Showing papers on "Ordinary differential equation published in 1983"

Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria

Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

The nonlinear development of Görtler vortices in growing boundary layers

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

On the Construction of Monopoles

Abstract: We show that any self-dual SU (2) monopole may be constructed either by Ward's twistor method, or Nahm's use of the ADHM construction. The common factor in both approaches is an algebraic curve whose Jacobian is used to linearize the non-linear ordinary differential equations which arise in Nahm's method. We derive the non-singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions.

On solitary waves running down an inclined plane

Abstract: We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [gsim ] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.

Ballooning mode spectrum in general toroidal systems

Abstract: A WKB formalism for constructing normal modes of short‐wavelength ideal hydromagnetic, pressure‐driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed. No incompressibility approximation is made. A dispersion relation is obtained from the eigenvalues of a fourth‐order system of ordinary differential equations to be solved by integrating along a line of force. Higher‐order calculations are performed to find the amplitude equation and the phase change at a caustic. These conform to typical WKB results. In axisymmetric systems, the ray equations are integrable, and semiclassical quantization leads to a growth rate spectrum consisting of an infinity of discrete eigenvalues, bounded above by an accumulation point. However, each eigenvalue is infinitely degenerate. In the nonaxisymmetric case, the rays are unbounded in a four‐dimensional phase space, and semiclassical quantization breaks down, leading to broadening of the discrete eigenvalues and the accumulation point of the axisymmetric unstable spectrum into continuum bands. Analysis of a model problem indicates that the broadening of the discrete eigenvalues is numerically very small, the dominant effect being broadening of the accumulation point.

Elementary first integrals of differential equations

Abstract: It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.

Amplitude Equations for Systems with Competing Instabilities

Abstract: We describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities. The method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail. The basic idea is to extend the Krylov–Bogolyubov–Mitropolsky method of nonlinear mechanics to bifurcation problems.

Advanced Engineering Mathematics

Abstract: Part One - Ordinary Differential Equations. 1. First Order Differential Equations. 2. Second Order Differential Equations. 3. The Laplace Transform. 4. Series Solutions. 5. Numerical Approximation Of Solutions. 6. Sturm-Liouville Theory, Eigenfunction Expansions And Special Functions. Part Two - Vectors And Linear Algebra. 7. Vectors And Vector Space. 8. Matrices, Determinants And Systems Of Linear Equations. 9. Eigenvalues And Diagonalization. Part Three - Systems Of Differential Equations And Qualitative Methods. 10. Linear Systems Of Differential Equations. 11. Non-linear Differential Equations And Qualitative Methods. Part Four - Vector Analysis. 12. Vector Differential Calculus. 13. Vector Integral Calculus. Part Five - Fourier Analysis And Boundary Value Problems. 14. Fourier Series And Integrals. 15. Fourier Transforms. 16. Partial Differential Equations And Boundary Value Problems. Part Six - Complex Analysis. 17. Complex Numbers And Complex Fractions. 18. Complex Integration. 19. Conformal Mappings And Some Applications.

The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae

Abstract: A class of modified extended backward differentiation formulae (MEBDF) suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced. The computational aspects of this new class of formulae are examined in detail. In particular, algorithms for varying both order and stepsize are given and this leads to a variable step/variable order process using highly stable formulae of order 2–8. Extensive numerical results for the well known DETEST set are given and on this basis a comparison is made between a code incorporating MEBDF, a code due to Hindmarsh (based on conventional backward differentiation formulae) and a code due to Skeel and Kong (based on blended linear multistep methods). It is shown that on this test set the MEBDF code is very reliable and is competitive with the other two codes on a significant class of problems. A computer listing of the MEBDF code used to obtain the results presented in this paper is available from the author.

Contractivity in the numerical solution of initial value problems

Abstract: Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|?|U(0)| (t?0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh>0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp?1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp>1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.

Lie transformations, nonlinear evolution equations, and Painlevé forms

Abstract: We present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one‐parameter continuous (Lie) group of transformations. It is shown that, in general, the corresponding invariant variables (the subclass of which is the usual similarity variables) lead to ordinary differential equations of Painleve type in the case of inverse scattering transform solvable equations, as conjectured by Ablowitz, Ramani, and Segur. This is found to be also true for certain higher spatial dimensional versions such as the Kadomtsev–Petviashivilli, two dimensional sine–Gordon, and Ernst equations. For the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points.

Sustained Resonance for a Nonlinear System with Slowly Varying Coefficients

Abstract: J. Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364–373; 26 (1974), pp. 638–669] studied resonance for a spinning reentry vehicle using a model system of ordinary differential equations with slowly varying coefficients. He and L. Lewin [SIAM J. Appl. Math., 35 (1978), pp. 738–754] gave formal multiple-time-scale expansions and numerical results to give a description of a mechanism for capture in sustained resonance. J. Sanders [SIAM J. Math. Anal., 10 (1979), pp. 1220–1243] studied these equations more rigorously using the method of averaging, but still could not prove the existence of sustained resonance. This paper continues the study of these equations using higher order averaging and the Melnikov method and shows rigorously that capture in sustained resonance does take place for some initial conditions. The Melnikov method measures the opening of a saddle connection for a small perturbation in terms of an integral. Since it has usually been applied to perturbations which depend periodically on time,...

Steepest descent moment method for three-dimensional magnetohydrodynamic equilibria

Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

On the qualitative behaviour of populations with age-specific interactions

Abstract: A mathematical model describing the age-dependent dynamics of a population is studied. Under very general assumptions, existence of nonzero equilibrium solutions is proven. For some special kind of rate law it is shown that this model can asymptotically be described by means of a simpler system of ordinary differential equations. An example is given where the model admits nonconstant time-periodic solutions.

Additive Runge-Kutta methods for stiff ordinary differential equations

Abstract: Certain pairs of Runge-Kutta methods may be used additively to solve a system of n differential equations x' = J(t)x + g(t, x). Pairs of methods, of order p < 4, where one method is semiexplicit and A-stable and the other method is explicit, are obtained. These methods require the LU factorization of one n X n matrix, and p evaluations of g, in each step. It is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations. 1. Introduction. In a recent article (2) the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x' = f(t, x), x(a) = xo, a - t - b,

On Spline Basis Selection for Solving Differential Equations

Abstract: The suitability of B-splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged. Two alternative local solution representations are considered in the context of collocating ordinary differential equations: “Hermite-type” and “Lmonomial”. Both are much easier and shorter to implement and somewhat more efficient than B-splines.A new condition number estimate for the B-splines and Hermite-type representations is presented. One choice of the Hermite-type representation is experimentally determined to produce roundoff errors at most as large as those for B-splines. The monomial representation is shown to have a much smaller condition number than the other ones, and correspondingly produces smaller roundoff errors, especially for extremely nonuniform meshes. The operation counts for the two local representations considered are about the same, the Hermite-type representation being slightly cheaper. It is concluded that both representations are preferable,...

Branch switching in bifurcation problems for ordinary differential equations

Abstract: The problem of switching branches in boundary-value problems of ordinary differential equations is considered. Three non-local methods for calculating emanating solutions near a nontrivial bifurcation point are proposed. These methods calculate one solution on an emanating branch (without a priori exact knowledge of the bifurcation point). Other solutions on the branch can be obtained by global continuation. The methods are convenient as they consist in solving boundary-value problems by standard software. The construction of an initial approximation of the emanating solution is outlined. A characteristic feature of the proposed methods is that they can be easily automated; the user can avoid nearly all preparatory work. The methods are tested on several examples arising in different application areas.

Chaos and Arnold diffusion in dynamical systems

Abstract: Chaotic motion refers to complicated trajectories in dynamical systems. It occurs even in deterministic systems governed by simple differential equations and its presence has been experimentally verified for many systems in several disciplines. A technique due to Melnikov provides an analytical tool for measuring chaos caused by horseshoes in certain systems. The phenomenon of Arnold diffusion is another type of complicated behavior. Since 1964, it has been playing an important role for Hamiltonian systems in physics. We present a tutorial treatment of this work and its place in dynamical systems theory, with an emphasis on results that can be checked in specific systems. A generalization of the Melnikov technique has been recently developed to treat n -degree of freedom Hamiltonian systems when n > 3 . We extend the Melnikov technique to certain non-Hamiltonian systems of ordinary differential equations. The extension is made with a view to applications in the physical sciences and engineering.

Order of Convergence of One-Step Methods for Volterra Integral Equations of the Second Kind

Abstract: Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to “extended” one-step methods for Volterra integral equations of the second kind. Furthermore, the convergence of “mixed” one-step methods is investigated. For both types general Volterra–Runge–Kutta methods are considered as examples.

On the detonation of a combustible gas

Abstract: This paper is concerned with the existence of detonation waves for a combustible gas. The equations are those of a viscous, heat conducting, polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas (see (1)). The reaction is assumed to be of the simplest form: A B, i.e., there is a single product and a single reactant. The main result (see Theorem 2.1) is a rigorous existence theorem for strong, and under certain conditions, weak detonation waves for explicit ranges of the viscosity, heat conduction, and species diffusion coefficients. In other words, a class of admissible " viscosity matrices" is determined. The problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points. The proof employs topological methods, including Conley's index of isolated invariant sets.

On strongly indefinite functionals with applications

Abstract: Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach-avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem. 0. Introduction. This paper is devoted to the critical point theory of strongly indefinite functionals in a real Hilbert space H. Here a strongly indefinite functional is a Cl-map H -* R which is neither bounded from above or below not even modulo a finite-dimensional subspace. Such functionals arise for example in the study of periodic solutions of the one-dimensional wave equation, in the study of periodic solutions of Hamiltonian systems of ordinary differential equations, in the existence theory for systems of elliptic equations, etc. To find the critical points of strongly indefinite functionals different methods were used. For example, Rabinowitz studied the restrictions of the functional to finite-dimensional subspaces, [1, 2], and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems. Amann [3] reduced the indefinite functionals by means of a saddlepoint-reduction to a finite-dimensional problem. Of course, this is only possible under some strong hypotheses. Under certain assumptions the functional can be replaced by a dual variational problem, having some better properties. (See for example [4 or 5].) The first direct approach was given by Benci and Rabinowitz [6]. The essential in their paper is the observation that more detailed information about the special form of the deformations (a substitute for the gradient flow) associated to the functional is needed. They constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation. This method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets. A main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly Received by the editors March 30, 1981 and, in revised form, November 30, 1981. 1980 Mathematics Subject Classification. Primary 58E05; Secondary 47H25, 35L05.