a novel numerical method for the time variable fractional order mobile immobile advection dispersion model evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations recently researchers have found that many physical processes exhibit fractional order behavior that varies with time or space the continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable in this paper we consider the mobile immobile advection dispersion model with the coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint a novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated as for the convergence of the numerical method we only consider a special case i e the time fractional derivative is independent of the time variable t the case where the time fractional derivative depends on both the time variable t and the space variable x will be considered in a future work finally numerical examples are provided to show that the implicit difference approximation is computationally efficient

soliton solutions of the kadomtsev petviashvili ii equation we study a general class of line soliton solutions of the kadomtsev petviashvili ii kpii equation by investigating the wronskian form of its tau function we show that in addition to the previously known line soliton solutions of kpii this class also contains a large variety of multisoliton solutions many of which exhibit nontrivial spatial interaction patterns we also show that in general such solutions consist of unequal numbers of incoming and outgoing line solitons from the asymptotic analysis of the tau function we explicitly characterize the incoming and outgoing line solitons of this class of solutions we illustrate these results by discussing several examples

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

a functional approach to the stationary waiting time and idle period distributions of the gi g 1 queue the gi g 1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest such as the stationary waiting time distribution for the case where the input distributions have densities techniques from infinite dimensional analysis are used to obtain derivatives and taylor series expansions for the functionals these yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

a renewal theorem for random walks in multidimensional time soit x x 1 x 2 une famille de variables aleatoires i i d a valeurs entieres moyenne positive et variance finie positive soit sn x 1 x 2 xn on etudie le comportement asymptotique de la somme ponderee r k a n p sn k avec sommation sur n 1 lorsque k

second order asymptotic behaviour of subordinated sequences with longtailed subordinator abstract suppose that a n is a discrete probability distribution on the set n 0 0 1 2 and p n is some non negative sequence defined on the same set the equation b n k 0 p k a k n defines a new sequence b n here a k n denotes the k fold convolution of the distribution a n in the paper the asymptotic behaviour of the sequence b n is investigated it is known that for the large classes of the sequences a n and p n b n p n where 1 is the mean of the distribution a n the main object of the present work is to estimate the difference b n p n for some classes of the sequences a n and p n

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

resonance and web structure in discrete soliton systems the two dimensional toda lattice and its fully discrete and ultra discrete analogues we present a class of solutions of the two dimensional toda lattice equation its fully discrete analogue and its ultra discrete limit these solutions demonstrate the existence of soliton resonance and web like structure in discrete integrable systems such as differential difference equations difference equations and cellular automata ultra discrete equations

soliton fusion and fission in a generalized variable coefficient fifth order korteweg de vries equation in fluids under investigation in this paper is a generalized variable coefficient fifth order korteweg de vries equation which describes the interaction between a water wave and a floating ice cover or the gravity capillary waves via the hirota method bell polynomial approach and symbolic computation bilinear forms n soliton solutions backlund transformation and lax pair are derived infinitely many conservation laws are obtained based on the bell polynomial typed backlund transformation soliton fusion and fission and influence of the variable coefficients from the equation are analyzed both variable coefficients c t and n t are in direct proportion to the soliton velocities but have no effect on the amplitudes while another constant coefficient can affect the types of the interactions in the sense of the elastic or inelastic elastic inelastic interactions among the three solitons are presented as well

resonance of solitons in a coupled higher order ito equation abstract in this letter we propose a new coupled higher order ito equation and present its n soliton solutions in pfaffian form furthermore some interesting examples of soliton resonance related to two solitons near the resonant state are pointed out

characterizations of jacobians of curves with automorphisms we obtain a characterization of theta functions of jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties p p a v we first give a characterization in terms of finite dimensional orbits for a suitable action in the sato grassmannian secondly the introduction of formal baker akhiezer functions and formal t functions attached to a p p a v for the multipuncture case allows us to characterize in terms of bilinear identities those baker akhiezer functions that are baker akhiezer functions of jacobians of curves with automorphisms further in the case of automorphisms with fixed points we rewrite the previous result as a hierarchy of partial differential equations for the function of a p p a v finally since baker akhiezer and t functions are written in terms of theta functions these results give rise to characterizations of p p a v in terms of their theta functions

n soliton solutions to the dkp equation and weyl group actions we study soliton solutions to the dkp equation which is defined by the hirota bilinear form where 0 1 the functions n are given by the pfaffians of a certain skew symmetric matrix we identify a one soliton solution as an element of the weyl group of d type and discuss a general structure of the interaction patterns among the solitons soliton solutions are characterized by a 4n 4n skew symmetric constant matrix which we call the b matrix we then find that one can have m soliton solutions with m being any number from n to 2n 1 for some of the 4n 4n b matrices having only 2n nonzero entries in the upper triangular part the number of solitons obtained from those b matrices was previously expected to be just n

a high order exponential scheme for solving 1d unsteady convection diffusion equations in this paper a high order exponential hoe scheme is developed for the solution of the unsteady one dimensional convection diffusion equation the present scheme uses the fourth order compact exponential difference formula for the spatial discretization and the 2 2 pade approximation for the temporal discretization the proposed scheme achieves fourth order accuracy in temporal and spatial variables and is unconditionally stable numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature the results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

on harmonic renewal measures let be a probability and nu h sum limits n 1 infty frac 1 n mu n the corresponding harmonic renewal measure complementing earlier results where is concentrated on a halfline we investigate the behaviour ofvh x x 1 and the harmonic renewal functiong x vh x asx ifm1 x dx 0 we also consider the casem1 0

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

loewner theory in annulus i evolution families and differential equations loewner theory based on dynamical viewpoint is a powerful tool in complex analysis which plays a crucial role in such important achievements as the proof of famous bieberbach s conjecture and well celebrated schramm s stochastic loewner evolution sle recently bracci et al 10 11 16 have proposed a new approach bringing together all the variants of the deterministic loewner evolution in a simply connected reference domain we construct an analogue of this theory for the annulus in this paper the first of two articles we introduce a general notion of an evolution family over a system of annuli and prove that there is a 1 to 1 correspondence between such families and semicomplete weak holomorphic vector fields moreover in the non degenerate case we establish a constructive characterization of these vector fields analogous to the nonautonomous berkson porta representation of herglotz vector fields in the unit disk 10

on various solutions of the coupled kp equation the coupled kp ckp equation possesses n soliton solutions with many more degrees of freedom than the solitons for the usual kp equation have in comparison ckp solutions can therefore be expected to model far more complex interactions than their kp counterparts in this paper we present some typical solutions for the ckp system and for some of its reductions a complex coupled kdv equation and a coupled boussinesq equation and we analyse their interaction and asymptotic properties

the n component kp hierarchy and representation theory it is the aim of the present article to give all formulations of the n component kp hierarchy and clarify connections between them the generalization to the n component kp hierarchy is important because it contains many of the most popular systems of soliton equations like the davey stewartson system for n 2 the two dimensional toda lattice for n 2 the n wave system for n 3 and the darboux egoroff system it also allows us to construct natural generalizations to the davey stewartson and toda lattice systems of course the inclusion of all these systems in the n component kp hierarchy allows us to construct their solutions by making use of vertex operators

a fourth order extrapolated compact difference method for time fractional convection reaction diffusion equations with spatially variable coefficients abstract this paper is concerned with numerical methods for a class of time fractional convection reaction diffusion equations the convection and reaction coefficients of the equation may be spatially variable based on the weighted and shifted grunwald letnikov formula for the time fractional derivative and a compact finite difference approximation for the spatial derivative we establish an unconditionally stable compact difference method the local truncation error and the solvability of the resulting scheme are discussed in detail the stability of the method and its convergence of third order in time and fourth order in space are rigorously proved by the discrete energy method combining this method with a richardson extrapolation we present an extrapolated compact difference method which is fourth order accurate in both time and space a rigorous proof for the convergence of the extrapolation method is given numerical results confirm our theoretical analysis and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method

toda versus pfaff lattice and related polynomials the pfaff lattice was introduced by us in the context of a lie algebra splitting of gl infinity into sp infinity and an algebra of lower triangular matrices the pfaff lattice is equivalent to a set of bilinear identities for the wave functions which yield the existence of a sequence of tau functions the latter satisfy their own set of bilinear identities which moreover characterize them in the semi infinite case the tau functions are pfaffians in the same way that for the toda lattice the tau functions are hankel determinants interesting examples occur in the theory of random matrices where one considers symmetric and symplectic matrix integrals for the pfaff lattice and hermitian matrix integrals for the toda lattice there is a striking parallel between the pfaff lattice and the toda lattice and even more striking there is a map from one to the other mapping skew orthogonal to orthogonal polynomials in particular we exhibit two maps dual to each other mapping hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

a new high order adi numerical difference formula for time fractional convection diffusion equation abstract based on exponential transformation quadratic interpolation polynomial and pade approximation a new high order finite difference scheme is proposed for solving the two dimensional 2d time fractional convection dominated diffusion equation of order 0 1 the resulting scheme is of 3 order accuracy in time and fourth order accuracy in space in order to reduce the amount of computation the alternating direction implicit adi scheme is further developed numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme

a limited memory block bi diagonal toeplitz preconditioner for block lower triangular toeplitz system from time space fractional diffusion equation abstract a block lower triangular toeplitz system arising from the time space fractional diffusion equation is discussed for efficient solutions of such the linear system the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited the main contribution of this paper has two aspects i a block bi diagonal toeplitz preconditioner is developed for the block lower triangular toeplitz system whose storage is of o n with n being the spatial grid number ii a new skew circulant preconditioner is designed to accelerate the inverse of the block bi diagonal toeplitz preconditioner multiplying a vector numerical experiments are given to demonstrate the effectiveness of our two proposed preconditioners

dispersionless dkp hierarchy and the elliptic lowner equation we show that the dispersionless dkp hierarchy the dispersionless limit of the pfaff lattice admits a suggestive reformulation through elliptic functions we also consider one variable reductions of the dispersionless dkp hierarchy and show that they are described by an elliptic version of the lowner equation with a particular choice of the driving function the latter appears to be closely related to the painleve vi equation with a special choice of parameters

algebraic geometrical solutions of some multidimensional nonlinear evolution equations the known 2 1 dimensional breaking soliton equation the coupled kp equation with three potentials and a new 3 1 dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the 1 1 dimensional akns equations the abel jacobi coordinates are introduced to straighten out the associated flows from which algebraic geometrical solutions of the 2 1 dimensional breaking soliton equation the coupled kp equation and the 3 1 dimensional evolution equation are explicitly given in terms of the riemann theta functions

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

on subordinated distributions and generalized renewal measures where an n eno is some sequence of nonnegative numbers sn neno is the sequence of partial sums s0 0 sn xflxk of another sequence xk ken of i i d random variables and a c r is a fixed borel set such as 0 1 or 0 oo examples of such convolution series are subordinated distributions f 0oan 1 which arise as distributions of random sums and harmonic and ordinary renewal measures a0 0 an 1 n for all n c n in the first an 1 for all n c no in the second case these examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes they are also of interest in connection with representation theorems such as the levy representation of infinitely divisible distributions a traditional approach to such problems is via regular variation if the underlying random variables are nonnegative we can use laplace transforms and the related abelian and tauberian theorems see e g stam 1973 in the context of subordination and feller 1971 xiv 3 in connection with renewal theory embrechts maejima and omey 1984 is a recent treatment of generalized renewal measures along these lines the approach of the present paper is based on the wiener levy gel fand theorem and has occasionally been called the banach algebra method in gruibel 1983 we gave a new variant of this method for the special case of lattice distributions showing that by using the appropriate banach algebras of sequences arbitrarily fine expansions are possible under certain assumptions on the higher order differences of p x1 n fnen here we give a corresponding treatment of nonlattice distributions we restrict ourselves to an analogue of first order differences and obtain a number of theorems which perhaps are described best as next term results to explain this let us consider a special case in more detail

on the soliton solutions of the two dimensional toda lattice the two dimensional toda lattice 2dtl is a well known 2 1 dimensional integrable system that admits a large class of line soliton solutions we classify these soliton solutions by exploiting the structure of the casoratian expression for its tau function in general these solutions consist of unequal numbers of incoming and outgoing line solitons we classify the incoming and outgoing line solitons based on asymptotic analysis of the tau function of the 2dtl as the discrete variable tends to infinity we also identify various subclasses of solutions and characterize some of them in terms of the amplitudes and directions of the interacting solitons as a special case we obtain the reduction in the soliton solutions of the one dimensional toda lattice throughout we point out the similarities with and the differences from the corresponding results for the kadomtsev petviashvili equation

a strong renewal theorem for generalized renewal functions in the infinite mean case letf x be a nonarithmetic c d f on 0 such that 1 f x x l x wherel x is slowly varying and 0 1 leta x be regularly varying with exponent 1 a strong renewal theorem of blackwell type for generalized renewal functions of the form g t equiv sum limits n 0 infty a n f n t is proved here thus extending the recent work of embrechts maejima and omey 1 and that of erickson 4

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

quantum painleve calogero correspondence for painleve vi this paper is a continuation of our previous paper where the painleve calogero correspondence has been extended to auxiliary linear problems associated with painleve equations we have proved for the first five equations from the painleve list that one of the linear problems can be recast in the form of the non stationary schrodinger equation whose hamiltonian is a natural quantization of the classical calogero like hamiltonian for the corresponding painleve equation in the present paper we establish the quantum painleve calogero correspondence for the most general case the painleve vi equation we also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding schlesinger system with rational spectral parameter

a high order exponential adi scheme for two dimensional time fractional convection diffusion equations in this paper a high order compact exponential alternating direction implicit adi scheme for two dimensional fractional convection diffusion equations is proposed with o t 3 c h 1 4 h 2 4 accuracy where t h 1 h 2 are the temporal and spatial step sizes respectively the convergence of the finite difference scheme is studied using its matrix form by the energy method difficulty arising from the convection term is overcome by an analysis based on the eigenvalue decomposition of nonsymmetric tridiagonal matrices

dispersionless pfaff toda hierarchy and elliptic lowner equation we show that one variable reductions of the pfaff toda integrable hierarchy in the dispersionless limit are described by a system of coupled elliptic lowner komatu goluzin equations

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

matrix integral solutions to the discrete kp hierarchy and its pfaffianized version matrix integrals used in random matrix theory for the study of eigenvalues of hermitian ensembles have been shown to provide functions for several hierarchies of integrable equations in this article we extend this relation by showing that such integrals can also provide functions for the discrete kp hierarchy and a coupled version of the same hierarchy obtained through the process of pfaffianization to do so we consider the first equation of the discrete kp hierarchy the hirota miwa equation we write the wronskian determinant solutions to the hirota miwa equation and consider a particular form of matrix integrals which we show is an example of those wronskian solutions the argument is then generalized to the whole hierarchy a similar strategy is used for the pfaffianized version of the hierarchy except that in that case the solutions are written in terms of pfaffians rather than determinants

a high order compact exponential scheme for the fractional convection diffusion equation a high order compact exponential finite difference scheme for solving the fractional convection diffusion equation is considered in this paper the convection and diffusion terms are approximated by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative for this fully discrete implicit scheme the local truncation error is analyzed and the fourier method is used to discuss the stability the error estimate is given by the discrete energy method numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

pfaff tau functions consider the two dimensional toda lattice with certain skew symmetric initial condition which is preserved along the locus s t of the space of time variables restricting the solution to s t we obtain another hierarchy called pfaff lattice which has its own tau function being equal to the square root of the restriction of 2d toda tau function we study its bilinear and fay identities w and virasoro symmetries relation to symmetric and symplectic matrix integrals and quasiperiodic solutions

harmonic renewal measures and bivariate domains of attraction in fluctuation theory each probability measure c on a first orthant is associated with a harmonic renewal measure g specifically we consider n s n the ladder time place of a random walk sn using bivariate g we show that when s1 is in a domain of attraction so is n sn this unifies and generalizes results of sinai rogosin

analysis of mixed finite element method mfem for solving the generalized fractional reaction diffusion equation on nonrectangular domains abstract in the current manuscript we consider a generalized fractional reaction diffusion equation the considered model is based on the time fractional derivative the developed scheme is based on two procedures at first we obtain a semi discrete scheme for the temporal direction the time fractional derivative is discretized by using a difference scheme with second order accuracy thus the final time discrete formula has second order accuracy in the temporal direction then we obtain a fully discrete scheme by applying the mixed finite element method mfem for the constructed numerical technique we prove the unconditional stability and also obtain an error bound for the full discrete scheme by using the energy method we employ some test problems to show the accuracy of the proposed technique

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

partial skew orthogonal polynomials and related integrable lattices with pfaffian tau functions skew orthogonal polynomials sops arise in the study of the n point distribution function for orthogonal and symplectic random matrix ensembles motivated by the average of characteristic polynomials of the bures random matrix ensemble studied in forrester and kieburg commun math phys 342 1 151 187 2016 we propose the concept of partial skew orthogonal polynomials psops as a modification of the sops and then the psops with a variety of special skew symmetric kernels and weight functions are addressed by considering appropriate deformations of the weight functions we derive nine integrable lattices in different dimensions as a consequence the tau functions for these systems are shown to be expressed in terms of pfaffians and the wave vectors psops in fact the tau functions also admit the multiple integral representations among these integrable lattices some of them are known while the others are novel to the best of our knowledge in particular one integrable lattice is related to the partition function of the bures ensemble besides we derive a discrete integrable lattice which can be used to compute certain vector pade approximants this yields the first example regarding the connection between integrable lattices and generalised inverse vector valued pade approximants about which hietarinta joshi and nijhoff pointed out that this field remains largely to be explored in the recent monograph hietarinta et al in discrete systems and integrability vol 54 cambridge university press cambridge 2016 section 4 4

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

string orthogonal polynomials string equations and 2 toda symmetries 1 the 2 toda lattice and its generic symmetries 2 a larger class of symmetries for special initial conditions 3 borel decomposition of moment matrices tau functions and string orthogonal polynomials 4 from string orthogonal polynomials to the 2 toda lattice and the string equation 5 virasoro constraints on two matrix integrals comment 45 pages latex

functions of discrete probability measures rates of convergence in the renewal theorem let and be finite measures on the set of integers such that the fourier transform of is an analytic function of the transform the central result shows how may be approximated by linear combinations of convolution powers of applications are given to renewal theory infinitely divisible measures and age dependent branching processes

compact alternating direction implicit scheme for the two dimensional fractional diffusion wave equation in this paper we consider the numerical method for solving the two dimensional fractional diffusion wave equation with a time fractional derivative of order alpha 1 alpha 2 a difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit adi method in the time stepping is proposed and analyzed the unconditional stability and h 1 norm convergence of the scheme are proved rigorously the convergence order is mathcal o tau 3 alpha h 1 4 h 4 2 where tau is the temporal grid size and h 1 h 2 are spatial grid sizes in the x and y directions respectively in addition a crank nicolson adi scheme is presented and the corresponding error estimates are also established numerical results are presented to support our theoretical analysis and indicate that the compact adi scheme reduces the storage requirement and cpu time successfully

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

numerical schemes with high spatial accuracy for a variable order anomalous subdiffusion equation in this paper we consider a variable order anomalous subdiffusion equation a numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed the convergence stability and solvability of the numerical scheme are discussed via the technique of fourier analysis another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

on moving frames and toda lattices of bkp and ckp types this paper is mainly concerned with the geometric formulations of toda lattices of bkp and ckp types as evolutions of invariants and their explicit expressions the theory for discrete curves in centro affine geometry is constructed by using the method of a moving frame with the help of orthogonal polynomial theory we choose the appropriate evolutions for the curves and the evolutions induced on invariants are related to these toda type lattices in addition the explicit expressions for the discrete invariant curve flows discrete moving frames and maurer cartan invariants are provided

a high order l2 compact difference method for caputo type time fractional sub diffusion equations with variable coefficients abstract a high order compact finite difference method is proposed for solving a class of time fractional sub diffusion equations the diffusion coefficient of the equation may be spatially variable and the time fractional derivative is in the caputo sense with the order 0 1 the caputo time fractional derivative is discretized by a 3 th order numerical formula called the l2 formula here which is constructed by piecewise quadratic interpolating polynomials but does not require any sub stepping scheme for the approximation at the first time level the variable coefficient spatial differential operator is approximated by a fourth order compact finite difference operator by developing a technique of discrete energy analysis a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all 0 1 the optimal error estimate is obtained in the l2 norm and shows that the proposed method has the temporal 3 th order accuracy and the spatial fourth order accuracy further approximations are also considered for enlarging the applicability of the method while preserving its high order accuracy applications are given to three model problems and numerical results are presented to demonstrate the theoretical analysis results

a fast second order implicit scheme for non linear time space fractional diffusion equation with time delay and drift term abstract in this paper a second order accurate implicit scheme based on the l2 1 formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time space fractional diffusion equations with time drift term and non linear delayed source function the stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions then we theoretically and numerically show that the proposed scheme converges in the l2 norm with the order o t 2 h 2 with time step t and mesh size h moreover it finds that the discreted linear systems are symmetric toeplitz systems in order to solve these systems efficiently the conjugate gradient method with suitable circulant preconditioners is designed in each iterative step the storage requirements and the computational complexity of the resulting equations are o n and o n log n respectively where n is the number of grid nodes numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners

soliton like behavior in automata abstract we propose a new kind of automaton that uses newly computed site values as soon as they are available we call them filter automata fa they are analogous to infinite impulse response iir digital filters whereas the usual cellular automata ca correspond to finite impulse response fir digital filters it is shown that as a class the fa s are equivalent to ca s in the sense that the same array of space generation values can be produced they must be generated in a different order however a particular class of irreversible totalistic fa s are described that support a profusion of persistent structures that move at different speeds and these particle like patterns collide in nondestructive ways they often pass through one another with nothing more than a phase jump much like the solitons that arise in the solution of certain nonlinear differential equations histograms of speed displacement and period are given for neighborhood radii from 2 to 6 and particles with generators up to 16 bits wide we then present statistics for neighborhood radii 2 to 9 which show that collisions which preserve the identity of particles are very common

backlund darboux transformations for the coupled kp hierarchy using the geometry of the infinite isotropic grassmannian related to the hirota ohta coupled kp hierarchy we construct its elementary backlund darboux transformations

an unconditionally stable compact adi method for three dimensional time fractional convection diffusion equation abstract a high order compact finite difference method is presented for solving the three dimensional 3d time fractional convection diffusion equation of order 1 2 the original equation is first transformed to a fractional diffusion wave equation then using fourth order pade approximation for spatial derivatives and the center difference method for time derivative respectively a fully discrete implicit compact scheme is obtained furthermore based on different splitting terms three unconditionally stable adi compact schemes with optimal convergence order are developed respectively the resulting schemes in each adi solution step corresponding to a strictly diagonally dominant matrix equation can be solved using the 1d tridiagonal thomas algorithm with a considerable saving in computing time numerical experiments show that these schemes can significantly improve the time accuracy

spider web solutions of the coupled kp equation the coupled kadomtsev petviashvili ckp equation possesses n soliton solutions with more parametric freedom than the solitons of the usual kp equation its solutions can therefore be expected to model far more complex interactions than their kp counterparts the existence of web like structures on a finite scale for ckp solutions isojima s willox r and satsuma j 2002 j phys a math gen 35 6893 6909 is a manifestation of this greater freedom in this paper we propose a new method to analyse the behaviour of solitons which we demonstrate in some examples in addition we discuss essentially three body collisions described by a two soliton solution of the ckp equation

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

painleve calogero correspondence revisited we extend the work of fuchs painleve and manin on a calogero like expression of the sixth painleve equation the painleve calogero correspondence to the other five painleve equations the calogero side of the sixth painleve equation is known to be a nonautonomous version of the rank one elliptic model of inozemtsev s extended calogero systems the fifth and fourth painleve equations correspond to the hyperbolic and rational models in inozemtsev s classification those corresponding to the third second and first are not included therein we further extend the correspondence to the higher rank models and obtain a multi component version of the painleve equations

similarities and differences of two exponential schemes for convection diffusion problems the fv cf and enate schemes abstract in this paper we present a comparison of two novel exponential schemes for convection diffusion problems an exponential scheme uses in one way or another the analytical solution of the flux of a one dimensional 1d transport equation thereby improving the results of the simulation in a multidimensional problem the 1d solution is combined with operator splitting the two approximations to be assessed are the finite volume complete flux fv cf and the enhanced numerical approximation of a transport equation enate schemes they were proposed by the two groups that co author the current paper both schemes share many similarities in 1d but differ especially in 2d in some aspects that will be highlighted in their derivation the algebraic coefficients of the computational stencil are integrals of flow parameters whose calculation is crucial for the accuracy of either method these factors and their various approximations will be analysed some test cases will be used to check the ability of both schemes to provide accurate results

compact exponential scheme for the time fractional convection diffusion reaction equation with variable coefficients high order compact exponential finite difference scheme for solving the time fractional convection diffusion reaction equation with variable coefficients is considered in this paper the convection diffusion and reaction coefficients can depend on both the spatial and temporal variables we begin with the one dimensional problem and after transforming the original equation to one with diffusion coefficient unity the new equation is discretized by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative we prove the solvability of this fully discrete implicit scheme and analyze its local truncation error for the fractional equation with constant coefficients we use fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate then we discuss the two dimensional problem give the compact adi scheme with the restriction that besides the time variable the convection coefficients can only depend on the corresponding spatial variables respectively numerical results are provided to verify the accuracy and efficiency of the proposed algorithm the time fractional convection diffusion reaction equation is numerically solved compact exponential scheme for the variable coefficients equation is given fourier method and matrix analysis are used for the analysis the compact exponential adi scheme is given for the two dimensional problem high accuracy and good efficiency of the proposed algorithm is demonstrated

on a family of solutions of the kadomtsev petviashvili equation which also satisfy the toda lattice hierarchy we describe the interaction pattern in the x y plane for a family of soliton solutions of the kadomtsev petviashvili kp equation the solutions considered also satisfy the finite toda lattice hierarchy we determine completely their asymptotic patterns for y and we show that all the solutions except the 1 soliton solution are of resonant type consisting of arbitrary numbers of line solitons in both asymptotics that is arbitrary n incoming solitons for y interact to form arbitrary n outgoing solitons for y we also discuss the interaction process of those solitons and show that the resonant interaction creates a web like structure having n 1 n 1 holes

on generalized renewal measures and certain first passage times by relating it to the family x t infln 1 sn t t 0 one of the major results is that up t oo for all t e r iff 0 t u0 t 1 a 1 ua i1as t a oo iff e xj 2 x1 oo provided 4 is ultimately increasing a 2 0 a related result is proved for u t h u t and up t e n 2 o4 n p mn t where mn max0 i n s our results form extensions of earlier ones by heyde kalma gut and others who either considered more specific functions 4 or used stronger moment assumptions the proofs are based on a regeneration technique from renewal theory and two martingale inequalities by burkholder davis and gundy

compact alternating direction implicit method for two dimensional time fractional diffusion equation high order compact finite difference scheme with operator splitting technique for solving two dimensional time fractional diffusion equation is considered in this paper a grunwald letnikov approximation is used for the riemann liouville time derivative and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme alternating direction implicit adi method is used to split the original problem into two separate one dimensional problems the local truncation error is analyzed and the stability is discussed by the fourier method the proposed scheme is suitable when the order of the time fractional derivative c lies in the interval 12 1 a correction term is added to maintain high accuracy when c 0 12 numerical results are provided to verify the accuracy and efficiency of the proposed algorithm