implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

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

asymptotic periodicity in diffusive logistic equations with discrete delays has been the object of intensive analysis by numerous authors l 6 as a model of single species population growth it is known that time delay has a tendency to produce oscillations 2 5 61 in 1 which is nonoscillatory when r 0 in an autonomous delay system modeling two competing species time delays can even lead to chaotic behavior 4 if the diffusion is taken into account in the growth rate as well then time delay may cause random and blow up effects in the parabolic logistic delay equation 7 nonlinear periodic diffusion equations arise naturally in population models s 12 where the birth and death rates rates of diffusion rates of interaction and environmental carrying capacities are periodic on a seasonal scale in this paper we study the following periodicparabolic logistic equation with instantaneous and delay effects duct x 3t au t x u t x a t x m x u t x

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

on the balancing principle for some problems of numerical analysis we discuss a choice of weight in penalization methods the motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution one example of such improvement is a regularization where a penalization substitutes an ill posed problem for a well posed one in modern numerical methods for pdes a penalization is used for example to enforce a continuity of an approximate solution on non matching grids a choice of penalty weight should provide a balance between error components related with convergence and stability which are usually unknown in this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components it is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates finally we successfully apply our strategy for self regularization of volterra type severely ill posed problems such as the sideways heat equation and for the choice of a weight in interior penalty discontinuous approximation on non matching grids numerical experiments on a series of model problems support theoretical results

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

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

time fractional ihcp with caputo fractional derivatives the numerical solution of the time fractional inverse heat conduction problem tfihcp on a finite slab is investigated in the presence of measured noisy data when the time fractional derivative is interpreted in the sense of caputo a finite difference space marching scheme with adaptive regularization using mollification techniques is introduced error estimates are derived for the numerical solution of the mollified problem and several numerical examples of interest are provided

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

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

some analytical solutions for the advection dispersion equation in this paper the parallel plate and cylindrical geometry are considered to model contaminant transport in a main fracture surrounded by a two dimensional rock matrix the transformation group theoretical approach is applied to present some new analytical solutions for the advection dispersion equation governing these models the application of this approach reduces the number of independent variables and consequently the governing equation is reduced to ordinary differential equations the general analysis developed in this study corresponds to the case of parabolic manner of the velocity within the main fracture

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

numerical methods for nonlinear integro parabolic equations of fredholm type abstract this paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction diffusion convection equations under either linear or nonlinear boundary conditions the discrete approximation of the problem is based on the finite difference method and the computation of the finite difference solution is by the method of upper and lower solutions three types of quasi monotone reaction functions are considered and for each type a monotone iterative scheme is obtained each of these iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system this monotone convergence leads to an existence uniqueness theorem as well as a computational algorithm for the computation of the solution an error estimate between the computed approximations and the true finite difference solution is obtained for each iterative scheme these error estimates are given in terms of the strength of the reaction function and the effect of diffusion convection and are independent of the true solution applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm

an inverse sturm liouville problem with a fractional derivative in this paper we numerically investigate an inverse problem of recovering the potential term in a fractional sturm liouville problem from one spectrum the qualitative behaviors of the eigenvalues and eigenfunctions are discussed and numerical reconstructions of the potential with a newton method from finite spectral data are presented surprisingly it allows very satisfactory reconstructions for both smooth and discontinuous potentials provided that the order a 1 2 of fractional derivative is sufficiently away from 2

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

on the adaptive selection of the parameter in regularization of ill posed problems we study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter as a result a rather general form of a selection criterion is proposed and its relation to the heuristical quasi optimality principle of tikhonov and glasko z vychisl mat mat fiz 4 1964 pp 564 571 and to an adaptation scheme proposed in a statistical context by lepskii theory probab appl 36 1990 pp 454 466 is discussed the advantages of the proposed criterion are illustrated by using such examples as self regularization of the trapezoidal rule for noisy abel type integral equations lavrentiev regularization for nonlinear ill posed problems and an inverse problem of the two dimensional profile reconstruction

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

asymptotic stability for an integrodifferential reaction diffusion system abstract a reaction diffusion system involving an integrodifferential linear parabolic equation and a nonlinear ordinary differential equation is studied under third type boundary conditions such a system may be used to describe the evolution of a class of man environment epidemics conditions are given for the existence and the asymptotic stability of a unique nontrivial equilibrium solution by using monotone iteration techniques

new advantages to obtain accurate matrix inversion this paper presents a refinement algorithm for improving the inverse of a given nonsingular matrix here we also improve the previously presented results by setting an efficient refinement algorithm we finally presents the performances of the algorithm supporting by the introduced refitment

a modified method for a non standard inverse heat conduction problem abstract a non standard inverse heat conduction problem is considered data are given along the line x 1 and the solution at x 0 is sought the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in order to solve the problem numerically it is necessary to employ some regularization method in this paper we study a modification of the equation where a fourth order mixed derivative term is added error estimates for this equation are given which show that the solution of the modified equation is an approximation of the heat equation a numerical implementation is considered and a simple example is given some numerical results show the usefulness of the modified method

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

numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data error estimates are derived for the discretization in space by finite elements and in time by the backward euler method of a parabolic equation with a semilinear memory term special attention is paid to the case of nonsmooth data and the computation of the memory term by quadrature

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

numerical analysis of coupled systems of nonlinear parabolic equations this paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method three monotone iteration processes for the finite difference system are presented and the sequences of iterations are shown to converge monotonically to a unique solution of the system including an existence uniqueness comparison theorem a theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary value problem an application to a reaction diffusion model in chemical engineering and combustion theory is given