a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

a hypothesis testing approach toward identifying active contrasts the problem of identifying the active contrasts in an unreplicated fractional factorial design is approached from a hypothesis testing point of view the null hypothesis that all contrasts are inactive is first tested and if rejected the active contrasts causing rejection are identified and estimated the method is illustrated with examples from box and meyer compared with the confidence interval approach of dong and lenth and found to be a worthy competitor

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

special solutions from the variable separation approach the davey stewartson equation a variable separation procedure for the davey stewartson ds equation is proposed by using a prior ansatz to its bilinear form the reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables the trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a boussinesq type equation system whenever a pair of solutions of the reduced fields are obtained a corresponding solution of the ds equation can be obtained algebraically the single dromion solution and some kinds of positon solutions are obtained explicitly

a new general algebraic method and its applications to the 2 1 dimensional broer kaup kupershmidt equations abstract using a computerized symbolic computation technique a new method named the repeated general algebraic method rgam is established in this study in order to find exact solutions of nonlinear partial differential equations nlpdes the new method is validated based on the 2 1 dimensional broer kaup kupershmidt bkk equations by using the rgam in various conditions a number of exact solutions of nlpdes have been obtained showing potential importance in future physical applications also it is anticipated that the rgam can be applied to other nonlinear evolution equations in mathematical physics to produce some interesting outcomes

on the time fractional navier stokes equations abstract this paper is concerned with the navier stokes equations with time fractional derivative of order 0 1 this type of equations can be used to simulate anomalous diffusion in fractal media we establish the existence and uniqueness of local and global mild solutions in h q meanwhile we also give local mild solutions in j q moreover we prove the existence and regularity of classical solutions for such equations in j q

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

identifying active contrasts by stepwise testing unreplicated fractional factorial designs are frequently used in industrial experimentation and are characterized by the absence of degrees of freedom for estimation of the error variance this article presents step down and step up testing procedures that efficiently identify the active contrasts significant effects in such experiments and simultaneously provide an accurate estimate of the error variance the methods allow the user to restrict the experimentwise error rate to a prespecified level use of the new methods is illustrated in terms of standard examples from the literature simulation shows that a version of the step up procedure outperforms its competitors

investigation of classical and fractional bose einstein condensation for harmonic potential abstract in this study classical and fractional gross pitaevskii gp equations were solved for harmonic potential and repulsive interactions between the boson particles using the homotopy perturbation method hpm to investigate the ground state dynamics of bose einstein condensation bec the purpose of writing fractional gp equations is to consider the system in a more realistic manner the memory effects of non markovian processes involving long range interactions between bosons with the restriction of the ergodic hypothesis and the effect of non gaussian distributions of bosons in the condensation can be taken into account with time fractional and space fractional gp equations respectively the obtained results of the fractional gp equations differ from the results of the classical one while the gauss distribution describing the homogeneous reversible and unitary system is obtained from the classical gp equation the probability density of the solution function of fractional gp equations is non conserved this situation describes the inhomogeneous irreversible and non unitary systems

generalized method and new exact wave solutions for 2 1 dimensional broer kaup kupershmidt system in this paper a new algebra method i e generalized elliptic equation rational expansion method is devised to uniformly construct a series of exact solutions for nonlinear partial differential equations compared with most existing tanh methods the proposed method not only recover some known solutions but also find some new and general solutions the efficiency of the method can be demonstrated on the 2 1 dimensional broer kaup kupershmidt system as a result we obtain many new types of solutions include rational formal solitary wave solutions rational formal triangular periodic wave solutions rational formal jacobi and weierstrass double period solutions and rational wave solutions

the numerical study for the ground and excited states of fractional bose einstein condensates abstract in this paper we study the ground and first excited states of the fractional bose einstein condensates bec which is modeled by fractional gross pitaevskii gp equation we first introduce the normalized gradient flow method and prove its energy diminishing property then the weighted shifted grunwald letnikov difference wsgd method is used to discretize the gross pitaevskii equation the corresponding normalization and energy diminishing property for the semi discrete scheme are proved for the time discretization we use the implicit integration factor iif method which decouples the diffusion and nonlinear terms separately finally the numerical methods are applied to compute the ground and first excited states of fractional bec with harmonic oscillator harmonic plus optical lattice and box potential our numerical results show that the ground and excited states in fractional gp equation differ from those of the standard non fractional gp equation

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

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

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

control of error rates in adaptive analysis of orthogonal saturated designs individual and simultaneous confidence intervals using the data adaptively are constructed for the effects in orthogonal saturated designs under the assumption of effect sparsity the minimum coverage probabilities of the intervals are equal to the nominal level 1

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

a permutation test for effects in an unreplicated factorial design a permutation test is proposed for examining the significance of effects in unreplicated factorial experiments the procedure tests each effect with a separate sampling distribution using a test statistic that is equivalent to the optimal invariant decision rule of birnbaum the proposed test is more flexible than other methods proposed for the same situation because it requires no a priori assumptions regarding the underlying distribution of the data nor does it impose any practical restriction on the number of potentially significant effects present

a novel approach of high accuracy analysis of anisotropic bilinear finite element for time fractional diffusion equations with variable coefficient abstract in this paper by using bilinear finite element and l 1 approximation a fully discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes unconditionally stable analysis of the proposed scheme are presented in both l 2 norm and h 1 norm moreover convergence superclose and superconvergence results are derived by combining interpolation with projection which is the key technique for the numerical analysis specifically by defining a novel projection operator the error estimate between the projection and the exact solution is obtained on anisotropic meshes furthermore high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique based on the related results about projection and interpolation the optimal error estimate in l 2 norm and superclose of interpolation in h 1 norm are deduced by skillfully dealing with fractional derivative at the same time the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

inference for effect saturated fractional factorials industrial scientists and engineers often use experimental designs in which all degrees of freedom are used to estimate effects and consequently no classical estimate of the error is possible robust scale estimates provide an alternative measure of the error in this study several such scale estimators are evaluated based on the power or related significance tests the pseudo standard error method of lenth provides the best overall performance lenth s t approximation for critical values was found to be inaccurate however so new tables are provided additional recommendations are made according to the experimenter s prior belief in the number of likely important factors

travelling wave solutions of nonlinear partial equations by using the first integral method in this paper the first integral method is employed for constructing the new exact travelling wave solutions of nonlinear partial differential equations the power of this manageable method is confirmed by applying it for two selected nonlinear partial equations this approach can also be applied to other systems of nonlinear differential equations

step up simultaneous tests for identifying active effects in orthogonal saturated designs a sequence of null hypotheses regarding the number of negligible effects zero effects in orthogonal saturated designs is formulated two step up simultaneous testing procedures are proposed to identify active effects nonzero effects under the commonly used assumption of effect sparsity it is shown that each procedure controls the experimentwise error rate at a given a level in the strong sense

quick and easy analysis of unreplicated factorials box and meyer 1986 introduced a method for assessing the sizes of contrasts in unreplicated factorial and fractional factorial designs this is a useful technique and an associated graphical display popularly known as a bayes plot makes it even more effective this article presents a competing technique that is also effective and is computationally simple an advantage of the new method is that the results are given in terms of the original units of measurement this direct association with the data may make the analysis easier to explain

efficient semi implicit schemes for stiff systems when explicit time discretization schemes are applied to stiff reaction diffusion equations the stability constraint on the time step depends on two terms the diffusion and the reaction the part of the stability constraint due to diffusion can be totally removed if the linear diffusions are treated exactly using integration factor if or exponential time differencing etd methods for systems with severely stiff reactions those methods are not efficient because the reaction terms in if or etd are still approximated with explicit schemes in this paper we introduce a new class of semi implicit schemes which treats the linear diffusions exactly and explicitly and the nonlinear reactions implicitly a distinctive feature of the scheme is the decoupling between the exact evaluation of the diffusion terms and implicit treatment of the nonlinear reaction terms as a result the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points it only depends on the number of original equations unlike the case in which standard implicit temporal schemes are directly applied to the reaction diffusion system the stability region for this class of methods is much larger than existing methods using an explicit treatment of reaction terms in particular the one with second order accuracy is unconditionally linearly stable with respect to both diffusion and reaction direct numerical simulations on test equations as well as morphogen systems from developmental biology show the new semi implicit schemes are efficient robust and accurate

existence and multiplicity results of homoclinic solutions for fractional hamiltonian systems abstract in this paper by the critical point theory we consider the existence and multiplicity of solutions for the following fractional differential equation t d d t u t l t u t w t u t t r where 1 2 1 d t and t d are left and right liouville weyl fractional derivatives of order on the whole axis r respectively u r n l t is positive definite symmetric matrix for all t r and w r r n r is a suitably chosen function

new exact solutions for some nonlinear differential equations using symbolic computation based on computerized symbolic computation and a new general ansatze a generalized tanh function method for constructing multiple travelling wave solutions of nonlinear evolution equations nees is presented and implemented in a computer algebraic system applying the generalized method with the aid of maple we consider some nees with physics interests as a result we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh function method and other more sophisticated methods more importantly for some equations we also obtain other new and more general solutions at same time the results include kink profile solitary wave solutions bell profile solitary wave solutions periodic wave solutions rational solutions singular solutions and new formal solutions the properties of the new soliton solutions for wbk equations are shown by some figures

a time space spectral method for the time space fractional fokker planck equation and its inverse problem abstract in the paper we consider a time space spectral method to get the numerical solution of time space fractional fokker planck initial boundary value problem the temporal discretization is constructed by jacobi polynomials and the spatial discretization is composed by legendre polynomials moreover we present the stability and convergence analysis strictly the main advantages of the provided method are spectrally accurate in time and space and high computational efficiency in addition we introduce the inverse problem based on the spectral form with high order accuracy of the direct problem for the first time the levenberg marquardt l m method is proposed to estimate the two fractional derivatives and 2 some numerical results presented are consistent with the theoretical analysis

on the continuum limit for discrete nls with long range lattice interactions we consider a general class of discrete nonlinear schrodinger equations dnls on the lattice h mathbb z with mesh size h 0 in the continuum limit when h 0 we prove that the limiting dynamics are given by a nonlinear schrodinger equation nls on mathbb r with the fractional laplacian as dispersive symbol in particular we obtain that fractional powers frac 1 2 alpha 1 arise from long range lattice interactions when passing to the continuum limit whereas the nls with the usual laplacian describes the dispersion in the continuum limit for short range or quick decaying interactions e g nearest neighbor interactions

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

fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and non smooth solutions abstract we consider the initial boundary value problem of the time fractional nonlinear sine gordon equation and the fractional derivative is described in caputo sense with the order 1

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results