Ordinary Initial Value Problems Research Articles

Optimal variational iteration method for parametric boundary value problem

<abstract> <p>Mathematical applications in engineering have a long history. One of the most well-known analytical techniques, the optimal variational iteration method (OVIM), is utilized to construct a quick and accurate algorithm for a special fourth-order ordinary initial value problem. Many researchers have discussed the problem involving a parameter <italic>c</italic>. We solve the parametric boundary value problem that can't be addressed using conventional analytical methods for greater values of <italic>c</italic> using a new method and a convergence control parameter <italic>h</italic>. We achieve a convergent solution no matter how huge <italic>c</italic> is. For the approximation of the convergence control parameter <italic>h</italic>, two strategies have been discussed. The advantages of one technique over another have been demonstrated. Optimal variational iteration method can be seen as an effective technique to solve parametric boundary value problem.</p> </abstract>

О нейронных обыкновенных дифференциальных уравнениях и их вероятностном расширении

This paper describes the transition from neural network architecture to ordinary differential equations and initial value problem. Two neural network architectures are compared: classical RNN and ODERNN, which uses neural ordinary differential equations. The paper proposes a new architecture of p-ODE-RNN, which allows you to achieve a quality comparable to ODE-RNN, but is trained much faster. Furthermore, the derivation of the proposed architecture in terms of random process theory is discussed.

Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation

The Lie group analysis or in other word the symmetry analysis method is extended to deal with a time-fractional order derivative nonlinear generalized KdV equation. Our research in this work aims to use transformation methods and their analysis to search for exact solutions to the nonlinear generalized KdV differential equation. It is shown that this equation can be reduced to an equation with Erdelyi-Kober fractional derivative. In this study, we research the initial and boundary conditions, considering them invariant, and so we get two ordinary initial value problems, i.e. two Cauchy problems. Conservation laws for the given equation are also investigated. In this work, we introduce symmetry analysis and find conservation laws for the nonlinear generalized time-fractional KdV equation by the Lie groups method using fractional derivatives in the Riemann-Liouville sense.

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

ABSTRACT(Chemical) kinetic models in terms of ordinary differential equations correspond to finite-dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the so called reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion and exploitation for deeper insight of unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We newly relate both of them within our variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as novel viewpoints on dimension reduction approaches that might open up new deep perspectives and approaches to the problem from dynamical systems theory.

Implementation of symmetrizers in ordinary differential equations

Two-step symmetrizers for the implicit midpoint and trapezoidal rules provide an alternative to the one-step smoothing formula for solving stiff ordinary differential equations. When used with the basic symmetric methods, these \(L\)-stable methods preserve the asymptotic error expansion in even powers of the step size and provide the necessary damping of oscillatory solutions. These new symmetrizers show effects similar to one-step smoothing but with the advantage of being order two. When generalized to higher order symmetric methods, such as the two-stage Gauss or the three-stage Lobatto IIIA, these symmetrizers can suppress order reduction for stiff problems. Here, we discuss one-step and two-step symmetrizers and their application in ordinary differential equations. We present numerical results with constant and variable step sizes that show the advantages of two-step symmetrizers over one-step symmetrizers of the implicit trapezoidal rule for stiff linear and nonlinear problems. References R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation method, Numer. Math. 8(1):1–13, 1966. doi:10.1007/BF02165234 R. P. K. Chan, Extrapolation of Runge–Kutta methods for stiff initial value problems. Thesis submitted for the degree of Doctor of Philosophy at the University of Auckland, 1989. https://researchspace.auckland.ac.nz/handle/2292/1842 R. P. K. Chan, A-stability of implicit Runge–Kutta extrapolations, App. Numer. Math. 22(1–3):179–203, 1996. doi:10.1016/S0168-9274(96)00031-1 R. P. K. Chan and N. Razali. Smoothing Effects on the IMR and ITR , Numer. Algorithms 65(3):401–420, 2013, doi:10.1007/s11075-013-9779-7 C. F. Curtiss, and J. O. Hirschfelder, Integration of stiff equations, P. Natl. Acad. Sci. 38:235–243, 1952. doi:10.1073/pnas.38.3.235 G. Dahlquist and B. Lindberg, On some implicit one-step methods for stiff differential equations , Royal Institute of Technology, 1973. H. T. Davis, Introduction to nonlinear differential and integral equations , Dover, New York, 1962. http://catalog.hathitrust.org/Record/000619023 R. Frank, J. Schneid and C. W. Ueberhuber. Order results for implicit Runge–Kutta methods applied to stiff systems, SIAM. J. Numer. Analysis , 22(3):515–534, 1985. doi:10.1137/0722031 A. Gorgey, Extrapolation of Symmetrized Runge-Kutta Methods. Thesis submitted for the degree of Doctor of Philosophy at the University of Auckland, 2012. https://researchspace.auckland.ac.nz/handle/2292/18996 W. B. Gragg, On extrapolation algorithm for ordinary initial value problems, SIAM J. Numer. Anal . 2(3):384–403 1965. doi:10.1137/0702030 E. Hairer and G. Wanner, Solving ordinary differential equations, II. Stiff and differential-algebraic problems , Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1991. doi:10.1007/978-3-662-09947-6 R. Holsapple, R. Iyer and D. Doman, Variable step-size selection methods for implicit integration schemes for ODEs, Int. J. Numer. Anal. Mod. , 4(2):210–240, 2007. http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-2-07/2007-02-04.pdf B. Lindberg, On the smoothing and extrapolation for the trapezoidal rule, BIT , 11(1):29–52, 1971. doi:10.1007/BF01935326 A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comput . 28:145–162, 1974. doi:10.1090/S0025-5718-1974-0331793-2 L. F. Richardson, J. A. Gaunt. The Deferred Approach to the Limit, Philos. T. R. Soc. A 226:299–361, 1927. doi:10.1098/rsta.1927.0008 H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations , Springer-Verlag, Berlin, 1973. http://www.springer.com/us/book/9783642654732

Four step in phase predictor corrector method for periodic initial value problems

In this paper, 4-step in phase and with minimal phase-lag predictor-corrector methods are developed for the numerical integration of periodic initial value problems using the implicit method of Raptis-Simos [23]. The 4-step in phase P-C method is of order six and it possesses an interval of periodicity of size 3.3874. We also obtain a 4-step sixth-order P-C method with minimal phase-lag of order eight. The numerical results for ordinary and partial differential initial value problems show that the 4-step predictor-corrector in phase method gives accurate results comparable to those of the implicit in phase method of Raptis-Simos [23].

Numerical comparison of methods for solving second-order ordinary initial value problems

In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C 2-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented.

State space model of the direct methanol fuel cell

In this report we propose a state space model of a direct methanol fuel cell. This dynamical model allows analysis and optimisation of the fuel cell in a control-theoretical framework, for instance to improve the overall dynamics of the fuel cell or to reject environmental disturbances. The model is particularly well suited to synthesise controllers, which are necessary to ensure stable, robust and efficient operation of a fuel cell. To yield a state space model of the direct methanol fuel cell, a system of nonlinear partial differential equations is transformed into a system of ordinary initial value problems by Laplace transforms, approximation in function spaces and the method of weighted residuals.

On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems

This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points s_{i-1+\\alpha}=x_{i-1}+\\alpha h , x_{i-1+\\beta}=x_{i-1}+\\beta h , x_{i-1+\ heta}=x_{i-1}+\ heta h and x_i=x_{i-1}+h in each subinterval [ x i @ 1 , x i ], i = 1(1) N for solving initial value problems in ordinary differential equations including stiff equations. Here 0 < f < g < è < 1 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: \\eqalign {&\\beta(\ heta -\ heta^2)+\\beta^2(\ heta^2-\ heta)+\\alpha^2[\ heta^2-\ heta +\\beta^2(1+4\ heta -5\ heta^2)+\ heta(4\ heta^2-4\ heta -1)]\\cr &\\qquad \\qquad \\qquad +\\alpha[\ heta -\ heta^2+\\beta (1+4\ heta -4\ heta^2)+\\beta^2(4\ heta^2+4\ heta -1)]\\le 0 and they are unstable if f , g , è < 0.7279115. Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035 h f < g < è < 1 the methods are A-stable while they are not if f h 0.5; on other hand, the sizes of regions of absolute stability increase when f , g , è M 1 m.

Sixth Order C 2 -Spline Collocation Method for Integrating Second Order Ordinary Initial Value Problems

A new procedure based on sixth degree (Hexic) C 2 -Spline for the numerical integration of the second order initial value problems (IVPs) y^{\\prime\\prime}=f(x,y) , including those possessing oscillatory solutions, is presented. The proposed method is essentially an implicit sixth order one-step method. Stability analysis shows that the method possesses (0, 75.3)\\bigcup (130.2, 201.9) as interval of periodicity and/or absolute stability. In addition, the method has phase-lag (dispersion) of order six with actual phase-lag H^{6}/774144 . Convergence results yield error bounds \\parallel\\! s^{(r)}-y^{(r)}\\!\\parallel\\,=O\\left(h^{6}\\right),r=0,1 , in the uniform norm, provided y\\in C^{8}[0,b] . Furthermore, it turns out that the method is a continuous extension of a sixth order four-stage Runge-Kutta (-Nyström) method. Numerical experiments will also be considered.

Sixth order non-dissipative C 1 spline collocation method for oscillatory ordinary initial value problems

In this paper we construct a global method, based on quintic C 1-spline, for the integration of first order ordinary initial value problems (IVPs) including stiff equations and those possessing oscillatory solutions as well. The method will be shown to be of order six and in particular is A-stable. Attention is also paid for the phase error (or dispersion) and it is proved that the method is dispersive and has dispersion order six with small phase-lag (compared with the extant methods having the same order (cf. [7])). Moreover, the method may be regarded as a continuous extension of the closed four-panel Newton–Cotes formula (NC4) (typically it is a continuous extension of an implicit Runge–Kutta method). In additiona priori error estimates, in the uniform norm, together with illustrative test examples will also be presented.

Quintic C2-spline integration methods for solving second-order ordinary initial value problems

In this paper we propose a global collocation method for the integration of the special second-order ordinary initial value problem (IVP) y″= f( x, y). The presented method is based on quintic C 2-splines s( x) as an approximation to the exact solution y( x) of the (IVP). Analysis of stability shows that the method possesses (0,36)∪(54,110.2) as interval of periodicity and absolute stability. Moreover, the method has phase-lag of order four with actual phase-lag H 4/18(6!). Error bounds, in the uniform norm, for ∥s (i)−y (i)∥= O(h 4), i=0(1)2 , if y∈ C 6 [0, b], together with illustrative test examples will also be considered.

Stabilized cubic c1-spline collocation method for solving first-order ordinary initial value problems

This paper is concerned with a construction of a “variable” one-parameter cubic C 1-spline collocation method for solving the initial value problem (IVP)viz where the collocation point . The presented method will be shown to be strongly unstable if β ≤ 1/2. It turns out that the proposed method is a continuous extension of the A-stabilized version of Simpson's rule introduced in [1], if β = 1. Moreover, the method is of order three, for β ∈ (1/2, 1) and is of order four, if β = 1 and y ∈ C 5[0,b].

A continuous implicit nyström method for solving ordinary second order initial value problems

This paper is concerned with the construction of a new formula for integrating special second order, periodic and nonperiodic, initial value problems in ordinary differential equations. The presented formula is based on C 2-cubic splines as an approximation to the exact solution of the initial value problem, It turns out that the proposed method is a continuous extension of the well-known second order 1-stage implicit Nyström method and hence possesses nonvanishing intervals of periodicity and absolute stability.

On the existence of C ∞ solutions to the asymptotic characteristic initial value problem in general relativity

The asymptotic characteristic initial value problem for Einstein's vacuum field equations is treated. It is shown that C$^{\infty}$ solutions exist and are unique for C$^{\infty}$ initial values. The proof is based on Friedrich's regular conformal vacuum field equations and Rendall's method of reducing the characteristic to an ordinary initial value problem.

Stability of Parallel Explicit ODE Methods

General-purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods, first formulated by J. C. Butcher. Among these the explicit methods are easiest to implement. For this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par- allel. Each such group of concurrent function evaluations can be regarded as a stage of the method. It is desirable that methods have a large region of stability so that stability-dictated restrictions on the stepsize will occur less frequently. In order to do a fair comparison of stability regions, one should consider the scaled stability region. This is defined using not the nominal stepsize but rather the stepsize per unit of work. Each stage counts as one work unit; in the context of parallel computing each stage may involve any number of independent function evaluations. Very important results on scaled stability regions were obtained by Jeltsch and Nevanlinna, in the context of serial computation. They identify a type of method, actually a type of alge- braic function, whose scaled stability region is optimal. This means that the region is not a proper subset of some other scaled stability region. The distinguished type of algebraic functions is one that satisfies Property C, which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method. It is shown that by modifying Property C the result on optimal stability regions holds also in the context of parallel computa- tion. An important consequence is that, although the use of parallelism enlarges the class of scaled stability regions, it does not result in scaled stability regions that properly contain those obtained without parallelism. Roughly stated, parallelism cannot improve the stability of an explicit linear method; but it can, even for the standard test problem, improve the accuracy.

Limits of parallelism in explicit ODE methods

Numerical methods for ordinary initial value problems that do not depend on special properties of the system are usually found in the class of linear multistage multivalue methods, first formulated by J.C. Butcher. Among these the explicit methods are easiest to implement. For these reasons there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in parallel. Each such group of concurrent function evaluations can be regarded as a stage of the method. However, it turns out that parallelism affords only limited opportunity for reducing the computing time with such methods. This is most evident for the simple linear homogeneous constant-coefficient test problem, whose solution is essentially a matter of approximating the exponential by an algebraic function. For a given number of stages and a given number of saved values, parallelism offers a somewhat enlarged set of algebraic functions from which to choose. However, there is absolutely no benefit in having the degree of parallelism (number of processors) exceed the number of saved values of the method. Thus, in particular, parallel one-step methods offer no speedup over serial one-step methods for the standard linear test problem. Although the implication of this result for general nonlinear problems is unclear, there are indications that dramatic speedups are not possible in general. Also given are some results relevant to the construction of methods.

Solution of Linear Systems of Ordinary Differential Equations on an INTEL Hypercube

In this paper there is developed and tested a parallel scheme for the solution of linear systems of ordinary initial value problems based on the box scheme and a modified recursive doubling technique. The box scheme may be replaced by any stable integrator. The algorithm can be modified to solve boundary value problems. Software for both problems is available upon request.

A Method of Skipping the Transient Phase in the Solution of Separably Stiff Ordinary Initial Value Problems

A Method of Skipping the Transient Phase in the Solution of Separably Stiff Ordinary Initial Value Problems

A method of skipping the transient phase in the solution of separably stiff ordinary initial value problems

Stiff systems of ordinary differential equations are characterized by an initial phase in which the solution changes rapidly. Often there is no interest in reproducing this transient phase. A method is proposed for modifying the initial value if the system of differential equations is separably stiff, i.e. is characterized by the occurrence of a few (typically one) large negative real eigenvalues which dominate the others. The modified system does not possess a transient phase, and in the constant coefficient linear case its solution does not differ from that of the original one in the nonstiff components.