approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

analytical and numerical solutions of a one dimensional fractional in space diffusion equation in a composite medium in this paper a one dimensional space fractional diffusion equation in a composite medium consisting of two layers in contact is studied both analytically and numerically since domain decomposition is the only approach available to solve this problem we at first investigate analytical and numerical strategies for a composite medium with the same fractal dimension in each layer to ascertain which domain decomposition approach is the most accurate and consistent with a global solution methodology which is available in this case we utilise a matrix representation of the fractional in space operator to generate a system of linear odes with the matrix raised to the same fractional exponent we show that the global and domain decomposition numerical strategies for this problem produce simulation results that are in good agreement with their analytic counterparts and conclude that the domain decomposition that imposes the neumann condition at the interface produces the most consistent results finally we carry this finding to study the composite problem with different fractal dimensions where we again favourably compare analytic and numerical solutions the resulting method can be naturally extended to space fractional diffusion in a composite medium consisting of more than two layers

numerical analysis of a new space time variable fractional order advection dispersion equation many physical processes appear to exhibit fractional order behavior that may vary with time and 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 a new space time variable fractional order advection dispersion equation on a finite domain the equation is obtained from the standard advection dispersion equation by replacing the first order time derivative by coimbra s variable fractional derivative of order x 0 1 x 0 1 and the first order and second order space derivatives by the riemann liouville derivatives of order x t 0 1 x t 0 1 and x t 1 2 x t 1 2 respectively we propose an implicit euler approximation for the equation and investigate the stability and convergence of the approximation finally numerical examples are provided to show that the implicit euler approximation is computationally efficient

numerical simulation of a new two dimensional variable order fractional percolation equation in non homogeneous porous media percolation flow problems are discussed in many research fields such as seepage hydraulics groundwater hydraulics groundwater dynamics and fluid dynamics in porous media many physical processes appear to exhibit fractional order behavior that may vary with time or space or space and time the theory of pseudodifferential operators and equations has been used to deal with this situation in this paper we use a fractional darcy s law with variable order riemann liouville fractional derivatives this leads to a new variable order fractional percolation equation in this paper a new two dimensional variable order fractional percolation equation is considered a new implicit numerical method and an alternating direct method for the two dimensional variable order fractional model is proposed consistency stability and convergence of the implicit finite difference method are established finally some numerical examples are given the numerical results demonstrate the effectiveness of the methods this technique can be used to simulate a three dimensional variable order fractional percolation equation

improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

co rotational beam elements with warping effects in instability problems the present paper investigates the formulation of 3d co rotational beam elements for the buckling and post buckling analysis of frame structures following pacoste and eriksson comput methods app

weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients a class of weighted finite difference methods wfdms for solving a class of initial boundary value problems of space fractional partial differential equations with variable coefficients is presented their stability and convergence properties are considered it is proven that the wfdms are unconditionally stable for email protected email protected 12 and conditionally stable for 12

the analytical solution and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation in this paper we consider the analytical and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation we derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate mittag leffler function an implicit difference approximation is constructed stability and convergence analysis of the numerical scheme are proved by the energy method numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis

analytical solution for the time fractional telegraph equation by the method of separating variables in this paper a method of separating variables is effectively implemented for solving a time fractional telegraph equation tfte we discuss and derive the analytical solution of the tfte with three kinds of nonhomogeneous boundary conditions namely dirichlet neumann and robin boundary conditions

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

a three dimensional non linear timoshenko beam based on the core congruential formulation a three dimensional geometrically non linear two node timoshenko beam element based on the total lagrangian description is derived the element behaviour is assumed to be linear elastic but no restrictions are placed on the magnitude of finite rotations the resulting element has twelve degrees of freedom six translational components and six rotational vector components the formulation uses the green lagrange strains and second piola kirchhoff stresses as energy conjugate variables and accounts for bending stretching and bending torsional coupling effects without special provisions the core congruential formulation ccf is used to derive the discrete equations in a staged manner core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level a sequence of matrix transformations carries these equations to beam cross sections and finally to the element nodal degrees of freedom the choice of finite rotation measure is made in the next to last transformation stage and the choice of over the element interpolation in the last one the tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations an extensive set of numerical examples are presented to test and validate the present element

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

solving linear and non linear space time fractional reaction diffusion equations by the adomian decomposition method in this paper we consider linear and non linear space time fractional reaction diffusion equations stfrde on a finite domain the equations are obtained from standard reaction diffusion equations by replacing a second order space derivative by a fractional derivative of order 1 2 and a first order time derivative by a fractional derivative of order 0 1 we use the adomian decomposition method to construct explicit solutions of the linear and non linear stfrde finally some examples are given

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

velocity based approach in non linear dynamics of three dimensional beams with enforced kinematic compatibility abstract in the paper we present a new finite element formulation for the dynamic analysis of geometrically exact three dimensional beams we limit our studies to implicit time integration schemes and possible approaches for increasing their robustness and numerical stability in contrast to standard displacement rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities to describe the rotational degrees of freedom quaternions are used the time integration scheme and the governing equations of the three dimensional beam are modified accordingly in the numerical implementation the galerkin type discretization is employed to obtain the finite element formulation of the problem the result of our studies is simple but accurate efficient and robust numerical model

a darboux transformation and an exact solution for the relativistic toda lattice equation a darboux transformation for the relativistic toda lattice hierarchy is constructed as an application an exact solution of the relativistic toda lattice equation is presented

recovering the historical distribution for nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space abstract in this paper we investigate the problem of recovering the historical distribution for a nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a fractional laplacian of order 1 2 1 which is usually used to model the anomalous diffusion the problem is severely ill posed to regularize the problem we propose a modified version of the tikhonov regularization method a stability estimate of holder type is established finally several numerical examples based on the finite difference approximation and the discrete fourier transform are presented to illustrate the theoretical results

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

a new regularization method for the time fractional inverse advection dispersion problem in this paper we consider the time fractional inverse advection dispersion problem tfiadp in a quarter plane the solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body such a problem is obtained from the classical advection dispersion equation by replacing the first order time derivative with the caputo fractional derivative of order alpha 0 alpha 1 we show that the tfiadp is severely ill posed and we further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under a priori bound assumptions for the exact solution finally one numerical example is given to show that the proposed numerical method is effective

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

on timoshenko like modeling of initially curved and twisted composite beams abstract a generalized finite element based cross sectional analysis for nonhomogenous initially curved and twisted anistropic beams is formulated from geometrically nonlinear three dimensional 3 d elasticity the 3 d strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one dimensional 1 d generalized strains and a 3 d warping displacement that is obtained from the formulation not assumed the warping is found in terms of the 1 d strains via the variational asymptotic method vam in this paper a timoshenko like model is presupposed for a beam with cross sectional characteristic length h wavelength of deformation given by l and the magnitude of the radius of initial curvature and or twist is taken to be of the order r first a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through o h2 r2 is obtained next the o h2 l2 correction is computed it is known that timoshenko like theory is not capable of capturing all the o h2 l2 corrections for generally anisotropic beams however if all the o h2 l2 terms are known then the corresponding timoshenko like theory is uniquely defined numerical results are presented to illustrate the trends of the various classical extension twist bending twist and extension bending and nonclassical couplings extension shear bending shear and shear torsion as the initial twist and curvatures are varied

new numerical methods for the riesz space fractional partial differential equations in this paper we consider the numerical solution of the riesz space fractional diffusion equation and advection dispersion equation first a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method furthermore we use the 2 2 pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation and get two difference schemes second using the matrix analysis method we prove that the two difference schemes are unconditionally stable finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

on the differentiation of the rodrigues formula and its significance for the vector like parameterization of reissner simo beam theory in this paper we present a systematic way of differentiating up to the second directional derivative i the rodrigues formula and ii the spin rotation vector variation relationship to achieve this goal several trigonometric functions are grouped into a family of scalar quantities which can be expressed in terms of a single power series these results are then applied to the vector like parameterization of reissner simo beam theory enabling a straightforward derivation and leading to a clearer formulation in particular and in contrast with previous formulations a relatively compact and obviously symmetric form of the tangent operator is obtained the paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples copyright 2002 john wiley sons ltd

solution of differential difference equations by using differential transform method abstract in this work we successfully extended differential transform method dtm by presenting and proving new theorems to the solution of differential difference equations ddes theorems are presented in the most general form to cover a wide range of ddes being linear or nonlinear and constant or variable coefficient in order to show the power and the robustness of the method and to illustrate the pertinent features of related theorems examples are presented

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

mixed function method for obtaining exact solution of nonlinear differential difference equations and coupled nddes in this paper we construct a new mixed function method for the first time by using this new method we study the two nonlinear differential difference equations named the generalized hybrid lattice and two component volterra lattice equations some new exact solutions of mixed function type such as discrete solitary wave solutions discrete kink and anti kink wave solutions and discrete breather solutions with kink and anti kink character are obtained and their dynamic properties are also discussed by using software mathematica we show their profiles

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

on finite element implementation of geometrically nonlinear reissner s beam theory three dimensional curved beam elements abstract finite element implementation of the three dimensional finite strain beam theory of reissner is considered in this work in contrast with some earlier works on the subject discussed here are the beam elements whose reference axes are arbitrary space curved lines we have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results however it also makes the choice of non locking finite element interpolations somewhat more delicate a hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena a very satisfying non locking performance is demonstrated for a set of problems in nonlinear elasto statics

kinematically exact curved and twisted strain based beam the paper presents a formulation of the geometrically exact three dimensional beam theory where the shape functions of three dimensional rotations are obtained from strains by the analytical solution of kinematic equations in general it is very demanding to obtain rotations from known rotational strains in the paper we limit our studies to the constant strain field along the element the relation between the total three dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed the analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential despite the analytical relationship between rotations and rotational strains the governing equations of the beam are in general too demanding to be solved analytically a finite element strain based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals some interesting connections between quantities and non linear expressions of the beam are revealed these relations can also serve as useful guidelines in the development of new finite elements especially in the choice of suitable shape functions

alternating direction implicit euler method for the two dimensional fractional evolution equation in this paper we present a fast and efficient numerical method to solve a two dimensional fractional evolution equation on a finite domain this numerical method combines the alternating direction implicit adi approach with the second order difference quotient in space the backward euler in time and order one convolution quadrature approximating the integral term by using the discrete energy method we prove that the adi scheme is unconditionally stable and the numerical solution converges to the exact one with order o k h x 2 h y 2 where k is the temporal grid size and h x h y are spatial grid sizes in the x and y directions respectively two numerical examples with known exact solution are also presented and the behavior of the error is analyzed to verify the order of convergence of the adi euler method

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

the quaternion based three dimensional beam theory this paper presents the equations for the implementation of rotational quaternions in the geometrically exact three dimensional beam theory a new finite element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam the formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form a strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented several numerical examples demonstrate the validity performance and accuracy of the proposed approach c 2009 elsevier b v all rights reserved

buckling of a twisted and compressed rod we consider the problem of determining the stability boundary for an elastic rod under thrust and torsion the constitutive equations of the rod are such that both shear of the cross section and compressibility of the rod axis are considered the stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations the type of bifurcation is determined for parameter values it is shown that the bifurcating branch is the branch with minimal energy finally by using the first integral the solution for one specific dependent variable is expressed in terms of elliptic integrals the solution pertaining to the complete set of equilibrium equations is obtained by numerical integration

solution of difference equations by using differential transform method abstract this work presents semi numerical analytical solutions to both linear and nonlinear difference equations with variable coefficients by applying differential transform method dtm and extending it with the related theorems and their proofs to make clear and illustrate the features and capabilities of the presented method examples are carried out and numerical results with comparison to exact solutions and other methods are given

a generalized ablowitz ladik hierarchy multi hamiltonian structure and darboux transformation starting from a discrete spectral problem we derive a hierarchy of nonlinear discrete equations which includes the ablowitz ladik hierarchy and a new hierarchy as special cases especially we investigate in detail the integrability and resolvability of the new hierarchy it is shown that the new hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure a darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its lax pair as applications of the darboux transformation new exact solutions of the discrete system are explicitly given

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

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

exact travelling wave solutions of the discrete sine gordon equation obtained via the exp function method abstract in this paper we generalize the exp function method which was used to find new exact travelling wave solutions of nonlinear partial differential equations npdes or coupled nonlinear partial differential equations to nonlinear differential difference equations nddes as an illustration two series of exact travelling wave solutions of the discrete sine gordon equation are obtained by means of the exp function method as some special examples these new exact travelling wave solutions can degenerate into the kink type solitary wave solutions reported in the open literature

a meshless method based on point interpolation method pim for the space fractional diffusion equation this paper aims to develop a meshless approach based on the point interpolation method pim for numerical simulation of a space fractional diffusion equation two fully discrete schemes for the one dimensional space fractional diffusion equation are obtained by using the pim and the strong forms of the space diffusion equation numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach

fitted mesh b spline collocation method for singularly perturbed differential difference equations with small delay abstract this paper deals with the singularly perturbed boundary value problem for a linear second order differential difference equation of the convection diffusion type with small delay parameter of o e whose solution has a boundary layer the fitted mesh technique is employed to generate a piecewise uniform mesh condensed in the neighborhood of the boundary layers b spline collocation method is used with fitted mesh parameter uniform convergence analysis of the method is discussed the method is shown to have almost second order parameter uniform convergence the effect of small delay on boundary layer has also been discussed several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution

geometrically nonlinear analysis of space frames by an incremental iterative technique the paper explores the analysis of the geometrically nonlinear behaviour of space structures using the modified arc length method of riks and crisfield several problems not previously documented in the literature are encountered in the solution procedure and means evolved for circumventing the same the resulting algorithm is robust and able to handle problems that exhibit several negative eigenvalues simultaneously several examples are given of the trace of load deflection paths of space frames

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

a symbolic computational method for constructing exact solutions to difference differential equations in this paper we extended tanh method to solve difference differential equations and pure difference equations with the projective riccati equation as an example we applied this method to a 2 1 dimensional toda lattice equation as a result many exact solutions are obtained with the help of symbolic system maple including soliton solutions presented by hyperbolic functions sinh and cosh periodic solutions presented by sin and cos and rational solutions this method can also be used to other nonlinear difference differential equations

on the determination of the path direction for arc length methods in the presence of bifurcations and snap backs abstract this paper discusses the choice of the sign of the incremental load factor in the predictor solution phase of arc length methods it is shown that the criterion recently proposed by y t feng d peric d r j owen determination of travel directions in path following methods mathl and comput modelling 21 1995 43 59 a new criterion for determination of initial loading parameter in arc length methods comput struct 58 1996 479 485 produces a reliable path direction prediction even in the presence of bifurcations and or snap backs when some popular criteria may fail to predict the continuation direction the robustness of the criterion is illustrated in a numerical example where finite element solutions are obtained to a structural stability problem characterised by the existence of several bifurcation paths and snap back phenomena

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

a rbf meshless approach for modeling a fractal mobile immobile transport model a fractional differential equation is used to describe a fractal model of mobile immobile transport with a power law memory function this equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times in this paper we firstly propose a finite difference method to discretize the time variable and obtain a semi discrete scheme then we discuss its stability and convergence secondly we consider a meshless method based on radial basis functions rbfs to discretize the space variable in contrast to conventional fdm and fem the meshless method is demonstrated to have distinct advantages calculations can be performed independent of a mesh it is more accurate and it can be used to solve complex problems finally the convergence order is verified from a numerical example which is presented to describe a fractal model of mobile immobile transport process with different problem domains the numerical results indicate that the present meshless approach is very effective for modeling and simulating fractional differential equations and it has good potential in the development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations

on a space fractional backward diffusion problem and its approximation of local solution abstract this article deals with a backward diffusion problem for an inhomogeneous backward diffusion equation with fractional laplacian in r u t x t u x t f x t x t r 0 t u x t g x x r lim x u x t 0 this problem is an ill posed problem due to the instability in solution the goal of this paper is not only to provide a simple but effective regularization scheme to obtain the holder convergence rate but also to give an approximation of solution of the equation with fractional diffusion to the one of the equation with laplacian in both l 2 r and l p r setting this result holds in particular when f x t is spatially compactly supported in which the difficulties due to the fractional laplacian have been successfully overcome thanks to an additional condition on fourier transform of f we further study the convergence of solution of inhomogeneous problem to that of the homogeneous problem finally numerical simulations with finite difference schemes based on discrete fourier transform dft algorithm are also presented to illustrate the theoretical results

finite element formulation of geometrically exact three dimensional beam theories based on interpolation of strain measures this paper presents a new finite element formulation of the geometrically exact finite strain beam theory the governing equations of the beam element are derived in which the strain vectors are the only unknown functions the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points the solution is found by a collocation algorithm the linearity of the strain space not only simplifies the application of newton s method on the non linear configuration space but also leads to the strain objectivity of the proposed method the accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples c 2003 elsevier b v all rights reserved

variational formulations instabilities and critical loadings of space curved beams abstract beam theories have been extensively studied for applications in structural engineering space curved beams with large displacements however have been explored to a much less extent not to mention explicit solutions concerning instabilities and critical loadings in this paper by carefully accounting for geometric nonlinearity and different scalings of kinematic variables we present a variational framework for large displacement space curved beams we show that the variational formulation is consistent with the classic field equations derive the appropriate boundary value problems for a variety of loading conditions and kinematic constraints and generalize the kirchhoff s helical solutions explicit planar solutions for semi circular arches are obtained upon linearization further two nonlinear asymptotic theories are proposed to address ribbon like and moderately deformed curved beams respectively based on the method of trial solutions we obtain explicit approximate solutions to critical loadings for semi circular arches losing stabilities due to twisting and out of plane displacement the variational framework nonlinear asymptotic theories stability analysis and explicit solutions are anticipated to have novel applications in stretchable electronics and biological macromolecules

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

the linearized three dimensional beam theory of naturally curved and twisted beams the strain vectors formulation this paper presents the equations of the linearized geometrically exact three dimensional beam theory of naturally curved and twisted beams a new finite element formulation for the linearized theory is proposed in which the strain vectors are the only unknown functions the linear form of the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points an arbitrary curved and twisted axis of the beam is taken into account which demands proper consideration of the non linearity of spatial rotations the accuracy and the efficiency of the derived numerical algorithm are demonstrated by comparing present numerical results with various analytical and numerical results

non linear spatial timoshenko beam element with curvature interpolation the paper presents a spatial timoshenko beam element with a total lagrangian formulation the element is based on curvature interpolation that is independent of the rigid body motion of the beam element and simplifies the formulation the section response is derived from plane section kinematics a two node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence the formulation is extended to n node elements with polynomial curvature interpolation models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step generalized second order stress resultants are identified and the section response takes into account non linear material behaviour green lagrange strains are expressed in terms of section curvature and shear distortion whose first and second variations are functions of node displacements and rotations a symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials the comparison of analytical results with numerical simulations in the literature demonstrates the consistency accuracy and superior numerical performance of the proposed element copyright 2001 john wiley sons ltd

an intrinsic beam model based on a helicoidal approximation part i formulation the focus of the present work is directed towards the one dimensional non linear analysis of space curved and twisted beams undergoing large displacements and finite rotations according to cosserat s model a beam is intended here as a continuum generated by the rigid motion of a cross section along a curve the novelty of the proposed methodology is implied in the assumption that the reference line of the beam both in the undeformed and in the deformed configurations is a helicoid in space this formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields in fact within the framework of helicoidal geometry a powerful description of space curved beam kinematics with large displacements and finite rotations can be developed furthermore the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non linear analysis the formulation discussed in this work adheres to a mixed variational principle the independent fields being the generalized displacements and the sectional stress resultants in a companion paper we address the linearization of the governing weak form the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations and the development of a helicoidal model for rigid body dynamics using finite elements in time

on the choice of finite rotation parameters in this work we discuss some aspects of the three dimensional finite rotations pertinent to the formulation and computational treatment of the geometrically exact structural theories among various possibilities to parameterize the finite rotations special attention is dedicated to a choice featuring an incremental rotation vector some computational aspects pertinent to the implementation of the newton iterative scheme and the newmark time stepping algorithm applied to solving these problems are examined representative numerical simulations are presented in order to illustrate the performance of the proposed formulation

stability and convergence of a new explicit finite difference approximation for the variable order nonlinear fractional diffusion equation in this paper we consider the variable order nonlinear fractional diffusion equation u x t t b x t xr a x t u x t f u x t where xr a x t is a generalized riesz fractional derivative of variable order a x t 1 a x t 2 and the nonlinear reaction term f u x t satisfies the lipschitz condition f u 1 x t f u 2 x t

chebyshev collocation method for static intrinsic equations of geometrically exact beams abstract a chebyshev collocation method is presented for the static analysis of the geometrically exact beams undergoing arbitrary large deflections using intrinsic formulation in the intrinsic formulation of the governing equations of the spatial beams neither displacement nor rotation variables appear the proposed collocation technique is based on the chebyshev points as the collocation points and the orthogonal chebyshev polynomials as the trial functions this method is successfully applied to the static solution of the strong form of the intrinsic governing equations of the nonlinear spatial beam and in order to show the applicability accuracy and computational efficiency of the proposed method the numerical results are compared to the experimental analytical and other numerical results for a number of test cases