Growth and oscillation theory of non-homogeneous linear differential equations

The Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations.

Exact and numerical solutions of linear and non-linear systems of fractional partial differential equations

The present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. The proposed scheme is based on Laplace transform and new homotopy perturbation methods. The fractional derivatives are considered in Caputo sense. To illustrate the ability and reliability of the method some examples are provided. The results ob...

Oscillation of Solutions of Linear Differential Equations

In this note we study the zeros of solutions of differential equations of the form u + pu = 0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given. §1. Number of zeros. Consider the linear differential equation u′′(x) + p(x) u(x) = 0 , where p(x) = 1 (1− x) , (1) on the interval −1 < x < 1. Two independent solutions are

On the Growth of Solutions of Some Non-homogeneous Linear Differential Equations

In this paper, we investigate the growth of solutions to the non-homogeneous linear differential equation f (k) +Ak−1e k−1f (k−1) + · · ·+A1ef ′ +A0ef = Fe, where Aj(z) 6≡ 0 (j = 0, 1, . . . , k − 1), F (z) 6≡ 0 are entire functions and a 6= 0, bj 6= 0 (j = 0, 1, . . . , k − 1) are complex numbers.

Growth of Solutions of Certain Non-homogeneous Linear Differential Equations with Entire Coefficients

In this paper, we investigate the growth of solutions of the differential equation f (k) +Ak−1 (z) f (k−1) + · · ·+A1 (z) f ′ +A0 (z) f = F, where A0 (z) , . . . , Ak−1 (z) , F (z) / ≡ 0 are entire functions, and we obtain general estimates of the hyper-exponent of convergence of distinct zeros and the hyper-order of solutions for the above equation.