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

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

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

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 divide and conquer fast finite difference method for spacetime fractional partial differential equation fractional partial differential equations fpdes provide better modeling capabilities for challenging phenomena with long range time memory and spatial interaction than integer order pdes do a conventional numerical discretization of spacetime fpdes requires o n2 mn memory and o mn3 m2n computational work where n is the number of spatial freedoms per time step and m is the number of time steps we develop a fast finite difference method fdm for spacetime fpde i we utilize the toeplitz like structure of the coefficient matrix to develop a matrix free preconditioned fast krylov subspace iterative solver to invert the coefficient matrix at each time step ii we utilize a divide and conquer strategy a recursive direct solver to handle the temporal coupling of the numerical scheme the fast method has an optimal memory requirement of o mn and an approximately linear computational complexity of o nm logn log2m without resorting to any lossy compression numerical experiments show the utility of the method

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

variational image registration by a total fractional order variation model in this paper a new framework of nonlocal deformation in non rigid image registration is presented it is well known that many non rigid image registration techniques may lead to unsteady deformation e g not one to one if the dissimilarity between the reference and template images is too large we present a novel variational framework of the total fractional order variation to derive the underlying fractional euler lagrange equations and a numerical implementation combining the semi implicit update and conjugate gradients cg solution to solve the nonlinear systems numerical experiments show that the new registration not only produces accurate and smooth solutions but also allows for a large rigid alignment the evaluations of the new model demonstrate substantial improvements in accuracy and robustness over the conventional image registration approaches

multigrid methods for space fractional partial differential equations we propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations sfpdes it is shown that our multigrid methods are optimal which means the convergence rates of the methods are independent of the mesh size and mesh level moreover our theoretical analysis and convergence results do not require regularity assumptions of the model problems numerical results are given to support our theoretical findings

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

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

fast upwind and eulerian lagrangian control volume schemes for time dependent directional space fractional advection dispersion equations abstract we develop control volume methods for two dimensional time dependent advection dominated directional space fractional advection dispersion equations with the directional space fractional derivative weighted in all the directions by a probability measure in the unit circle which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media we develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space centered numerical discretizations of advection term which are relatively straightforward to implement we also develop a eulerian lagrangian control volume method for the governing equation which symmetrizes the governing equation by combining the time derivative term and the advection term into a material derivative term along characteristic curves both methods are locally mass conservative which are essential in these applications due to the nonlocal nature of the directional space fractional differential operators corresponding numerical discretizations usually generate full stiffness matrices conventional direct solvers tend to require o n 2 memory requirement and have o n 3 computational complexity per time step where n is the number of spatial unknowns which is computationally significantly more expensive than the numerical approximations of integer order advection diffusion equations based on the analysis of the structure of stiffness matrix we propose a fast krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and eulerian lagrangian control volume methods which reduce computational complexity from o n 3 by a direct solver to o n log n per krylov subspace iteration per time step and a memory requirement from o n 2 to o n numerical results are presented to show the utility of the methods

a preconditioned numerical solver for stiff nonlinear reaction diffusion equations with fractional laplacians that avoids dense matrices the numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators the dense coefficient matrices that arise from spatial discretisation of these operators mean that even one dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more in this work we address this issue of efficiency for one dimensional nonlinear space fractional reaction diffusion equations with fractional laplacian operators we apply variable order variable stepsize backward differentiation formulas in a jacobian free newton krylov framework to advance the solution in time a key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional laplacian operator we show how a banded approximation to this matrix which can be formed and factorised efficiently can be used as part of an effective preconditioner that accelerates convergence of the krylov subspace iterative solver our approach also captures the full contribution from the nonlinear reaction term in the preconditioner which is crucial for problems that exhibit stiff reactions numerical examples are presented to illustrate the overall effectiveness of the solver

periodic review stochastic inventory problem with forecast updates worst case bounds for the myopic solution abstract muth first considered the linear cost periodic review inventory problem in which the mean demand in a period undergoes a non observed random walk assuming the random walk variance and the within period demand variance to be known and stationary he showed that the best linear unbiased estimate blue for the mean is given by exponential smoothing and derived the formula for the optimal steady state smoothing constant in terms of the variances we first show that the corresponding bayesian analysis is useful under transient conditions and converges to the muth results under steady state for the steady state solution we prove that the myopic policy is near optimal using the concepts of p myopic and d myopic introduced in this paper in the sense that the worst case bounds on policy errors in using it may be given analytically and are small for reasonable values of parameters as further validation we use dynamic programming dp to compute optimal policies and compare them with myopic policies policy errors are very small by considering analogous results in recent literature it is conjectured that additions of such factors as non stationarity lead times perishability setup costs would produce near myopic results

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

a computationally effective alternating direction method for the space and time fractional bloch torrey equation in 3 d the space and time fractional bloch torrey equation st fbte has been used to study anomalous diffusion in the human brain numerical methods for solving st fbte in three dimensions are computationally demanding in this paper we propose a computationally effective fractional alternating direction method fadm to overcome this problem we consider st fbte on a finite domain where the time and space derivatives are replaced by the caputo djrbashian and the sequential riesz fractional derivatives respectively the stability and convergence properties of the fadm are discussed finally some numerical results for st fbte are given to confirm our theoretical findings

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

a finite element method for space time directional fractional diffusion partial differential equations in the plane and its error analysis abstract we present and analyze a fast finite element method for space time fractional directional partial differential equations in a bounded domain in the plane the fast solver significantly reduces the computational work of solving the discrete linear algebraic systems from o m n 3 m 2 n by a direct solver to o m n log m n per krylov subspace iteration and a memory requirement from o m n 2 to o n log m an error estimate is proved numerical results are presented to show the utility of the method

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

a fast preconditioned iterative method for two dimensional options pricing under fractional differential models abstract in recent years fractional partial differential equation fpde has been widely applied in options pricing problems which better explains many important empirical facts of financial markets however the vast majority of the literatures focus on pricing the single asset option under the fpde framework in this paper a two dimensional fpde governing the valuation of rainbow options is established where two underlying assets are assumed to follow independent exponential levy processes and its boundary conditions are determined by solving one dimensional fpdes a second order accurate finite difference scheme is proposed to discretize the two dimensional fpde given the block toeplitz with toeplitz block structure of the coefficient matrix a fast preconditioned krylov subspace method is developed for solving the resulting linear system with o n log n computational complexity per iteration where n is the matrix size the proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

fast numerical solution for fractional diffusion equations by exponential quadrature rule after spatial discretization to the fractional diffusion equation by the shifted grunwald formula it leads to a system of ordinary differential equations where the resulting coefficient matrix possesses the toeplitz like structure an exponential quadrature rule is employed to solve such a system of ordinary differential equations the convergence by the proposed method is theoretically studied in practical computation the product of a toeplitz like matrix exponential and a vector is calculated by the shift invert arnoldi method meanwhile the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift invert arnoldi method numerical results are given to demonstrate the efficiency of the proposed method the matrix of the spatial discretization of the fractional diffusion equation is toeplitz like an exponential quadrature rule is employed to solve the system of ordinary differential equations the toeplitz like matrix exponential is calculated by the shift invert arnoldi method the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift invert arnoldi method numerical results show the efficiency of the exponential quadrature rule

domain decomposition methods for space fractional partial differential equations abstract a two level additive schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations sfpdes it is shown that the condition number of the preconditioned system is bounded by c 1 h where h is the maximum diameter of subdomains and is the overlap size among the subdomains numerical results are given to support our theoretical findings

fractional integral associated to generalized cookie cutter set and its physical interpretation this paper is based on nigmatullin s study when the residual memory set is a generalized cookie cutter set on using various hypotheses it is proved that the fractional exponent of a fractional integral is not uniquely determined by the fractal dimension of the generalized cookie cutter set it is determined by of self similar measure or infinite self similar measure on this generalized cookie cutter set and can run over all positive real numbers

finite difference approximations for a fractional advection diffusion problem the use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed fractional space derivatives are used to model anomalous diffusion or dispersion where a particle plume spreads at a rate inconsistent with the classical brownian motion model when a fractional derivative replaces the second derivative in a diffusion or dispersion model it leads to enhanced diffusion also called superdiffusion we consider a one dimensional advection diffusion model where the usual second order derivative gives place to a fractional derivative of order a with 1 a 2 we derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection diffusion equation we present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions in the end we present a test problem

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