Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations

We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Padé approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Padé and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Padé-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution sought and show how this distribution controls the efficiency of the two methods. In general the method based on Padé approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.     

Construction of ODE Curves

This paper deals with a construction problem of free-form curves from data constituted by some approximation points and a boundary value problem for an ordinary differential equation (ODE). The solution of this problem is called an ODE curve. We discretize the problem in a space of B-spline functions. Finally, we analyze a graphical example in order to illustrate the validity and effectiveness of our method.     

Une méthode de décomposition en temps avec des schémas dʼintégration réversible pour la résolution de système dʼéquations différentielles ordinaires

We propose a time domain decomposition method that breaks the sequentiality of the integration scheme for systems of ODE. Under the condition of differentiability of the flow, we transform the initial value problem into a well-posed boundary values problem using the symmetrization of the interval of time integration and time-reversible integration scheme. For systems of linear ODE, we explicitly construct the block tridiagonal system satisfied by the solutions at the time sub-intervals extremities. We then propose an iterative algorithm of Schwarz type for updating the interfaces conditions which can extend the method to systems of nonlinear ODE.     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

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Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.     

一类二阶常系数非齐次线性微分方程及边值问题的解法

利用非齐次方程通解方法和Green函数法给出了非齐次项为点源函数的二阶常系数线性常微分方程及边值问题的求解方法和公式.然后以渗流力学一类具体问题为例进行了论证.结果表明这两种方法在本质上是一致的,所得到的结果是相互吻合的.该点源解可用于分析相关边值问题,并可用来求解具有一般非齐次项的微分方程及相关定解问题.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

The Design and Implementation of Usable ODE Software

In the last decade it has become standard for students and researchers to be introduced to state-of-the-art numerical software through a problem solving environment (PSE) rather than through the use of scientific libraries callable from a high level language such as Fortran or C. In this paper we will identify the constraints and implications that this imposes on the ODE software we investigate and develop. In particular, the way a numerical solution is displayed and viewed by a user dictates that new measures of performance and quality must be adopted. We will use the MATLAB environment and ODE software for initial value problems, boundary value problems and delay problems to illustrate the issues that arise and the progress that has been made. One of the major implications is the expectation that accurate approximations at off-mesh points must be provided. Traditional numerical methods for ODEs have produced approximations to the underlying solution on an associated discrete, adaptively chosen mesh. In recent years it has become common for the ODE software to also deliver approximations at off-mesh values of the independent variable. Such a feature can be extremely valuable in applications and leads to new measures of quality and performance which are more meaningful to users and more consistently interpreted and implemented in contemporary ODE software. Numerical examples of the robust and reliable behaviour of such software will be presented and the cost/reliability trade-offs that arise will be quantified.     

Quadratic Störmer‐type methods for the solution of the Boussinesq equation by the method of lines

We study the numerical treatment of Boussinesq PDE equation using the method of lines. For the space discretization, we choose either classical finite differences or Fourier pseudospectral methods. Both cases result in a system of second‐order ordinary differential equations (ODEs) that is quadratic. In order to take advantage of this special feature, we choose to solve the ODE system using a new type of hybrid Numerov method specially constructed for such problems. Other efficient ODE solvers taken from the literature are used to solve the system of ODEs as well. By taking all the combinations of space discretization methods and ODE solvers, we discuss the stability and accuracy features revealed from the numerical tests. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Symmetric collocation methods for linear differential-algebraic boundary value problems

Summary. We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally by index reduction techniques) and apply Gau?-type (for the differential part) and Lobatto-type (for the algebraic part) collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions, we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence when using the combination of Gau? and Lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods. Received September 22, 2000 / Revised version received February 7, 2001 / Published online August 17, 2001     

Galerkin methods in time for semi-discrete viscoelastodynamics

In semi-discrete nonlinear elastodynamics, higher order energy and momentum conserving time stepping schemes turned out to be well suited for computing long time motions [1]. In comparison to standard ODE integrators, conserving schemes exhibit superior stability properties which are of utmost importance in a nonlinear .nite element framework. We show that conserving schemes are particularly well suited as starting point for the development of energy consistent schemes for dissipative dynamical systems. In particular, viscoelastic material behaviour is considered. A key advantage of energy consistent schemes lies in the fact that the equilibrium state of viscoelastic systems can be de.nitely reached, independent of the material parameters. In the paper, we compare two Galerkin methods for the temporal discretisation of semi-discrete nonlinear viscoelastodynamics: the standard continuous Galerkin (cG) method and an enhanced continuous Galerkin (eG) method which ful.ls the total energy balance exactly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Memory-efficient Implementation of Riccati Approach for Time-Dependent Optimal Control Problems in ODEs

We consider a time-dependent optimal control problem, where the state evolution is described by an ODE. There is a variety of methods for the treatment of such problems. We prefer to view them as boundary value problems and apply to them the Riccati approach for non-linear BVPs with separated boundary conditions. There are many relationships between multiple shooting techniques, the Riccati approach and the Pantoja method, which describes a computationally efficient stage-wise construction of the Newton direction for the discrete-time optimal control problem. We present an efficient implementation of this approach. Furthermore, the well-known checkpointing approach is extended to a ‘nested checkpointing’ for multiple transversals. Some heuristics are introduced for an efficient construction of nested reversal schedules. We discuss their benefits and compare their results to the optimal schedules computed by exhaustive search techniques. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time integration of index 1 DAEs with Rosenbrock methods using Krylovsubspace techniques

The derivation of Rosenbrock‐Krylov methods for index 1 DAEs involves two well known techniques: a limit process which transforms a singular perturbed ODE to an index 1 DAE and the use of Krylov iterations instead of direct linear solvers for the stage equations. We show that our derived class of Rosenbrock‐Krylov schemes is independent of the order in which we apply these techniques. We also conclude that for convergence a rather accurate solution of the algebraic part is always needed.     

Breaking the limits: The Taylor series method

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge-Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.     

Properties of Difference Schemes with Oblique Stencils for Hyperbolic Equations

In this paper, various difference schemes with oblique stencils, i.e., schemes using different space grids at different time levels, are studied. Such schemes may be useful in solving boundary value problems with moving boundaries, regular grids of a non-standard structure (for example, triangular or cellular ones), and adaptive methods. To study the stability of finite difference schemes with oblique stencils, we analyze the first differential approximation and dispersion. We study stability conditions as constraints on the geometric locations of stencil elements with respect to characteristics of the equation. We compare our results with a geometric interpretation of the stability of some classical schemes. The paper also presents generalized oblique schemes for a quasilinear equation of transport and the results of numerical experiments with these schemes.     

Similar constructing method for solving nonhomogeneous mixed boundary value problem of n‐interval composite ode and its application

This paper studies a simple method—Similar Constructing Method (SCM)—for constructing the exact solutions of the nonhomogeneous mixed boundary value problem for sets of n‐interval composite second‐order ordinary differential equation (ODE) with variable coefficient. Then this paper proves the correctness of the solution obtained by SCM. After that, this paper has done simulation experiment. This section uses the SCM to solve the nonhomogeneous boundary value problem of three‐interval composite Bessel equation. Solutions are presented in graphical form for various parameter values, and the influence of parameters on the solution is analyzed. The example shows that using SCM to solve the class of nonhomogeneous mixed boundary value problems of n‐interval composite second‐order linear ODE is easy, convenient, and effective.