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In this paper we define unconditional stability properties of exponential Runge-Kutta methods when they are applied to semi-linear systems of ordinary differential equations characterized by a stiff linear part and a nonstiff non-linear part. These properties are related to a class of systems and to a specific norm. We give sufficient conditions in order that an explicit method satisfies such properties. On the basis of such conditions we analyze some of the popular methods.

In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.

Inspired by the theory of modified equations (backward error analysis), a new approach to high-order, structure-preserving numerical integrators for ordinary differential equations is developed. This approach is illustrated with the implicit midpoint rule applied to the full dynamics of the free rigid body. Special attention is paid to methods represented as B-series, for which explicit formulae for the modified differential equation are given. A new composition law on B-series, called substitution law, is presented.

In this paper, we consider an explicit exponential method of classical order two for the time discretisation of quasi-linear parabolic problems. The numerical scheme is based on a Magnus integrator and requires the evaluation of two exponentials per step. Our convergence analysis includes parabolic partial differential equations under a Dirichlet boundary condition and provides error estimates in Sobolev spaces. In an abstract formulation the initial boundary value problem is written as an initial value problem on a Banach space X u'(t) = A(u(t))u(t), 0 < t < T, u(0) given, involving the sectorial operator A(v): D → X with domain D C X independent of v ∈ V C X. Under reasonable regularity requirements on the problem, we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between X and D. Various applications and a numerical experiment illustrate the theoretical results.

Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact using Gr\"{o}bner basis techniques and give a direct method for finding a polynomial over $K$ that $y'/y$ satisfies. Moreover, we provide explicit degree bounds and extend the result to fields with positive characteristic. Finally, we give an application of our method to a class of nonlinear differential equations., Comment: 9 pages, Proceedings of the AMS

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity., Comment: 18 pages, 1 figure, AMSLaTeX, published electronically at http://www.ams.org/journal-getitem?pii=S0002-9947-03-03223-9

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

The differential equation \[ d x ( t ) = a ( x ( t ) , t ) d Z ( t ) + b ( x ( t ) , t ) d t dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt \] for fractal-type functions Z ( t ) Z(t) is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation x ( t ) = h ( y ( t ) + Z ( t ) , t ) x(t)\, =\, h(y(t)+Z(t),t) for certain C 1 C^1 -functions h h and y y . The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.

Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

This paper is concerned with a class of complete second order linear differential equations in a Banach space. We show the existence and uniqueness of classical solutions of (SE) { u ( t ) = A ( t ) u ′ ( t ) + B ( t ) u ( t ) + f ( t ) for t ∈ [ 0 , T ] u ( 0 ) = x and u ′ ( 0 ) = y . \begin{equation}\tag {SE} \begin {cases} u(t) = A(t)u’(t) + B(t)u(t) + f(t) \text {for $t \in [0,T]$} \ u(0) = x \text {and} u’(0) = y. \end{cases} \end{equation}

We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection Φ \Phi of compactly supported functions in L p ( R s ) ( 1 ⩽ p ⩽ ∞ ) {L_p}({\mathbb {R}^s})\quad (1 \leqslant p \leqslant \infty ) , we consider the shift-invariant space S S generated by Φ \Phi and the family ( S h : h > 0 ) ({S^h}:h > 0) , where S h {S^h} is the h h -dilate of S S . We prove that ( S h : h > 0 ) ({S^h}:h > 0) provides L p {L_p} -approximation order r r only if S S contains all the polynomials of total degree less than r r . In particular, in the case where Φ \Phi consists of a single function φ \varphi with its moment ∫ φ ≠ 0 \int {\varphi \ne 0} , we characterize the approximation order of ( S h : h > 0 ) ({S^h}:h > 0) by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.

We apply Runge-Kutta methods to linear partial differential equations of the form u t ( x , t ) = L ( x , ∂ ) u ( x , t ) + f ( x , t ) {u_t}(x,t) = \mathcal {L}(x,\partial )u(x,t) + f(x,t) . Under appropriate assumptions on the eigenvalues of the operator L \mathcal {L} and the (generalized) Fourier coefficients of f, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the L r {L^r} -norm used to estimate the global error. The analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.

The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. A variance reduction technique for the Monte Carlo procedures is considered. Layer methods for linear and semilinear SPDEs are constructed and the corresponding convergence theorems are proved. The approach developed is supported by numerical experiments.

This paper presents an analysis of the modified Newton method as it is used in codes implementing implicit formulae for integrating stiff ordinary differential equations. We prove that near a smooth solution of the differential system, when the Jacobian is essentially negative dominant and slowly varying, the modified Newton iteration is contractive, converging to the locally unique solution-whose existence is hereby demonstrated-of the implicit equations. This analysis eliminates several common restrictive or unrealistic assumptions, and provides insight for the design of robust codes. 1. BACKGROUND, RESULTS, SIGNIFICANCE 1.1. Prototype stiff problems. Their salient properties. Note the structure o the solution of the model differential equation (1.1) y' =Ay+cost, A< -1, namely, y(t) = eit (y(0) + 2) + 1 2 (sin t A cos t). There is an initial transient of duration O(JA 1 l log JI)), after which the tern eit is not active and the solution is as smooth as cos t. Under suitable conditions, see [29] and infra, solutions of the stiff time varying linear system / ~~~~~N (1.2) y =B(t)y + g(t), y E R have the same structure: y(t) is the sum of a smooth particular solution y,(t' and a transient v(t). The transient, a solution of the homogeneous equation v (t) = B(t)v(t), v(O) = Y(O) (0) expires after a short time. Meanwhile, ys(t) and its derivative have bounds expressible in terms of 1 dvB dvg (1.3) B t)W B (t) v=O,)1. _ _ _ _ _ _ _ _ ~ ~~~~~~dtv Received September 28, 1989; revised August 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65L05, 65H10.

Let T f Tf denote any one of the usual classical or generalized Littlewood-Paley functions. This paper derives a BLO norm estimate for ( T f ) 2 {(Tf)^2} and a pointwise estimate for T f Tf .

The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80% may be achieved by using two-step Runge-Kutta methods instead of the classical onestep methods. These two-step Runge-Kutta methods were first introduced by Byrne and Lambert in 1966. They are designed to have the same number of function evaluations as the equivalent one-step schemes, and thus they are potentially more efficient. By solving a nonlinear programming problem, which is specified by stability requirements, optimal two-step schemes are designed. The optimization technique is applicable for stability regions of any shape.

Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional V V is constructed for the linear Volterra integrodifferential equation \[ x ˙ ( t ) = A x ( t ) + ∫ 0 T B ( t − τ ) x ( τ ) d t , t ≥ t 0 , x ( t ) = f ( t ) , 0 ≤ t ≤ t 0 \dot x\left ( t \right ) = Ax\left ( t \right ) + \int _0^T {B\left ( {t - \tau } \right )x\left ( \tau \right ) dt, \qquad t \ge {t_0}, \\ x\left ( t \right ) = f\left ( t \right ), \qquad 0 \le t \le {t_0}} \] . This functional, and its derivative V ˙ \dot V , is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for ordinary differential equations. The method of construction used suggests functionals which are useful for more general equations.

Using what may be considered as a natural generalization of the exponential function, some of the formalism of the theory of ordinary linear differential equations is extended to a class of linear partial differential equations among which are some important equations of mathematical physics. In \S 1 {\text {\S }}1 we give the definitions of the generalized exponentials and derive expressions for them. § 2 \S 2 is devoted to the study of some of the properties of the exponential in two independent variables. In § 3 \S 3 we derive the general solutions of some key partial differential equations using the method of recursion. The last section is devoted to extending the formalism of the method of variation of parameters to a class of linear partial differential equations.

An asymptotic theory for weakly nonlinear, highly oscillatory systems of ordinary differential equations leads to methods which are suitable for accurate computation with large time steps. The theory is developed for systems of the form \[ Z ′ = ( A ( t ) / ε ) Z + H ( Z , t ) , Z ( 0 , ε ) = Z 0 , 0 > t > T , 0 > ε ≪ 1 , \begin {array}{*{20}{c}} {{\mathbf {Z}}’= (A(t)/\varepsilon ){\mathbf {Z}} + {\mathbf {H}}({\mathbf {Z}},t),} \hfill \\ {{\mathbf {Z}}(0,\varepsilon ) = {{\mathbf {Z}}_0},\quad 0 > t > T,0 > \varepsilon \ll 1,} \hfill \\ \end {array} \] where the diagonal matrix A ( t ) A(t) has smooth, purely imaginary eigenvalues and the components of H ( Z , t ) {\mathbf {H}}({\mathbf {Z}},t) are polynomial in the components of Z with smooth t-dependent coefficients. Computational examples are presented.

The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval I is approximated, on each subinterval σ k {\sigma _k} corresponding to a partition π N {\pi _N} of I, by a linear combination U k ( x ) {U_k}(x) of exponential functions. The function U k ( x ) {U_k}(x) will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for σ k {\sigma _k} . The unknown vectors in U k ( x ) {U_k}(x) are computed recursively by requiring that U k ( x ) {U_k}(x) satisfy the given system at certain suitable points in σ k {\sigma _k} (collocation), with the additional condition that the collection of these functions { U k } \{ {U_k}\} represent a continuous function on I satisfying the given initial conditions.

This paper is concerned with the analysis of the stability of a class of one-step integration methods, originated by the Lanczos tau method and applicable to particular linear differential systems. It is proved that these methods are A-stable for every order.

Two examples of nonsolvable partial differential operators with multiple characteristics are presented. They illustrate the possibility that certain terms in the principal part may play no role in determining the solvability properties of the operator. This situation cannot occur for simple characteristics, where solvability is determined by the principal part.

A polynomial representation of the hybrid methods for solving ordinary differential equations is presented. The advantages of the representation are briefly discussed. Also it is shown that one step taken using a hybrid method is equivalent to two steps of the usual multistep methods; one step taken using an explicit method and the other taken using an implicit method. Therefore, the hybrid methods are really a special case of cyclic methods.

It is well known that difference schemes for hyperbolic equations display dispersion of waves. For a general dissipative difference scheme, we present a dispersive wave equation and show that the dispersions are essentially the same when the initial data is a step function.

This paper considers the problem of stabilizing a control system governed by a combination of partial and ordinary differential equations. The partial differential equations govern the evolution of the system in the interior of some spatial domain, the ordinary differential equations describe the evolution of the boundary data; the control enters through the boundary ordinary differential equations in a bilinear fashion. We provide sufficient conditions for feedback stabilization of such 'hybrid' systems. Two examples to wave equations with dynamic boundary conditions are provided. (Author)

This paper deals with the analysis of general one-step methods for the numerical solution of initial (-boundary) value problems for stiff ordinary and partial differential equations. Restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size. These restrictions are related to disks contained in the stability region of the method, and the errors are measured with arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of a diffusion-convection problem where the error growth is measured with the maximum norm.

A local stability analysis is given for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. The standard linear stability analysis is reviewed, then it is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and of the one-leg k-step numerical solution can be approximated computationally. Correspondence between the one-leg k-step solution and its associated linear k-step solution is shown, and two examples are given.

The paper presents decomposition formulas for solutions of a class of singular partial differential equations. The equations consist of products of iterated differential operators each of which involves a real parameter. The decomposition is given in terms of solutions corresponding to each operator.

Let R R be a constant N × N N \times N matrix and g ( t ) g(t) an N × N N \times N matrix of functions representable as absolutely convergent Laplace-Stieltjes transforms for t > 0 t > 0 . The paper gives sufficient conditions for certain solutions of the system y ′ = ( R + g ( t ) ) y y’ = (R + g(t))y to be expressed as Laplace-Stieltjes transforms or as linear combinations of such transforms with coefficients which are powers of t t .

Some L-stable fourth-order explicit one-step numerical integration formulas which require no matrix inversion are proposed to cope effectively with systems of ordinary differential equations with large Lipschitz constants (including those having highly oscillatory solutions). The implicit integration procedure proposed in Fatunla [11] is further developed to handle a larger class of stiff systems as well as those with highly oscillatory solutions. The same pair of nonlinear equations as in [11] is solved for the stiffness/oscillatory parameters. However, the nonlinear systems are transformed into linear forms and an efficient computational procedure is developed to obtain these parameters. The new schemes compare favorably with the backward differentiation formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].

The reasons a class of codes for solving ordinary differential equations might want to use an extremely small step size are investigated. For this class the likelihood of precision difficulties is evaluated and remedies examined. The investigation suggests a way of selecting automatically an initial step size which should be reliably on scale.

The existence of maximal and minimal solutions for initial-boundary value problems and the Cauchy initial value problem associated with L u = f ( x , t , u , ∇ u ) Lu = f(x,t,u,\nabla u) where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows f to be highly nonlinear, i.e., f locally Hölder continuous with almost quadratic growth in | ∇ u | |\nabla u| .

The recursive formulation of the Tau method is extended to the case of systems of ordinary differential equations, and an error analysis is given. Upper and lower error bounds are given in one of the examples considered. The asymptotic behavior of the error compares in this case with that of the best approximant by algebraic polynomials for each of the components of the vector solution. 1. Introduction. Interest in the Tau method (see (2), (3), (5)), for a long time regarded only as a tool for the construction of accurate approximations of a very restricted class of functions, has been enhanced by the availability of software for its computer implementation and by the possibility of using it in the numerical solution of complex nonlinear differential equations over extended intervals. The approxima- tion of the solution of such type of equations is achieved as a result of finding Tau approximants of a sequence of problems defined by linear differential equations. Details of this technique are given in (6). The subject of this paper is the extension of Ortiz' recursive formulation of Lanczos' Tau method (5) to the case of systems of differential equations and, more particularly, to its error analysis for such systems. Our error estimation technique is applied to three model examples for which the exact solution is readily available. It is discussed in general and with more detail when applied to the first of these examples. For the second example we show how to get upper and lower error bounds; we then compare these bounds with those given by Meinardus (4) for the best uniform approximation of each of the components of the vector solution by algebraic polynomials, to find that they are asymptotically equivalent. The third example is a differential equation with variable coefficients and a nonempty subspace of residuals; see (5). Results of numerical experiments on the use of the Tau method for the approxi- mate solution of systems of ordinary differential equations, with particular reference to stiff systems, are reported in (8). The problems discussed in this paper can also be considered in the framework of simultaneous approximation of a function and its derivatives with the Tau method; see (1).

The numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered. We discuss a finite-difference method which was already investigated by Fox in 1954 and Fox and Mitchell in 1957. Hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown. We further propose and discuss new Galerkin methods for initial value problems along the lines of the boundary value approach.

In [1] the Reduction to Scalar CDS method for the solution of separably stiff initial value problems is proposed. In this paper an improved version is given that is equivalent for linear problems but considerably superior for nonlinear problems. A naturally arising numerical example is given, for which the old version fails, yet the new version yields very good results. The disadvantage of the new version is that in the case of several dominant eigenvalues s > 1 s > 1 , say, a system of s nonlinear equations has to be solved, whereas the old version gives rise to s uncoupled nonlinear equations.

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from x n + k − 1 {x_{n + k - 1}} to x n + k {x_{n + k}} by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given. The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.

A theory for linear multistep schemes apolied to the initial value problem for a nonlinear first order system of differential equations with a singularity of the first kind is developed. Predictor-corrector schemes are also considered. The specific examples given are systems derived from partial differential equations in the presence of symmetry.

Asymptotic error expansions have been obtained for certain numerical methods for linear Volterra integro-differential equations. These results permit the application of extrapolation procedures. Computational examples are presented.

The differential equation u ( m ) ( t ) + p ( t ) u ( g ( t ) ) = 0 {u^{(m)}}(t) + p(t)u(g(t)) = 0 . where P is one-signed, is broken into four cases, according to the parity of m and the sign of p. In each case, an analysis is given of the effect g can have on oscillation properties, and oscillation and nonoscillation criteria are given.

A polynomial representation for the second-derivative linear multistep methods for solving ordinary differential equations is presented. This representation leads to an implementation of the second-derivative methods using the Nordsieck polynomial representation. Possible advantages of such an implementation are then discussed.

A class of A-stable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of y' and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementation of the fourth order method is described and the method applied to some stiff equations. Numerical comparisons are made with Gear's method. 1. Introduction. Recently, particular attention has been given to the study of A-stable methods for the solution of the m ordinary differential equations