Paolo Amore

Improved Lindstedt–Poincaré method for the solution of nonlinear problems

Abstract We apply the Linear Delta Expansion (LDE) to the Lindstedt–Poincare (“distorted time”) method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator (Duffing equation), of the non linear pendulum and of more general anharmonic potentials. The approximate solutions found with this method converge more rapidly to the exact ones than in the simple Lindstedt–Poincare method.

Alternative perturbation approaches in classical mechanics

We discuss two alternative methods, based on the Lindstedt–Poincare technique, for the removal of secular terms from the equations of perturbation theory. We calculate the period of an anharmonic oscillator by means of both approaches and show that one of them is more accurate for all values of the coupling constant. We believe that present discussion and comparison may be a suitable exercise for teaching perturbation theory in advanced undergraduate courses on classical mechanics.

Exact and approximate expressions for the period of anharmonic oscillators

In this paper, we present a straightforward systematic method for the exact and approximate calculation of integrals that appear in formulae for the period of anharmonic oscillators and other problems of interest in classical mechanics.

A variational sinc collocation method for strong-coupling problems

We have devised a variational sinc collocation method (VSCM) which can be used to obtain accurate numerical solutions to many strong-coupling problems. Sinc functions with an optimal grid spacing are used to solve the linear and nonlinear Schrodinger equations and a lattice 4 model in (1 + 1). Our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision.

Presenting a new method for the solution of nonlinear problems

Abstract We present a method for the resolution of (oscillatory) nonlinear problems. It is based on the application of the linear delta expansion to the Lindstedt–Poincare method. By applying it to the Duffing equation, we show that our method substantially improves the approximation given by the simple Lindstedt–Poincare method.

Collocation method for fractional quantum mechanics

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on little sinc functions, which discretizes the Schrodinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a Wentzel–Kramers–Brillouin analysis is performed.

Solving the Helmholtz equation for membranes of arbitrary shape: numerical results

I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions. The matrix representation of the PDE obtained using this method is explicit and does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples, part of them are taken from the literature, and part of them new. When possible, I have tested the accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the L-shaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.

Asymptotic and exact series representations for the incomplete Gamma function

Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a,x) for any value of a or x. Applications of these formulas are discussed.

Comparison of alternative improved perturbative methods for nonlinear oscillations

Abstract We discuss and compare two alternative perturbation approaches for the calculation of the period of nonlinear systems based on the Lindstedt–Poincare technique. As illustrative examples we choose one-dimensional anharmonic oscillators and the Van der Pol equation. Our results show that each approach is better for just one type of model considered here.

Analytical formulas for gravitational lensing

In this paper we discuss a new method which can be used to obtain arbitrarily accurate analytical expressions for the deflection angle of light propagating in a given metric. Our method works by mapping the integral into a rapidly convergent series and provides extremely accurate approximations already to first order. We have derived a general first order formula for a generic spherically symmetric static metric tensor and we have tested it in four different cases.