Scalar Ordinary Differential Equation Research Articles

Analytical and numerical treatment of a singular initial value problem in avalanche modeling

We discuss a leading-edge model used in the computation of the run-out length of dry-flowing avalanches. The model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics. Existence, uniqueness and smoothness properties of the analytical solution are shown. We also prove the existence of a unique root of the solution. Moreover, we present a FORTRAN 90 code for the numerical computation of the run-out length. The code is based on a solver for singular initial value problems which is an implementation of the acceleration technique known as iterated defect correction based on the implicit Euler method.

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q(3)=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q(3)=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q(3)==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n≥5) ODEs which admitnontrivial contact symmetry algebras.

Total Stability of Scalar Differential Equations Determined from Their Limiting Functions

Stability properties of the zero solution of a scalar ordinary differential equation are studied using functions, called fences, which act as bounds to the solutions of the differential equation. It is shown, under simple restrictions, that these fences are bounds for the solutions of the perturbed differential equations as well. Hence one finds that total stability of the zero solution of a scalar ordinary differential equation is equivalent to the existence of these functions.

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n≥3, $$\frac{{\partial ^{2_n } u}}{{\partial x^{2_n } }} = f\left( {x,u,\frac{{\partial u}}{{\partial x}}, \ldots ,\frac{{\partial ^{2_{n - 1} } u}}{{\partial x^{2_{n - 1} } }}} \right),$$ we construct a contact two-form Π such that dΠ≡0 modΠ, if and only if Equation (1) admits a nondegenerate Lagrangian of order n. We show that the space of all nondegenerate Lagrangians for (1) is at most one-dimensional. The necessary and sufficient conditions for sixth-order and eighth-order scalar ordinary differential equation to admit a variational multiplier are found in terms of vanishing of a certain set of functions. The exact relationship between the Lie algebra of the classical infinitesimal contact symmetries of a variational Equation (1) and its the Lie subalgebra of infinitesimal divergence symmetries is established.

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.

Data Assimilation in Models with Convective Adjustment

Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified. In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution. Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented. Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.

CANONICAL FORMS FOR PARTICLE LAGRANGIANS OF THE LINEAR SECOND-ORDER EQUATION

Abstract A particle Lagrangian of a linear scalar second-order ordinary differential equation can admit maximally one of 1,2,3 or 5 Noether point symmetries. Moreover, canonical forms of particle Lagrangians of the linear equation are presented according to the number (and algebra) of Noether point symmetries they admit.

Travelling waves in radially symmetric parabolic systems

SynopsisIt is shown that all travelling or standing plane-wave solutions to certain radially symmetric parabolic systems may be found by solving a related scalar ordinary differential equation (ODE). The radially symmetric systems considered here are those whose reaction term is radially directed and points inward near infinity. The stability of these waves is also discussed. Many systems arising in the physical sciences are included in the class studied and so the classification and stability of the travelling waves has physical significance.

A periodically forced scalar ordinary differential equation

We use topological and classical methods to study the first order periodically excited differential equation, x ̇ = ƒ(x) + μg(t) . Given suitable conditions on ƒ(x), we prove results on the number, stability and bifurcations of periodic orbits. We give several examples to illustrate both the use of qualitative techniques and errors which may arise from the application of classical methods.

Arithmetic tests forA-stability,A[α]-stability, and stiff-stability

Arithmetic tests forA-stability,A[α]-stability, and stiff-stability are presented as special cases of a general stability test for numerical integration methods. The test evolves from extracted properties of the characteristic polynomial (in two variables) of the numerical method applied to the prototype scalar ordinary differential equation\(\dot x = qx\), Re {q}<0. The several steps of the test impose root clustering conditions — such as being Hurwitz — on a polynomial of one variable.