Some solutions of the nonhomogeneous Bagley–Torvik equation

In this study, nonhomogeneous differential equation of the second order is considered, which contains fractional derivative (Bagley–Torvik equation), where the derivative order ranges within 1 and 2. This equation is applied in mechanics of oscillation processes. To study the equation, we use the Laplace transform, which allows us to obtain an image of the solution in an explicit form. Two typical kinds of functions of the right-hand side of the equation are considered. Numerical solutions are constructed for each of them. The solutions obtained are compared with experimental information on polymer concrete samples. The comparison allows for the conclusion about the adequacy of the numerical and analytical solutions to the nonhomogeneous Bagley–Torvik equation.

Differential Equation in a Banach Space Multiplicatively Perturbed by Random Noise

We consider the problem of finding the moment functions of the solution of the Cauchy problem for a first-order linear nonhomogeneous differential equation with random coefficients in a Banach space. The problem is reduced to the initial problem for a nonrandom differential equation with ordinary and variational derivatives. We obtain explicit formula for the mathematical expectation and the second-order mixed moment functions for the solution of the equation.

On the Singular Perturbations for Fractional Differential Equation

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.