An improved class of generalized Runge–Kutta methods for stiff problems. Part I: The scalar case

A new family of p-stage methods for the numerical integration of some scalar equations and systems of ODEs is proposed. These methods can be seen as a generalization of the explicit p-stage Runge–Kutta ones, while providing better order and stability results. We will show in this first part that, at the cost of losing linearity in the formulas, it is possible to obtain explicit A-stable and L-stable methods for the numerical integration of scalar autonomous ODEs. Scalar autonomous ODEs are of very little interest in current applications. However, be begin studying this kind of problems because most of the work can be easily extended to a more general situation. In fact, we will show in a second part (entitled ‘The separated system case'), that it is possible to generalize our methods so that they can be applied to some non-autonomous scalar ODEs and systems. We will obtain linearly implicit L-stable methods which do not require Jacobian evaluations. In both parts, some numerical examples are discussed in order to show the good performance of the new schemes.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

On the Numerical Solution of Involutive Ordinary Differential Systems: Higher Order Methods

We analyse some Taylor and Runge—Kutta type methods for computing one-dimensional integral manifolds, i.e. solutions to ODEs and DAEs. The distribution defining the solutions is taken to be defined only on the relevant manifold and hence all the intermediate points occuring in the computations are projected orthogonally to the manifold. We analyse the order of such methods, and somewhat surprisingly there does not appear any new order conditions for the Runge—Kutta methods in our context, at least up to order 4. The analysis shows that some terms appearing in the error expansions can be quite naturally expressed in terms of standard notions of Riemannian geometry. The numerical examples show that the methods work reliably and moreover produce qualitatively correct results for Hamiltonian systems although the methods are not symplectic.This revised version was published online in October 2005 with corrections to the Cover Date.     

Numerical solution of nonlinear Jaulent–Miodek and Whitham–Broer–Kaup equations

In this work we investigate the numerical solution of Jaulent–Miodek (JM) and Whitham–Broer–Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in JM equation is diagonal but in WBK equation is not diagonal. However for WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

A verified inexact implicit Runge–Kutta method for nonsmooth ODEs

Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science and a scholarship from Egyptian Government.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Comparative analysis of two asymptotic approaches based on integral manifolds

A comparative analysis of the two powerful asymptotic methods,ILDM and MIM (intrinsic low-dimensional manifolds; method ofinvariant manifold), is presented in the paper. The two methodsare based on the general theory of integral manifolds. The ILDMmethod is able to handle large systems of ODEs, whereas theMIM method treats systems with a limited number of unknown variables.The MIM method allows one to conduct analytical explorationof the original system and to obtain final expressions in compactform, whereas the ILDM method is a numerical approach that yieldsthe numerical form of the desired surface. The ILDM method workswell in a region where a rough splitting of the initial systemexists. Regions of the phase space where splitting does notexist are problematic for the ILDM method. In these regionsthe MIM method provides additional information regarding thedynamical behaviour of the system. A number of simple examplesare considered and analysed. It is shown that for the Semenovmodel (singularly perturbed system of ODEs) the ILDM methodgives a surface which appears close to the first order (withrespect to the corresponding small parameter) approximationof the stable (attracting) invariant manifolds. The complementaryproperties of the two asymptotic approaches suggests a feasiblecombination of the two methods, which is the subject of a futurework.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Morse indices of critical manifolds generated by min-max methods with compact lie group actions and applications

We derive some upper and lower bounds for Morse indices of critical manifolds generated by min-max principles for functionals invariant under a general compact Lie group or a finite group action. The results generalize the similar results in the nonequivariant (no group action) case. In doing so, we also generalize the extension theorem of Dugundji type in the nonequivariant case to the equivariant (group action) case. As an application, we obtain a precise growth estimate for the whole sequence of critical values given by the min-max procedure for some superquadratic second-order differential equations. It is well-known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed Hamiltonian systems and equations.     

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.     

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Branches of forced oscillations in degenerate systems of second-order ODEs

We use fixed point index methods to study the set of forced oscillations in periodically perturbed systems of ODEs on manifolds. We prove the existence of branches of periodic solutions for a particular class of system where, contrary to the usual ‘nondegeneracy’ assumption, the leading vector field is neither trivial nor has a set of compact zeros.     

Piecewise finite series solution of nonlinear initial value differential problem

In this paper, we apply a piecewise finite series as a hybrid analytical-numerical technique for solving some nonlinear systems of ordinary differential equations. The finite series is generated by using the Adomian decomposition method, which is an analytical method that gives the solution based on a power series and has been successfully used in a wide range of problems in applied mathematics. We study the influence of the step size and the truncation order of the piecewise finite series Adomian (PFSA) method on the accuracy of the solutions when applied to nonlinear ODEs. Numerical comparisons between the PFSA method with different time steps and truncation orders against Runge-Kutta type methods are presented. Based on the numerical results we propose a low value truncation order approach with small time step size. The numerical results show that the PFSA method is accurate and easy to implement with the proposed approach.     

A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem

In recent time, Runge-Kutta methods that integrate special third order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Albeit, the methods require evaluation of three set of equations to proceed with the numerical integration. In this paper, we propose a class of multistep-like Runge-Kutta methods (hybrid methods), which integrates special third order ODEs directly. The method is completely derivative-free. Algebraic order conditions of the method are derived. Using the order conditions, a four-stage method is presented. Numerical experiment is conducted on some test problems. The method is also applied to a practical problem in Physics and engineering to ascertain its validity. Results from the experiment show that the new method is more accurate and efficient than the classical Runge-Kutta methods and a class of direct Runge-Kutta methods recently designed for special third order ODEs.