimensional and two dimensional time fractional advection dispersion equations are solved by the variable weights particle tracking method compared to traditional particle tracking method the variable weights version may eliminate the fluctuation and improve the accuracy by orders of magnitude without more computational cost

a meshless method for solving the time fractional advection diffusion equation with variable coefficients abstract in this paper an efficient and accurate meshless method is proposed for solving the time fractional advection diffusion equation with variable coefficients which is based on the moving least square mls approximation in the proposed method firstly the time fractional derivative is approximated by a finite difference scheme of order o t 2 0 1 and then the mls approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the caputo sense also the validity of the proposed method is investigated in error analysis discussion the main aim is to show that the meshless method based on the mls shape functions is highly appropriate for solving fractional partial differential equations fpdes with variable coefficients the efficiency and accuracy of the proposed method are verified by solving several examples

a cubic trigonometric b spline collocation approach for the fractional sub diffusion equations an algorithm for fractional sub diffusion model using cubic trigonometric b spline approach is presented the proposed scheme is unconditionally stable using von neumann approach error norms confirm the accuracy and efficiency of presented scheme the proposed method provides more accurate results as compare to cubic b spline method a cubic trigonometric b spline collocation approach for the numerical solution of fractional sub diffusion equation is presented in this paper the approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the second order derivative with respect to space is obtained by the cubic trigonometric b spline functions with the help of grunwald letnikov discretization of the riemann liouville derivative the scheme is shown to be stable using the fourier method and the accuracy of the scheme is tested by application to a test problem the results of the numerical test verify the accuracy and efficiency of the proposed algorithm

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

efficient reduction of polynomial zero one optimization to the quadratic case we address the problem of optimizing a polynomial with real coefficients over binary variables we show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem the number of variables in the latter linearization is only slightly larger than in the former if polynomial constraints are present in the original problem then their linearized counterparts carry over to the linearized quadratic problem unchanged if the original problem formulation does not contain any constraints we obtain a reduction to unconstrained quadratic zero one optimization which is equivalent to the well studied max cut problem the separation problem for general unconstrained polynomial zero one optimization thus reduces to the separation problem for the cut polytope this allows us to transfer the entire knowledge gained for the latter polytope by intensive research and in particular the sophisticated separation techniques that have been developed we report preliminary experimental results obtained with a straightforward implementation of this approach

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

continuous time random walks on moving fluids the scheme of the continuous time random walk ctrw is generalized to include the possibility of a moving background it is shown that this generalization reproduces in the macroscopic limit the usual diffusion advection equation and the properties of standard diffusion in a shear flow the new formalism is then used to derive the corresponding macroscopic equation for ctrw s with infinite mean squared step length and with infinite mean waiting time in a moving fluid for these two ctrw s we finally include an analysis of the dispersion in three different two dimensional linear shear flows

a modified memory based mathematical model describing fluid flow in porous media abstract this study presents a modified memory based mathematical model describing fluid flow in porous media for the first time such model is derived employing the grunwald letnikov g l definition of the riemann liouville r l time fractional operator via a generalized darcy s equation the proposed mathematical model is suitable to describe the anomalous diffusion behavior observed in a medium of fractal geometry as well as in disordered and highly heterogeneous porous media a numerical scheme based on existing discretization method is employed to handle the modified memory based mathematical model the accuracy of the numerical model is validated through the analytical solution for a simplified problem in addition numerical experiments are presented to demonstrate the effect of the anomalous diffusion exponent on the predicted reservoir pressure wellbore pressure and volumetric flux considering the flow of an under saturated oil in a hydrocarbon reservoir as an example results show that decrease in magnitude of the anomalous diffusion exponent results in larger pore and wellbore pressure this study would open a door to apply the g l interpretation of the time fractional operator in numerical modeling of fluid flow through porous media

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

locating objects in the plane using global optimization techniques we address the problem of locating objects in the plane such as segments arcs of circumferences arbitrary convex sets their complements or their boundaries given a set of points we seek the rotation and translation for such an object optimizing a very general performance measure which includes as a particular case the classical objectives in semi obnoxious facility location in general the above mentioned model yields a global optimization problem whose resolution is dealt with using difference of convex dc techniques such as outer approximation or branch and bound

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

weak solutions of the time fractional navier stokes equations and optimal control abstract in this paper we deal with the navier stokes equations with the time fractional derivative of order 0 1 which can be used to simulate anomalous diffusion in fractal media we firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of galerkin approximations in the case that the dimension n 4 moreover a complete proof of the uniqueness is given when n 2 at last we give a sufficient condition of optimal control pairs

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

a positivity preserving high order discontinuous galerkin scheme for convection diffusion equations abstract for constructing high order accurate positivity preserving schemes for convection diffusion equations we construct a simple positivity preserving diffusion flux discontinuous galerkin dg schemes with such a positivity preserving diffusion flux are nonlinear schemes which can be regarded as a reduction of the high order positivity preserving dg schemes for compressible navier stokes equations in 1 to scalar diffusion operators in this paper we focus on the local dg method to discuss how to apply such a flux a limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity preserving in the sense that dg schemes with this flux satisfies a weak positivity property together with a positivity preserving limiter high order dg schemes with strong stability preserving time discretizations can be rendered positivity preserving without losing conservation or high order accuracy for convection diffusion problems with periodic boundary conditions or a special class of dirichlet or neumann boundary conditions numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

integrablization of time fractional pdes abstract here we present a numerical method for the solution of time fractional partial differential equations fpdes the method is based on constructing a sequence of integrable approximations to the fpde such that in the appropriate limit the original fractional pde is obtained convergence results are obtained in general and examples are given for simple cases such as the fractional heat equation and the fractional diffusion wave equation

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

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

some high order difference schemes for the distributed order differential equations two difference schemes are derived for both one dimensional and two dimensional distributed order differential equations it is proved that the schemes are unconditionally stable and convergent in an l 1 l norm with the convergence orders o 2 h 2 2 and o 2 h 4 4 respectively where h and are the step sizes in time space and distributed order variables several numerical examples are given to confirm the theoretical results

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

solving the ordered one median problem in the plane in this paper we propose a general approach solution method for the single facility ordered median problem in the plane all types of weights non negative non positive and mixed are considered the big triangle small triangle approach is used for the solution rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

alternating direction implicit spectral element method adi sem for solving multi dimensional generalized modified anomalous sub diffusion equation abstract the main aim of the current paper is to solve the multi dimensional generalized modified anomalous sub diffusion equation by using a new spectral element method at first the time variable has been discretized by a finite difference scheme with second order accuracy the stability and convergence of the time discrete scheme have been investigated we show that the time discrete scheme is unconditionally stable and the convergence order is o 2 in the temporal direction secondly the galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable the unconditional stability and convergence of the full discrete scheme have been proved by developing the proposed scheme we need to calculate one dimensional integrals for two dimensional problems and two dimensional integrals for three dimensional problems thus the used cpu time for the presented numerical procedure is lower than the two and three dimensional galerkin spectral element methods also the proposed method is suitable for computational domains obtained from the tensor product finally two examples are analyzed to check the theoretical results

quenching study of two dimensional fractional reaction diffusion equation from combustion process abstract the quenching phenomenon and its physical characters are critical issues in the study of combustion process in this paper a two dimensional temporal fractional combustion model with caputo derivative is considered we design an ingenious adaptive finite difference discontinuous galerkin method to solve this fractional combustion model to catch quenching moment more accurately an explicit bisection adaptive strategy is proposed for discretization of fractional derivative moreover discontinuous galerkin method is employed for spatial discretization to settle large gradient changes and non smoothness property of solution in adjacent area of quenching moment the physical properties of combustion process such as positivity monotonicity and stability are theoretically or numerically investigated finally we carry out systematic numerical experiments for such abstract combustor with different shapes and various parameter settings which illustrate that the dynamics of such model depends on the intensity of initial inputs the order of fractional derivative and the area of spatial domain