a simple proof of existence of weak and statistical solutions of navier stokes equations the galerkin approximation to the navier stokes equations in dimension n where n is an infinite non standard natural number is shown to have standard part that is a weak solution this construction is uniform with respect to non standard representation of the initial data and provides easy existence proofs for statistical solutions

a method for solving differential equations of q fractional order abstract in this work we considered the q differential equations of order 0 x 0 x 0 the derivative with fractional order considered here is the caputo time since it allows the use of initial conditions an analytical method to obtain exact solution of this class of fractional differential equations is provided some illustrative examples are provided to underpin and illustrate the efficiency of the used method

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

fractional laplace transform method in the framework of the ctit transformation we propose an adapted laplace transform method that gives the solution of a linear fractional differential equation with constant coefficients in terms of exponential function after we mention what the utilized transformation the ctit transformation is based on we explain how it can reduce the problem from fractional form to ordinary form when it is used with laplace transformation via some examples for 0 2 where is the order of fractional derivative finally we illustrate the applications of our results

a method for solving differential equations of fractional order in this paper we consider caputo type fractional differential equations of order 0 a 1 with initial condition x 0 x 0 we introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations generalization of the technique to finite systems is also given finally we give some examples to illustrate the applications of our results

asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions to define a discretization we introduce a set f h of grid points see 2 2 and replace the derivatives in 1 1 by symmetric divided differences with respect to those grid points this device is unproblematic as long as the direct grid neighbours of interior grid points are either interior grid points or boundary points by a proper choice of h that may be obtained e g for rectangular shape regions f2 for all inner grid points for general domains f2 this assumption is no longer satisfied and direct grid neighbours will not always be in and therefore symmetric divided differences are not directly applicable in some points so called irregular mesh points there is a long history to solve this problem 1930 gerschgorin 153 has proposed to use for the unknown function value in a grid neighbour outside of f2 next to an irregular grid point the boundary value of the next boundary point on that grid line obtaining a method of order 1 compared with order 2 available in cases without irregular grid points in 1933 collatz 11 improved this method by using linear interpolation between the irregular grid point and its neighbouring boundary point obtaining a method of order 2 wasow 22 showed that for the methods in 11 15 no asymptotic expansions of the discretization error can exist the 1938 approach of shortley weller 20 may be interpreted as using polynomials of degree 2 interpolating in the

stochastic nonhomogeneous incompressible navier stokes equations we construct solutions for 2 and 3 d stochastic nonhomogeneous incompressible navier stokes equations with general multiplicative noise these equations model the velocity of a mixture of incompressible fluids of varying density influenced by random external forces that involve feedback that is multiplicative noise weak solutions for the corresponding deterministic equations were first found by kazhikhov a v kazhikhov solvability of the initial and boundary value problem for the equations of motion of an inhomogeneous viscous incompressible fluid soviet phys dokl 19 6 1974 331 332 english translation of the paper in dokl akad nauk sssr 216 6 1974 1240 1243 a stochastic version with additive noise was solved by yashima h f yashima equations de navier stokes stochastiques non homogenes et applications thesis scuola normale superiore pisa 1992 the methods here extend the loeb space techniques used to obtain the first general solutions of the stochastic navier stokes equations with multiplicative noise in the homogeneous case m capinski n j cutland stochastic navier stokes equations applicandae math 25 1991 59 85 the solutions display more regularity in the 2d case the methods also give a simpler proof of the basic existence result of kazhikhov

two phase entropy solutions of a forward backward parabolic equation this article deals with the cauchy problem for a forward backward parabolic equation which is of interest in physical and biological models considering such an equation as the singular limit of an appropriate pseudoparabolic third order regularization we consider the framework of entropy solutions namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation moreover we restrict the attention to two phase solutions that is solutions taking values in the intervals where the parabolic equation iswell posed proving existence and uniqueness of such solutions

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

gaussian collocation via defect correction for the numerical integration of boundary value problems for first order ordinary differential systems collocation on gaussian points is known to provide a powerful method in this paper we introduce a defect correction method for the iterative solution of such high order collocation equations the method uses the trapezoidal scheme as the basic discretization and an adapted form of the collocation equations for defect evaluation the error analysis is based on estimates of the contractive power of the defect correction iteration it is shown that the iteration produceso h 2 convergence rates for smooth starting vectors a new result is that the iteration damps all kind of errors so that it can also handle non smooth starting vectors successfully

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

b convergence properties of defect correction methods i b convergence properties of defect correction methods based on the implicit euler and midpoint schemes are discussed the property ofb convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness it turns out that the orders ofb convergence of these methods coincide with the conventional orders of small wherel is a lipschitz constant of the right hand side in part i these assertions are reduced to the validity of the so called hypothesisa which is discussed in greater detail in part ii numerical experiments confirming the theoretical analysis are also given in part ii

a remark on the fractional order differential equations abstract an important solving method was presented in article demirci and ozalp 2012 which shows that the exact solution of a fractional order differential equation can be obtained by means of the solution of an integer order differential equation this article and its method have been cited by many researchers in this paper we will show that this solving method is wrong and then we use several counter examples to disconfirm this solving method

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

asymptotic expansions for the discretization error of least squares solutions of linear boundary value problems on presente des developpements asymptotiques pour l erreur de discretisation des solutions des moindres carres des problemes aux limites lineaires

discrete newton methods and iterated defect corrections in this paper we present a general theory for discrete newton methods iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error

grid functions of nonstandard analysis in the theory of distributions and in partial differential equations abstract we introduce the space of grid functions a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of young measures we will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches as an example we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions applications to nonlinear partial differential equations will be discussed in a subsequent paper

a remark on the q fractional order differential equations abstract an analytical method to obtain the exact solution of q fractional order differential equations was presented in article koca 2015 which shows that solving a q fractional order differential equation is equivalent to solve a q integer order differential equation in this article we first show that the analytical method given in 3 is wrong then we propose analytical and numerical methods for the caputo type q fractional differential equations by using the q beta function some examples are given to disconfirm the solving method in 3 and valid the effectiveness of our methods

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

attractors for three dimensional navier stokes equations for the three dimensional navier stokes equations we propose two new approaches to the notion of an attractor they involve multi valued semiflows constructed via the nonstandard framework used for solving the equations where even in dimension three we have uniqueness of solution for the corresponding equation