Second-order Homogeneous Linear Differential Equation Research Articles

Solving Second-Order Homogeneous Linear Differential Equations in Terms of the Tri-Confluent Heun’s Function

In this paper, we state an algorithm that checks whether a given second-order linear differential equation can be reduced to the tri-confluent Heun’s equation. The algorithm provides a method for finding solutions of the form exp(∫r(x)dx)·HeunT(q,α,γ,δ,ϵ,f(x)), where the parameters α,β,λ∈C, the functions r,f∈C(x), and f are not constant.

Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

We propose a new method to compute the unipotent radical Ru(H) of the differential Galois group H associated to a parameterized second-order homogeneous linear differential equation of the form∂2∂x2Y−qY=0, where q∈F(x) is a rational function in x with coefficients in a Π-field F of characteristic zero, and Π is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of Ru(H) to solving a creative telescoping problem, whose effective solution depends on the assumption that the maximal reductive quotient H/Ru(H) is a Π-constant linear differential algebraic group. When this condition is not satisfied, one can effectively compute a new set of parametric derivations Π′ such that the associated differential Galois group H′ has the property that H′/Ru(H′) is Π′-constant, and such that Ru(H) is defined by the same differential equations as Ru(H′). Thus the computation of Ru(H) is reduced to the effective computation of Ru(H′). We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.

On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method

On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method

ANHARMONIC OSCILLATOR ENERGIES BY THE ASYMPTOTIC ITERATION METHOD

In a previous paper (J. Phys. A36, 11807 (2003)), the "asymptotic iteration method" was introduced and developed for solving second-order homogeneous linear differential equations. In this paper we use the method to study the quantum quartic anharmonic oscillator problem. We obtain accurate eigenvalues for both small and large coupling parameters.

Perturbation theory in a framework of iteration methods

In a previous paper [J. Phys. A 36 (2003) 11807], we introduced the ‘asymptotic iteration method’ for solving second-order homogeneous linear differential equations. In this Letter, we study perturbed problems in quantum mechanics and we use the method to find the coefficients in the perturbation series for the eigenvalues and eigenfunctions directly, without first solving the unperturbed problem.

Construction of exact solutions to eigenvalue problems by the asymptotic iteration method

We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schroedinger problems with Coulomb, harmonic oscillator, or Poeschl-Teller potentials, as well as the special eigenproblems studied recently by Bender et al [J. Phys. A: Math. Gen. 34 9835 (2001)] and generalized in the present paper to higher dimensions.

On the Siegel Conjecture for Second-Order Homogeneous Linear Differential Equations

The structure of the set of E-functions satisfying second-order homogeneous linear differential equations with coefficients from \(\mathbb{C}(z)\) is described.

Asymptotic iteration method for eigenvalue problems

An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.

On the WKBJ approximation

A simple approach employing properties of solutions of differential equations is adopted to derive an appropriate extension of the WKBJ method. Some of the earlier techniques that are commonly in use are unified, whereby the general approximate solution to a second-order homogeneous linear differential equation is presented in a standard form (SF) that is valid for all orders. In comparison to other methods, the present one is shown to be leading in the order of iteration, and thus possibly has the ability of accelerating the convergence of the solution.

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.