Boundary Value Problems For ODEs Research Articles

Numerical Solution of 2nd Order Boundary Value Problems with Dirichlet, Neumann and Robin Boundary Conditions using FDM

In many fields of science and engineering, to determine the harmonic motion, damped and forced variation, current from electric circuit, 2nd order ODE is required to solve. Solving the ODE with complicated boundary condition that occur in engineering problems is a great challenges analytically. Therefore, numerical technique finite difference method (FDM) is very popular and important for solving the boundary value problems. In this article three different conditions as Dirichlet, Neumann and Robin (mixed) boundary conditions are applied in initial-boundary problem. FDM is used to solve ODE boundary value problems. Error calculation, stability, convergence are also explained. To test the accuracy numerical solutions are verified with analytical solution and error is calculated at each point for different mesh grid size as mesh grid size is decreased result will give the accuracy.

The Novelty of Thermo-Diffusion and Diffusion-Thermo, Slip, Temperature and Concentration Boundary Conditions on Magneto–Chemically Reactive Fluid Flow Past a Vertical Plate with Radiation

The significance of radiation, Soret and Dufour’s effects on MHD flow in a porous media near a stagnation point past a vertical plate with slip, temperature, and concentration boundary conditions were investigated. Local similarity variables are used in the solution, which reduces the PDEs into analogous boundary value problem for ODEs. Symmetry analysis can be used to detect these variations in local similarity. To numerically explain the problem, a shooting approach and the MATLAB bvp4c solver are utilized. As the magnetic field and porous medium parameters are raised, the skin friction increases, and the temperature increases as the radiation pointer is increased. As the Soret number grows, the concentration profile rises.

Meanders, Zero Numbers and the Cell Structure of Sturm Global Attractors

We study global attractors \(\mathcal {A}=\mathcal {A}_f\) of semiflows generated by semilinear partial parabolic differential equations of the form \(u_t = u_{xx} + f(x,u,u_x), 0<x<1\), satisfying Neumann boundary conditions. The equilibria \(v\in \mathcal {E}\subset \mathcal {A}\) of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of \(\mathcal {A}\) into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors \(\mathcal {A}\) as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation \(\sigma =\sigma _f\) derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries \(x=0,1\), respectively, completely determines the Sturm global attractor \(\mathcal {A}\). Equivalently, we use a planar curve, the meander \(\mathcal {M}=\mathcal {M}_f\), associated to the the ODE boundary value problem by shooting. In the previous paper (Fiedler and Rocha in J Dyn Differ Equ, 2020. https://doi.org/10.1007/s10884-020-09836-5), we set up to determine the boundary neighbors of any specific unstable equilibrium \(\mathcal {O}\), based exclusively on the information on the corresponding signed hemisphere complex. In addition, a certain minimax property of the boundary neighbors was established. In the signed hemisphere decomposition of the cell boundary of \(\mathcal {O}\), this property identifies the equilibria which are closest to, or most distant from, \(\mathcal {O}\) at the boundaries \(x=0,1\), in each hemisphere. The main objective of the present paper is to derive this minimax property directly from the Sturm permutation \(\sigma \), or equivalently from the Sturm meander \(\mathcal {M}\), based on the Sturm nodal properties of the solutions of the ODE boundary value problem. This minimax result simplifies the task of identifying the equilibria on the cell boundary of each unstable equilibrium, directly from the Sturm meander \(\mathcal {M}\). We emphasize the local aspect of this result by an example for which the identification of the equilibria is obtained from the knowledge of only a segment of the Sturm meander \(\mathcal {M}\).

A Riemann solution approximation based on the zero diffusion–dispersion limit of Dafermos reformulation type problem

In the present work, a new numerical strategy is designed to approximate the Riemann solutions of systems of conservation laws. Here, the main difficulty comes from the definition of the discontinuous solutions. Indeed, the shock solutions are no longer selected by entropy criterion but they are defined as the zero limit of a diffusive–dispersive system. As a consequence, the solutions of interest may contain non classical shocks. In order to derive a suitable numerical approach, the Dafermos diffusion technique is adopted here. Then, the PDE initial value problem is reformulated as an ODE boundary value problem. A fourth-order finite difference scheme is introduced to approximate the solution of this ODE boundary value problem. In this work, a particular attention is paid on the existence of discrete solutions and several numerical experiments illustrate the relevance of the derived numerical strategy.

Sturm 3-Ball Global Attractors 2: Design of Thom–Smale Complexes

This is the second of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x)\,, \end{aligned}$$ on the unit interval $$0< x<1$$ with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom–Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom–Smale complex turns out to be a regular cell complex. Our geometric description is slightly more refined. It involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is in terms of the Sturm permutation, alias the meander properties, of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $$x=0$$ and $$x=1$$ , respectively, and the overlapping reach of polar serpents in the shooting meander. In the first paper we showed the implications $$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom}{-}\text {Smale complex} \quad \Longrightarrow \quad \text {meander}\,. \end{aligned}$$ The present part 2 closes the cycle of equivalences by the implication $$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}\,. \end{aligned}$$ In particular this cycle allows us to construct a unique Sturm 3-ball attractor for any prescribed Thom–Smale complex which satisfies the geometric properties of the bipolar orientation and the hemisphere decomposition. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy, however, is just another step towards a still elusive geometric and combinational characterization of all Sturm global attractors in arbitrary dimensions.

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, \begin{document}$u_t = u_{xx} + f(x, u, u_x)\, , $ \end{document} on the unit interval \begin{document}$0 with Neumann boundary conditions. Equilibria \begin{document}$v_t = 0$\end{document} are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} as 3-cell templates \begin{document}$\mathcal{C}$\end{document} . The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries \begin{document}$x = 0$\end{document} and \begin{document}$x = 1$\end{document} , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.

Sturm 3-ball global attractors 1: Thom–Smale complexes and meanders

This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x), \end{aligned}$$ on the unit interval $$0< x<1$$ with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom–Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom–Smale complex turns out to be a regular cell complex. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $$x=0$$ and $$x=1$$ , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we show the implications $$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom--Smale complex} \quad \Longrightarrow \quad \text {meander}. \end{aligned}$$ The sequel, part 2, closes the cycle of equivalences by the implication $$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}. \end{aligned}$$ Each implication, or mapping, involves certain constructions which are tuned such that the final 3-ball Sturm global attractor defined by the meander combinatorics coincides with the originally given Sturm 3-ball. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.

The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems

We consider perturbed Hammerstein integral equations of the form: $$\begin{aligned} y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )+\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\ \mathrm{d}s \end{aligned}$$ in the case, where $$H_1$$ and $$H_2$$ are continuous functions, which can be either linear or nonlinear subject to some restrictions, and $$\varphi _1$$ and $$\varphi _2$$ are linear functionals. We demonstrate that by introducing a specially constructed order cone, one can equip $$\varphi _1$$ and $$\varphi _2$$ with coercivity conditions that are useful in improving existence results for both the integral equation and associated boundary value problems for ODEs and elliptic PDEs on annuli with nonlocal boundary conditions. We illustrate this theory with some specific examples.

Multiple shooting approach for computing approximately shortest paths on convex polytopes

In this paper, we use a multiple shooting approach in solving boundary value problems for ODE to introduce a novel iterative algorithm for computing an approximate shortest path between two points on the surface of a convex polytope in 3D. Namely, the polytope is partitioned into subpolytopes, shooting points and a Straightness condition are established. The algorithm specifies how to combine shortest paths between shooting points in subpolytopes to become the required shortest path by the Straightness condition. In particular, the algorithm does not rely on Steiner points and graph tools on the entire polytope. It is implemented in C++ and a comparison with Agarwal, Har-Peled, and Karia’s algorithm, on the accurate construction of the shortest path, is presented.

A Correction and an Extension of Stampacchia’s Work on the Geometric BVP

Abstract G. Stampacchia introduced the geometric boundary value problem for ODEs in his doctoral thesis and published four papers related to it. Here we point out that the proof of his last theorem on the subject is incorrect and we provide a substitute for it as well as a generalizations of some of his earlier results.

On "Solving differential equations in R" by K.~Soetaert, J.~Cash, and F.~Mazzia

There is a number of powerful software packages intended to solve numerical problems and differential equations in particular. First of all, we can mention here MATLAB, Maple and Mathematica. Recently to that list we can add R. The advantage of using R as compared to the previously mentioned programs, which are rather expensive, is that R is an open source software. At the same time, the recent development of R make it a very powerful and flexible software for solving differential equations. The book ''Solving differential equations in R by K. Soetaert (NIOO-CEME, Yerseke, Netherlands); J. Cash (Imperial College, London, UK); F. Mazzia (University of Bari, Italy), Springer, World Scientific, Publishers Series in Business, 2012. Approx. 250 p. 70 illus. (Use R!) Softcover. ISBN: 978-3-642-28069-6, describes the present stage of development of R packages intended to solve differential equations. Consecutive chapters focus on initial value problems for ODE, differential algebraic equations, boundary value problems for ODE, delayed differential equations and partial differential equations. From the point of view of the R software the book presents the following R packages: \textbf{deSolve}, \textbf{rootSolve}, \textbf{deTestSet}, \textbf{ReacTran} and \textbf{bvpSolve}. The package \textbf{deSolve} contains most of the integration methods used to solve differential equations. It contains a large number of Runge-Kutta methods, both explicit and implicit, and a large variety of multistep methods (Adams methods). These methods can be applied to solve initial value problems for both non-stiff and stiff ODE systems. Also delayed differential equations can be solved using specially designed functions form the package \textbf{deSolve}. Algebraic differential equations can be solved using special methods from \textbf{deSolve} and \textbf{deTestSet} packages. The package \textbf{bvpSolve} is for solving boundary value problems. This package implements 3 methods of solution: shooting, MIRK and collocation methods. Solving partial differential equations in R requires an additional work from the user. The idea is to reduce a particular PDE to a system of ODE's or a system of algebraic equations and then solve such a system using tools available in R. For simple parabolic or hyperbolic equations the method of line can be used and reduce the problem to a system of ODE's which can be solved by special functions from the \textbf{deSolve} package. For elliptic equations a finite difference approximations can be applied and the resulting system of algebraic equations can be solved by a function from the package \textbf{rootSolve}. A grid generation for diffusive and adjective transport terms in analyzed PDE's can be performed automatically by functions from the package \textbf{ReacTran}. Subsequent topics of the book are presented in a similar manner. First comes a chapter with relevant theoretical results about existence of solutions, limits of solvability, discretization methods and stability problems. The presentation is accompanied by a large number of references where more detailed information is available. Then follows a chapter with R implementation of described methods together with a detailed information about the use of the relevant packages. These implementations are illustrated by a large number of well-known examples. Most of important ODE's models are treated as illustrations to particular methods or peculiarities of solutions obtained. Logistic equation, Lorenz model, Arenstorf orbits, van der Pol equation, Josephson junctions, pendulum problem, Sturm-Liouville problem and a large number of models from biology and medicine -- all that can be found in some place in the book. To whom the book is intended? First of all to practitioners: researchers and graduate students who need solutions to differential equations. The R software gives them a possibility to obtain solutions without going into details of computer coding. In addition, a large number of R graphic and statistical functions can improve visualization of the solution obtained. There is however a word of warning to non-expert in numerical analysis. There is a number of traps in numerical solutions of differential equations related to solvability or numerical stability problems. Majority of these traps is addressed in theoretical chapters of the book. But these theoretical chapters are written in a very concise form. A mathematical theorem is sometime reported just in one sentence. Hence, when the reader is a non-expert he should read very carefully a relevant part of theoretical introduction before starting computations to avoid frustration when the result do not fit to expectations. The book can be also very interesting for experts in differential equations. I strongly recommend the book to all courses in ODE and numerical ODE as a valuable source of interesting examples and illustrations to the course. It can be as well an interesting source of problems for students enabling them to test lectured topics and experiment with new models. Finally even experts in the field can find in the book tasty remarks on the state of the art of some numerical methods for differential equations.

Nonlinear cyclic weak contractions in G-metric spaces and applications to boundary value problems

Abstract We present fixed point theorems for nonlinear cyclic mappings under a generalized weakly contractive condition in G-metric spaces. We furnish examples to demonstrate the usage of the results and produce an application to second-order periodic boundary value problems for ODEs. MSC:47H10, 34B15.

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009, where the abstract theory of fold completeness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may contain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary conditions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach L q -framework, in contrast to previously known results in the Hilbert L 2-framework. Some concrete mechanical problems are also presented.

Newton Interpolating Series at m Distinct Points with Coefficients in a Real Banach Algebra

A real Banach algebra of Newton interpolating series, used to approximate the solutions of multipoint boundary value problems for ODE′s, is studied.

Cubic string boundary value problems and Cauchy biorthogonal polynomials

Cauchy biorthogonal polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis–Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third-order ODE boundary value problem –f ′′′ = zgf which is a generalization of the inhomogeneous string problem studied by Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated with these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L2g.

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory

N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.

On estimation algorithms for ordinary differential equations

This article addresses the problem of estimating the parameters of a system of ordinary differential equations given data derived from noisy observations on the state variables. This problem is important in a range of applications in areas such as adaptive, real time control. There are two main classes of method for attacking this problem, and their equivalence and effectiveness (consistency) are discussed. Recent rate of convergence results for the major implementation techniques are summarized, and some matters requiring further consideration indicated. References H. G. Bock. Recent advances in parameter identification techniques in {ODE}. In P. Deuflhard and E. Hairer, editors, Numerical Treatment of Inverse Problems in Differential and Integral Equations , pages 95--121. Birkhauser, 1983. J. Nocedal and S. J. Wright. Numerical Optimization . Springer--Verlag, 1999. M. R. Osborne. Numerical questions in ODE boundary value problems. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006 , volume 48 of ANZIAM J. , pages C899--C926, February 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/79 [February 11, 2008]. M. R. Osborne. Fisher's method of scoring. Int. Stat. Rev. , 86:271--286, 1992. M. R. Osborne. The bock iteration for the ode estimation problem. 2008. in preparation. I. Tjoa and L. T. Biegler. Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic systems. Ind. Eng. Chem. Res. , 30:376--385, 1991.

A numerical approach to nonlinear two-point boundary value problems for ODEs

In this paper we propose a numerical approach to solve some problems connected with the implementation of the Newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two-point BVPs for ODEs with mixed linear boundary conditions by using the finite difference method.

Numerical questions in ODE boundary value problems

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood. References U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations , SIAM, Philadelphia, 1995. J. C. Butcher, Numerical methods for ordinary differential equations , John Wiley and Sons, 2003. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33--53. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27--43. F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well-conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), 89--105. P. Deuflhard, Newton methods for nonlinear problems , Springer--Verlag, Berlin Heidelberg, 2004. R. England and R. M. M. Mattheij, Boundary value problems and dichotomic stability, SIAM J. Numer. Anal. 25 (1988), 1037--1054. Z. Li, M. R. Osborne, and T. Prvan, Parameter estimation of ordinary differential equations, IMA J. Numer. Anal. 25 (2005), 264--285. J. Nocedal and S. J. Wright, Numerical optimization , Springer--Verlag, 1999. M. R. Osborne, On shooting methods for boundary value problems, J. Math. Analysis and Applic. 27 (1969), 417--433. M. R. Osborne, Cyclic reduction, dichotomy, and the estimation of differential equations, J. Comp. and Appl. Math. 86 (1997), 271--286. B. J. Quinn and E. J. Hannan, The estimation and tracking of frequency , Cambridge University Press, Cambridge, United Kingdom, 2001. J. O. Ramsay, G. Hooker, C. Cao, and C. Campbell, Estimating differential equations , preprint, Department of Psychology, McGill University, Montreal, Canada, 2005, p. 40.

Numerical questions in ODE boundary value problems

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood. References U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, SIAM, Philadelphia, 1995. J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley and Sons, 2003. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33--53. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27--43. F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well-conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), 89--105. P. Deuflhard, Newton methods for nonlinear problems, Springer--Verlag, Berlin Heidelberg, 2004. R. England and R. M. M. Mattheij, Boundary value problems and dichotomic stability, SIAM J. Numer. Anal. 25 (1988), 1037--1054. Z. Li, M. R. Osborne, and T. Prvan, Parameter estimation of ordinary differential equations, IMA J. Numer. Anal. 25 (2005), 264--285. J. Nocedal and S. J. Wright, Numerical optimization, Springer--Verlag, 1999. M. R. Osborne, On shooting methods for boundary value problems, J. Math. Analysis and Applic. 27 (1969), 417--433. M. R. Osborne, Cyclic reduction, dichotomy, and the estimation of differential equations, J. Comp. and Appl. Math. 86 (1997), 271--286. B. J. Quinn and E. J. Hannan, The estimation and tracking of frequency, Cambridge University Press, Cambridge, United Kingdom, 2001. J. O. Ramsay, G. Hooker, C. Cao, and C. Campbell, Estimating differential equations, preprint, Department of Psychology, McGill University, Montreal, Canada, 2005, p. 40.