fast and stable explicit operator splitting methods for phase field models numerical simulations of phase field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods in this paper we propose fast and stable explicit operator splitting methods for both one and two dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and the cahn hilliard equation the equations are split into nonlinear and linear parts the nonlinear part is solved using a method of lines together with an efficient large stability domain explicit ode solver the linear part is solved by a pseudo spectral method which is based on the exact solution and thus has no stability restriction on the time step size we demonstrate the performance of the proposed methods on a number of one and two dimensional numerical examples where different stages of coarsening such as the initial preparation alternating rapid structural transition and slow motion can be clearly observed

on a backward parabolic problem with local lipschitz source abstract we consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t a u t f u t t u 1 where a is a positive self adjoint unbounded operator and f is a local lipschitz function as known it is ill posed and occurs in applied mathematics e g in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology a new version of quasi reversibility method is described we show that the regularized problem with a regularization parameter 0 is well posed and that its solution u t converges on 0 1 to the exact solution u t as 0 these results extend some earlier works on the nonlinear backward problem

energy stable linear schemes for polymer solvent phase field models abstract we present new linear energy stable numerical schemes for numerical simulation of complex polymer solvent mixtures the mathematical model proposed by zhou et al 2006 consists of the cahn hilliard equation which describes dynamics of the interface that separates polymer and solvent and the oldroyd b equations for the hydrodynamics of polymeric mixtures the model is thermodynamically consistent and dissipates free energy our main goal in this paper is to derive numerical schemes for the polymer solvent mixture model that are energy dissipative and efficient in time to this end we will propose several problem suited time discretizations yielding linear schemes and discuss their properties

interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference information systems which contain only crisp data precise and unique attribute values for all objects have been widely investigated due to the fact that in realworld applications imprecise data are abundant uncertainty is inherent in real information systems in this paper information systems are called fuzzy information systems and formalized by objects attributes f in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values to interpret and extract fuzzy decision rules from fuzzy information systems the meta theory based on modal logic proposed by resconi et al is modified the modified meta theory not only expresses uncertainty between objects and their attributes but also uncertainty in the process of recognizing fuzzy information systems in addition according to perception computing proposed by zadeh granules of fuzzy information systems can be represented by fuzzy decision rules so that fuzzy inference methods can be used to obtain the decision attribute of a new object finally a novel way of combining evidences based on the modified meta theory is introduced which extends the concept of combining evidences based on dempster shafer theory

a nonlinear parabolic equation backward in time regularization with new error estimates abstract consider a nonlinear backward parabolic problem in the form u t a u t f t u t u t g where a is a positive self adjoint unbounded operator based on the fundamental solution to the parabolic equation we propose to solve this problem by the fourier truncated method which generates a well posed integral equation then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven our regularizing scheme can be considered a new regularization with the advantage of a relatively small amount of computation compared with the quasi reversibility or quasi boundary value regularizations error estimates for this method are provided together with a selection rule for the regularization parameter these errors show that our method works effectively

diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix a model for phase separation in multicomponent systems is studied in particular the possibility of a concentration dependence of the mobility matrix is taken into account this leads to a system of fourth order degenerate parabolic partial differential equations we derive these equations from balance laws show some properties of the model and prove a global existence result for the degenerate system

a modified quasi boundary value method for ill posed problems in this paper we study a final value problem for first order abstract differential equation with positive self adjoint unbounded operator coefficient this problem is ill posed perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter we show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution we also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions finally we give explicit convergence rates

stability of solutions of a class of nonlinear fractional laplacian parabolic problems abstract in this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional laplace operator with nonlinear source function the equations are generalized from lots of well known equations such as the burger equation the cahn hilliard equation herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem and then we establish a regularization method some numerical experiments are performed to verify efficiency and accuracy of our method

efficient numerical solution of cahn hilliard navier stokes fluids in 2d a finite element discretization of the variable density cahn hilliard navier stokes system is presented this system is then decoupled with the cahn hilliard part solved via the nonlinear multigrid method proposed by the authors in j comput phys 212 2006 pp 288 304 and the navier stokes part via a multigrid preconditioned gmres scheme this new solver is examined numerically and is shown to be efficient with respect to mesh refinement and robust with respect to the problem parameters

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

regularization of an inverse nonlinear parabolic problem with time dependent coefficient and locally lipschitz source term abstract we consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form u t a t a u t f t u t subject to the final condition u t here a is a positive self adjoint unbounded operator in a hilbert space h and f satisfies a locally lipschitz condition this problem is ill posed using quasi reversibility method we shall construct a regularized solution u e from the measured data a e and e we show that the regularized problems are well posed and that their solutions converge to the exact solutions error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts

logarithmic well posed approximation of the backward heat equation in banach space abstract in this paper we apply a logarithmic approximation introduced by boussetila and rebbani in order to study an ill posed cauchy problem in banach space the proposed method induces a well posed problem whose approximation parameter yields continuous dependence on modeling throughout properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in l p r 1 p

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

ill posed problems associated with sectorial operators in banach space abstract ill posed problems of the form u t a u h 0 t t u 0 are studied in banach space under two related scenarios both of which consider a well posed logarithmic approximation in the first scenario a generates a bounded holomorphic semigroup of angle 2 and so a is sectorial of angle 0 in the second where may be relaxed to any value in 0 2 we impose an additional assumption that e a is sectorial regularization results for the original ill posed problem are then established in the linear case where h h t and subsequently extended to nonlinear problems where h is replaced by h t u t we compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations both linear and nonlinear in l p r n 1 p

an approximate solution for nonlinear backward parabolic equations we consider the backward parabolic equation ut au f t u t 0 t t u t g where a is a positive unbounded operator and f is a nonlinear function satisfying a lipschitz condition with an approximate datum g the problem is severely ill posed using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces according to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type for all t 0 t an example on heat equations and numerical experiments are given

regularization for ill posed cauchy problems associated with generators of analytic semigroups this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator a in a banach space our main result is that if a is the generator of an analytic semigroup then there exists a family of regularizing operators for such an ill posed cauchy problem by using the quasi reversibility method fractional powers and semigroups of linear operators the applications to ill posed partial differential equations are also given

continuous dependence on modeling for nonlinear ill posed problems abstract the nonlinear ill posed cauchy problem d u d t a u t h t u t u 0 where a is a positive self adjoint operator on a hilbert space h h and h 0 t h h is a uniformly lipschitz function is studied in order to establish continuous dependence results for solutions to approximate well posed problems the authors show here that solutions of the problem if they exist depend continuously on solutions to corresponding approximate well posed problems if certain stabilizing conditions are imposed the approximate problem is given by d v d t f a v t h t v t v 0 for suitable functions f the main result is that u t v t c 1 t t m t t where c and m are computable constants independent of and 0 1 this work extends to the nonlinear case earlier results by the authors and by ames and hughes

regularization of a backward problem for a lotka volterra competition system abstract lotka volterra systems are used to describe the dynamics of biological systems we study the backward problem for the lotka volterra system to determine the population density of species at preceding times the problem is ill posed in the sense that if the solution exists it does not depend continuously on the given data we propose two stable regularization methods to regularize the system furthermore under some a priori assumptions on the sought solution we show that the corresponding regularized solutions converge to the exact solutions in l 2 norm numerical simulations are presented to demonstrate the theoretical results

a new approach to inference in approximate reasoning abstract we introduce a new approach to inference in approximate reasoning based on truth value restriction the degree to which the actual given value a of a variable x agrees with the antecedent value b in a production if x is b then y is c is represented as a fuzzy subset of a truth space we introduce a new form of implication based on the exponential operation a new compatibility relation based on the input value a and the antecedent value b is defined a fuzzy truth value true derived from the antecedent value b is also defined it is shown that by mapping this compatibility relation into the truth space a fuzzy truth value is generated which produces an exponential family of the fuzzy truth value true we show that the inferred consequent can be generated from this truth value theoretical results showing the validity of this method to 1 chain rules consisting of several single fuzzy conditional propositions and 2 conjunctive fuzzy conditional propositions are also presented theoretical and simulation results demonstrate that our method is superior to the existing methods when intuitively correct responses are required