efficient tests for top down termination of logical rules considered is the question of whether top down prolog like evaluation of a set of logical rules can be guaranteed to terminate the nail system is designed to process programs consisting of logical rules and to select for each fragment of the program the best from among many possible strategies for its evaluation in the context of such a system it is essential that termination tests be fast thus the uniqueness property of logical rules is introduced this property is satisfied by many of the common examples of rules and is easily recognized for rules with this property a set of inequalities whose satisfaction is sufficient for termination of the rules can be generated in polynomial time then a polynomial test for satisfaction of constraints generated by this process is given

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

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

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

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

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

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

existence and uniqueness of srb measure on c 1 generic hyperbolic attractors let m be a smooth riemannian manifold we show that for c1 generic f in rm diff 1 m if f has a hyperbolic attractor f then there exists a unique srb measure supported on f moreover the srb measure happens to be the unique equilibrium state of potential function psi f in c 0 lambda f defined by psi f x log det df e u x x in lambda f where e u x is the unstable space of txm

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

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

on the existence of nonterminating queries for a restricted class of prolog clauses abstract this work studies the existence of nonterminating queries for a restricted class of recursive predicates the class is restricted to predicates defined by a set of clauses containing only one direct recursive clause moreover this clause must be of the form p p for this restricted class of predicates we prove a necessary and sufficient condition for the existence of a query which is nonterminating in the absence of the occur check the condition is based upon concepts from rational trees

a novel analytical technique for the solution of time fractional ivancevic option pricing model abstract the ivancevic option pricing model is an alternative of the standard black scholes pricing equation which signifies a controlled brownian motion related to the nonlinear schrodinger equation even though many researchers have studied the applicability and practicality of this model but the analytical approach of this model is rarely found in the literature in this paper a novel semi analytical technique called fractional reduced differential transform method has been applied to solve the schrodinger type option pricing model which is characterized by the time fractional derivative two problems are explained to validate and prove the effectiveness of the proposed technique obtained results are compared with the solution of other existing methods for a particular case this comparison shows that the attained results are in good agreement with the existing solutions

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

controlling recursive inference abstract loosely speaking recursive inference occurs when an inference procedure generates an infinite sequence of similar subgoals in general the control of recursive inference involves demonstrating that recursive portions of a search space will not contribute any new answers to the problem beyond a certain level we first review a well known syntactic method for controlling repeating inference inference where the conjuncts processed are instances of their ancestors provide a proof that it is correct and discuss the conditions under which the strategy is optimal we also derive more powerful pruning theorems for cases involving transitivity axioms and cases involving subsumed subgoals the treatment of repeating inference is followed by consideration of the more difficult problem of recursive inference that does not repeat here we show how knowledge of the properties of the relations involved and knowledge about the contents of the system s database can be used to prove that portions of a search space will not contribute any new answers

monte carlo method for solving fredholm integral equations of the second kind in this paper we present a numerical method based on random sampling for the solution of fredholm integral equations of the second kind this method is a monte carlo method based on the simulation of a continuous markov chain to illustrate the usefulness of this technique we apply it to some test problems numerical results are performed in order to show the efficiency and accuracy of the present work

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

a compact lod method and its extrapolation for two dimensional modified anomalous fractional sub diffusion equations a crank nicolson type compact locally one dimensional lod finite difference method is proposed for a class of two dimensional modified anomalous fractional sub diffusion equations with two time riemann liouville fractional derivatives of orders 1 and 1 0 1 the resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one dimensional problems this property evidently enhances the simplicity of programming and makes the computations more easy the unconditional stability and convergence of the scheme are rigorously proved the error estimates in the standard h 1 and l 2 norms and the weighted l norm are obtained and show that the proposed compact lod method has the accuracy of the order 2 min in time and 4 in space a richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min 4 min if and min 1 4 if a comparison study of the compact lod method with the other existing methods is given to show its superiority numerical results confirm our theoretical analysis and demonstrate the accuracy and the effectiveness of the compact lod method and the extrapolation algorithm

the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation abstract in this paper the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation is studied most of the previous methods are low in temporal and spatial accuracy order based on the idea of monte carlo markov chain method and compact finite difference schemes a probability model for solving the modified anomalous fractional sub diffusion equation is established numerical examples are given to show the feasibility of the proposed scheme compared with the compact finite difference method the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order and it is also applied to solve partial differential equation in irregular domains

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