Logarithmic A-hypergeometric series

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular [Formula: see text]-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic [Formula: see text]-hypergeometric series solutions, we adopt a different perturbation in this paper.

The Picard-Lindel of’s theorem at a regular singular point

We consider initial value problems of the form..., where f : [−a, b] × U → Cn is a continuous function in its variables and U ⊂ Cn is an open set. D(x) is an n × n diagonal matrix whose first n − m diagonal entries are 1 and the last m diagonal entries are x, with m = 0, 1, 2, . . . or n. This is an initial value problem where the initial condition is given at a regular singular point of the system of differential equations. The main result of this paper is an existence and uniqueness theorem for the solution of this initial value problem. It is shown that this problem has a unique solution and the Picard-Lindelof’s expansion converges to that solution if the function ¨ F(y, x) := xD−1 (x)f (x, y) is Lipschitz continuous in the variables y with Lipschitz constant L of the form L = N + Mxp for a certain p > 0, M > 0 and 0 ≤ N < 1. When we add the condition y (s) ∈ C[−a, b], s ∈ N, to the formulation of the problem and the Taylor polynomial of y at x = 0 and degree s − 1 is available from the differential equation, then the same conclusion is true with a less restrictive condition upon N: 0 ≤ N < s + 1. The standard Picard-Lindelof’s ¨ theorem is the particular case of the problem studied here obtained for m = 0 (D(x) is the identity matrix), N = 0, p = 1 and M is the Lipschitz constant of f (x, y).

Numerical Simulation of Emden-Fowler Type Equations Using Variational Iteration Algorithm

The main objective of this article is to present a reliable algorithm to determine exact and approximate solutions of the generalized Emden-Fowler type equations. The algorithm mainly is based on He’s variational iteration method (VIM) with an alternative framework designed to overcome the difficulty of the regular singular point at x = 0. In this method, general Lagrange multipliers are introduced to construct a correction for the problem. The multipliers in the functional can be identified optimally via the variational theory. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.