adaptive approximation of solutions to problems with multiple layers by chebyshev pseudo spectral methods we develop and analyze a family of mappings which enhance the accuracy of chebyshev pseudo spectral methods in approximating functions with multiple regions of localized rapid variation layers the mapping family depends on 3n minus 1 free parameters where n is the number of layers n parameters depend upon the locations of the layers and on the widths of the layers while the other n minus 1 parameters depend on the resolution of each layer relative to the first layer the parameters can be determined adaptively by minimizing a functional which measures the error of the approximation techniques to simplify the minimization process are developed we further demonstrate that the appropriate choice of mappings can lead to a significant reduction in the condition number of matrices associated with chebyshev pseudo spectral differentation we illustrate the effectiveness of the proposed mapping and adaptive procedure by examples in which we approximate i given functions exhibiting multiple layers and ii the solution of a system of partial differential equations modeling combustion in counterflowing jets so that two distinct flames occur 24 refs 8 figs 5 tabs

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

the approximate solutions of some stochastic differential equations using transformations in this paper the transformation technique together with a numerical technique is used to obtain an approximate probability density function pdf for the solution process of some stochastic differential equations the approach is applied on some first order and second order differential equations with making comparisons with the exact solution if possible

adaptive pseudo spectral domain decomposition and the approximation of multiple layers abstract when chebyshev pseudo spectral methods are used with domain decomposition procedures in the numerical solution of partial differential equations the use of multiple domains can significantly affect the accuracy of the approximation this is particularly true when the solution exhibits layer type behavior i e there are narrow regions of rapid variation accuracy may be enhanced if the interfaces between adjacent subdomains are such that large gradients occur near the interfaces while accuracy can be degraded if the rapid variations occur in the interior of the subdomains the use of appropriate mappings within each subdomain can improve the accuracy of the approximation by choosing mappings so that the transformed function is more readily approximated by a low order polynomial the particular choice of mappings however depends critically on whether the solution exhibits boundary layer or interior layer behavior within each subdomain we analyze the relationship between interface location and mappings required to obtain an efficient approximation of such functions we compare two strategies both based on constructing subdomains so that each subdomain contains only one layer in the first strategy interface locations are chosen so that the rapid variations occur as interior layers and mappings are employed which enhance the resolution of such layers strategy i for interior in the second strategy interface locations are chosen so that rapid variations occur as boundary layers and mappings are employed which enhance resolution of boundary layers strategy b for boundary both strategies lead to adaptive domain decomposition procedures based entirely on the locations of the layers we demonstrate that strategy b offers superior accuracy for a given computational effort and employ this strategy in developing an adaptive domain decomposition method for problems with multiple layers both strategies are comparable regarding spectral radii of the resulting matrices and we conclude that domain decomposition itself cannot result in larger stable time steps when accuracy of the approximation of the layer is considered the adaptive domain decomposition method is illustrated by the computation of both axisymmetric and cellular flames with a sequential reaction mechanism involving two reaction zones

a spectral collocation method for two dimensional compressible convection abstract a fourier chebyshev spectral simulation of two dimensional compressible convection is presented the fluid is a perfect gas with constant dynamic viscosity and thermal conductivity both slippery and rigid boundary conditions for the velocity are used here the temperature is maintained fixed at the upper and lower boundaries an explicit adams bashforth predictor corrector numerical scheme is used in order to overcome the courant friedriechs lewy condition the nonlinear diffusion terms are handled by an iterative method with finite differences or spectral preconditioning steady state solutions have been obtained for both types of boundary conditions critical exponents are found to be the same as in the incompressible case at least for a weak value of the stratification parameter

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

second order linear differential equations with analytic uncertainties stochastic analysis via the computation of the probability density function abstract this paper concerns the analysis of random second order linear differential equations usually solving these equations consists of computing the first statistics of the response process and that task has been an essential goal in the literature a more ambitious objective is the computation of the solution probability density function we present advances on these two aspects in the case of general random non autonomous second order linear differential equations with analytic data processes the frobenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series we demonstrate that the convergence requires the boundedness of the random input coefficients further the mean square error of the frobenius method is proved to decrease exponentially with the number of terms in the series although not uniformly in time regarding the probability density function of the solution at a given time which is the focus of the paper we rely on the law of total probability to express it in closed form as an expectation for the computation of this expectation a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set we prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation the pointwise convergence turns out to be exponential under a lipschitz hypothesis as the density functions are expressed in terms of expectations we propose a crude monte carlo sampling algorithm for their estimation this algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper after that the efficiency of the algorithm is improved by employing the control variates method numerical examples corroborate the variance reduction of the monte carlo approach

accuracy resolution and stability properties of a modified chebyshev method while the chebyshev pseudospectral method provides a spectrally accurate method integration of partial differential equations with spatial derivatives of order m requires time steps of approximately o n 2m for stable explicit solvers theoretically time steps may be increased to o n m with the use of a parameter alpha dependent mapped method introduced by kosloff and tal ezer em j comput phys 104 1993 pp 457 469 our analysis focuses on the utilization of this method for reasonable practical choices for n namely n lesssim 30 as may be needed for two or three dimensional modeling results presented confirm that spectral accuracy with increasing n is possible both for constant alpha hesthaven dinesen and lynov em j comput phys 155 1999 pp 287 306 and for alpha scaled with n alpha sufficiently different from 1 don and solomonoff em siam j sci comput 18 1997 pp 1040 1055 theoretical bounds however show that any realistic choice for alpha in which both resolution and accuracy considerations are imposed permits no more than a doubling of the time step for a stable explicit integrator in time much less than the o n improvement claimed by kosloff and tal ezer on the other hand by choosing alpha carefully it is possible to improve on the resolution of the chebyshev method in particular one may achieve satisfactory resolution with fewer than pi points per wavelength moreover this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one dimensional wave equations specifically while alpha can be chosen such that the mapped method improves the accuracy and resolution of the chebyshev method for practical choices of n it is not possible to achieve both single precision accuracy and gain the advantage of an o n m time step

an adaptive multidomain chebyshev method for nonlinear eigenvalue problems application to self similar solutions of gas dynamics equations with nonlinear heat conduction an adaptive multidomain chebyshev collocation method for handling nonlinear eigenvalue problems such as those arising in the computation of self similar solutions is presented the algorithm is made of two stages the second one using an iterative method the method is applied to the time dependent one dimensional self similar solutions of the euler equations with nonlinear heat conduction in which a thermal front follows a shock wave it leads to highly accurate solutions for solutions stiffness ranging from 1 to 10 7 the method is quite general and may be applied to a large class of problems

a comparative study of the numerical approximation of the random airy differential equation the aim of this paper is twofold first we deal with the extension to the random framework of the piecewise frobenius method to solve airy differential equations this extension is based on mean square stochastic calculus second we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario this includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and monte carlo simulations to conduct this comparative study we have chosen the airy random differential equation because it has highly oscillatory solutions this feature allows us to emphasize differences between all the considered approaches

boundary layer resolving pseudospectral methods for singular perturbation problems pseudospectral methods are investigated for singularly perturbed boundary value problems for ordinary differential equations odes which possess boundary layers it is well known that if the boundary layer is very small then a very large number of spectral collocation points is required to obtain accurate solutions we introduce here a new effective procedure based on coordinate stretching and the chebyshev pseudospectral method to resolve the boundary layers stable and accurate results are obtained for very thin boundary layers with a fairly small number of spectral collocation points

random differential operational calculus theory and applications in this article we obtain a product rule and a chain rule for mean square derivatives an application of the chain rule to the mean square solution of random differential equations is shown however to achieve such mean square differentiation rules fourth order properties were needed and therefore we first studied a mean fourth order differential and integral calculus results are applied to solve random linear variable coefficient differential problems

heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems a heterogeneous domain decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems the method which is parallelizable at various levels uses several ideas of asymptotic analysis the subdomains match the domains of validity of the local inner and outer asymptotic expansions and cut off functions are used to match solutions in neighboring subdomains the positions of the interfaces as well as the mesh widths depend on the small parameter varepsilon on the subdomains iterative solution techniques are used which may vary from one subdomain to another the global convergence rate depends on varepsilon it generally increases like some power of log varepsilon 1 1 as varepsilon downarrow 0 the method is illustrated on several two dimensional singular perturbation problems

network simplex algorithm for the general equal flow problem abstract in this paper the general equal flow problem is considered this is a minimum cost network flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the same value this model can be applied to approach water resource system management problems or multiperiod logistic problems in general involving policy restrictions which require some arcs to carry the same amount of flow through the given study period although the bases of the general equal flow problem are no longer spanning trees it is possible to recognize a similar structure that allows us to take advantage of the practical computational capabilities of network models after characterizing the bases of the problem as good r 1 forests a simplex primal algorithm is developed that exploits the network structure of the problem and requires only slight modifications of the well known network simplex algorithm

a proposed technique of sfem on solving ordinary random differential equation the stochastic finite element sfem is a new method to solve random differential equation using the known deterministic finite element technique fem adapted to stochastic techniques for solving stochastic differential equations in this paper a proposed technique of fem and random variable transformation rvt is used to solve a differential equation with random excitation the technique shows high accuracy when solving a case study compared with the exact solution

random fractional generalized airy differential equations a probabilistic analysis using mean square calculus abstract the aim of this paper is to study a generalization of fractional airy differential equations whose input data coefficient and initial conditions are random variables under appropriate hypotheses assumed upon the input data we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense afterwards we provide reliable explicit approximations for the main statistical information of the solution process mean variance and covariance further we show a set of numerical examples where our obtained theory is illustrated more precisely we show that our results for the random fractional airy equation are in full agreement with the corresponding to classical random airy differential equation available in the extant literature finally we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional airy differential equation by combining the knowledge of the mean and the variance and the principle of maximum entropy

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

the equal flow problem abstract this paper presents a new algorithm for the solution of a network problem with equal flow side constraints the solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible the proposed algorithm makes use of lagrangean relaxation to obtain a lower bound and decomposition by right hand side allocation to obtain upper bounds the lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound but also allows an initial allocation of equal flows for the upper bound the algorithm has been tested on problems with up to 1500 nodes and 6000 arcs computational experience indicates that solutions whose objective function value is well within 1 of the optimum can be obtained in 1 65 of the mpsx time depending on the amount of imbalance inherent in the problem incumbent integer solutions which are within 99 99 feasible and well within 1 of the proven lower bound are obtained in a straightforward manner requiring on the average 30 of the mpsx time required to obtain a linear optimum

adaptive pseudospectral solution of a diffuse interface model a diffuse interface type model using an energy based variational formulation with a free energy that is a function of the density and its gradients is presented all of the boundary terms are retained and related to external surface forces which can be of particular interest when considering the fluid fluid solid region the numerical solution of these types of problems can be troublesome if a thin transition layer is desired here chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied a mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived in order to achieve high precision for problems exhibiting thin transition layers a modified version of the arc length monitor function is proposed which yields a sufficiently smooth coordinate transformation at every step of the continuation process a fixed number of iterations is implemented so that the equidistribution equations are not solved completely at each step which saves a considerable amount of computational effort numerical results for the static phase field model exhibiting thin transition layers are presented

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

the efficient generation of simple two dimensional adaptive grids we present an efficient adaptive procedure for solving steady partial differential equations based on the simple two dimensional quadrilateral grid generator of huang and sloan siam j sci comput 15 1994 pp 776 797 the mesh equations are derived using an equidistribution principle and are discretized in the physical domain this allows the application of the adaptive grid generator to finite element and finite volume methods to efficiently solve the nonlinear algebraic equations describing the adapted grid we propose an alternating line gauss seidel relaxation procedure the beneficial effect of adaptive grids is demonstrated by coupling the grid generator to finite volume and finite difference approximations of advection and advection diffusion test problems for some examples the proposed method is shown to be 50 times more efficient than previously used solution procedures

algorithms for the simple equal flow problem the minimum cost flow problem is to determine a least cost shipment of a commodity through a network g n a in order to satisfy demands at certain nodes from available supplies at other nodes in this paper we study a variant of the minimum cost flow problem where we are given a set math not displayed of arcs and require that each arc in r must carry the same amount of flow this problem which we call the simple equal flow problem arose while modeling a water resource system management in sardinia italy we consider the simple equal flow problem in a directed network with n nodes m arcs and where all arc capacities and node supplies are integer and bounded by u we develop several algorithms for the simple equal flow problem the network simplex algorithm the parametric simplex algorithm the combinatorial parametric algorithm the binary search algorithm and the capacity scaling algorithm the binary search algorithm solves the simple equal flow problem in o log nu applications of any minimum cost flow algorithm the capacity scaling algorithm solves it in o m m n log n log nu time which is almost the same time needed to solve the minimum cost flow problem by the capacity scaling algorithm these algorithms can be easily modified to obtain an integer solution of the simple equal flow problem

solving second order linear differential equations with random analytic coefficients about ordinary points a full probabilistic solution by the first probability density function abstract this paper deals with the approximate computation of the first probability density function of the solution stochastic process to second order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so called random variable transformation technique the validity of the proposed method is shown through several illustrative examples

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

a new parallel solver for the nonperiodic incompressible navier stokes equations with a fourier method we present a specific use of domain decomposition and decomposition in function space combined with asymptotic analytical qualitative results to obtain on parallel computers efficient and accurate solvers 3 for rapidly varying quasi planar unsteady combustion fronts in liquids in particular we give anew parallel direct solverof the unsteady incompressible navier stokes equations in the stream function formulation this solver is based on an embedding technique that allows us to generalize our previous results from the case with periodic boundary conditions 6 7 to thenonperiodiccase with wall boundary conditions in a direction perpendicular to front propagation the solution is decomposed into a particular solution suitable for a fourier method and the general homogeneous solution calculated from an analytic solution with high precision to satisfy the boundary conditions the algorithm is implemented for parallel computers and results in a very effective code results on the effect of the convection onto the front propagation are provided

solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post processing a coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions the work is an extension of a method presented by mulholland huang and sloan for the adaptive pseudospectral solution of steady problems a coarse grid is generated by a moving mesh finite difference method that is based on equidistribution and this grid is used to construct a time dependent coordinate transformation a sequence of spatial transformations may be generated at discrete points in time or a single transformation may be generated as a continuous function of space and time the differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time numerical results are presented for unsteady problems in one space dimension

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

accuracy and speed in computing the chebyshev collocation derivative we study several algorithms for computing the chebyshev spectral derivative and compare their roundoff error for a large number of collocation points the elements of the chebyshev differentiation matrix if constructed in the usual way are not computed accurately a subtle cause is found to account for the poor accuracy when computing the derivative by the matrix vector multiplication method methods for accurately computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately the roundoff error of the matrix vector multiplication is as small as that of the transform recursion algorithm furthermore results of the cpu time usage are shown for several different algorithms for computing the derivative by the chebyshev collocation method for a wide variety of two dimensional grid sizes on both an ibm mainframe and a cray 2 computer we find that which algorithm is fastest on a particular machine depends not only on the grid size but also on small

polynomial time decision algorithms for probabilistic automata deciding in an efficient way weak probabilistic bisimulation in the context of probabilistic automata is an open problem for about a decade in this work we close this problem by proposing a procedure that checks in polynomial time the existence of a weak combined transition satisfying the step condition of the bisimulation this enables us to arrive at a polynomial time algorithm for deciding weak probabilistic bisimulation and also branching probabilistic bisimulation we furthermore present several extensions to interesting related problems in particular weak and branching probabilistic simulation setting the ground for the development of more effective and compositional analysis algorithms for probabilistic systems

random analytic solution of coupled differential models with uncertain initial condition and source term this paper deals with the construction of random power series solution of vector initial value problems containing uncertainty in both initial condition and source term under appropriate hypothesis on the data we prove that the random series solution constructed by a random frobenius method is convergent in the mean square sense also the main statistical functions of the approximating stochastic process solution generated by truncation of the exact series solution are given finally we apply the proposed technique to several illustrative examples

solving the random legendre differential equation mean square power series solution and its statistical functions in this paper we construct by means of random power series the solution of second order linear differential equations of legendre type containing uncertainty through its coefficients and initial conditions by assuming appropriate hypotheses on the data we prove that the constructed random power series solution is mean square convergent in addition the main statistical functions of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed method to some illustrative examples to compare the numerical results for the average and the variance with respect to those obtained by the monte carlo approach

analytic numerical approximating processes of diffusion equation with data uncertainty this paper deals with the construction of approximating processes for solving mixed problems related to the diffusion equation where the initial condition is assumed to be a stochastic process and the diffusion coefficient is a random variable expectation and standard deviation of the approximation processes are given

a dynamical pseudo spectral domain decomposition technique a dynamical spectral domain decomposition method is presented in each subdomain a transformation of coordinate is used both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theh 2 norm of the calculated solution we show on some functions that the total norm of the chebyshev series depends on the location of the interfaces moreover there exists a minimum that defines the best location of the interface this defines a dynamical generation of chebyshev collocation points this numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation this algorithm is then used for the numerical solution of the time dependent full navier stokes equations the solution technique consists in a fourier chebyshev collocation method combined with a matching method the computational domain is decomposed into subdomains in the vertical direction in each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined the elliptic problems coming from the viscous terms are solved by means of the chebyshev acceleration method density is matched with an upwind procedure whereas the velocities the temperature and the concentration are handled with the influence matrix method numerical examples are carried out on the compressible kelvin helmholtz and rayleigh taylor flows

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

on the solution of convection diffusion boundary value problems using equidistributed grids the effect of using grid adaptation on the numerical solution of model convection diffusion equations with a conservation form is studied the grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc length of the solution in particular a parameter dependent monitor function is introduced which incorporates fixed meshes approximate arc length equidistribution and equidistribution of the absolute value of the solution in a single framework thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values a class of singularly perturbed problems including burgers s equation in the limit of small viscosity is studied singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations and numerical results are compared with the results from the analysis computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory it is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that furthermore the spurious solutions can be stable for an appropriate moving mesh method

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

random airy type differential equations mean square exact and numerical solutions this paper deals with the construction of power series solutions of random airy type differential equations containing uncertainty through the coefficients as well as the initial conditions under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent over the whole real line in addition the main statistical functions such as the mean and the variance of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples which show substantial speed up and improvement in accuracy compared to other approaches such as monte carlo simulations