Problems For Second-order Differential Equations Research Articles

High order adaptive methods of Nyström-Cowell type

A class of unconditionally stable multistep methods is discussed for solving initial-value problems of second-order differential equations which have periodic or quasiperiodic solutions. This situation frequently occurs in celestial mechanics, in nonlinear oscillations and various other situations. The methods depend upon a parameter ω > 0, and integrate exactly trigonometric functions along with algebraic polynomials. In this paper we show a procedure for the construction of adaptive Nyström-Cowell formulas of arbitrarily high order of accuracy, and reduce to the classical Nyström-Cowell methods for ω = 0. Our methods compare advantageously with other methods.

Differential Operators, Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract With each second-order differential equation Z in the evolution space J 1(M n+1) we associate, using the natural f(3, −1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J 1(M n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J 1(M n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincare–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.

Iterative method for solving singular spectral problems for second-order differential equations

A fast-converging iterative method is proposed for determining the eigenvalues of singular problems for differential equations of second order. Convergence conditions are derived and a bound on the rate of convergence is given.

Equivalence of a part of derived chains of boundary-value problems for second-order ordinary differential equations

A method is proposed for obtaining criteria for the equivalence of a part of derived chains of boundary-value problems for second-order differential equations with a spectral parameter in boundary conditions.

Matrix methods for solving the boundary-value problem for a second-order differential equation with delayed argument

Zero-rank matrix numerical differentiation algorithms are applied to construct efficient numerical-analytical methods (so-called zero-rank matrix methods) to find the eigenvalues and eigenfunctions of boundary-value problems for second-order differential equations with a delayed argument. The proposed methods are analyzed.

Nonuniform transmission-line synthesis using inverse eigenvalue analysis

Recent research on inverse eigenvalue problems for second-order differential equations on a finite interval has made it possible to reconstruct the coefficients in such an equation from spectral data. These data consist of the eigenvalues and the end values of the normalized eigenfunctions. This result is applied to reconstruct the axial profile of a finite transverse electromagnetic transmission line, and it is shown that the normalized scattering matrix of the line can also be expressed, in closed form, in terms of the spectral data. Asymptotic considerations indicate that for any continuous line the eigenvalues and end values will converge to those of a uniform line. This suggests that one should consider lines for which all the eigenvalues and all but the first n end values are equal to those of the uniform line. Numerical results show that this family of nonuniform lines contains members with useful and controllable transmission characteristics. Experimental results confirm the accuracy of the analysis. >

Geometric Lagrangian approach to first-order systems and applications

The geometric theory of presymplectic systems is developed to study both the Lagrangian and Hamiltonian formulations of a system described by a Lagrangian linear in the velocities. The results are used to study some related problems for second-order differential equations and regular Lagrangians.

On a method of solving problems of the bending of rods and plates of piecewise-constant stiffness

By using an effective procedure /2/ expansions were constructed in /1/ of an arbitrary function in the eigenfunction integral of a boundary value problem for second-order differential equations given in an interval with different sections of stepwise-changing physical characteristics. This procedure is applied below to construct the expansion of an arbitrary function in an eigenfunction series for fourth-order equations given in a finite interval with an arbitrary quantity of stepwise-varying characteristics. The finite integral transforms obtained in such a manner can be utilized in solving different bending problems for rods or plates with piecewise-varying stiffness, which enables us to formalize the procedure for solving specific problems. The need to solve differential equations given in a composite interval also arises in a study of the state of stress of layered elastic media. A general approach to the solution of problems of mechanics for layered media is proposed in /3/, and is based on extending the method of initial parameters and results in an effective calculational procedure.

A Sixth-order P-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

A Sixth-order <i>P</i>-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

A Note on Finite Difference Methods for Solving the Eigenvalue Problems of Second-Order Differential Equations

A Note on Finite Difference Methods for Solving the Eigenvalue Problems of Second-Order Differential Equations

A note on finite difference methods for solving the eigenvalue problems of second-order differential equations

A note on finite difference methods for solving the eigenvalue problems of second-order differential equations