Answer to Question #51. Applications of third-order and fifth-order differential equations

Oscillation of Third-order Functional Differential Equations

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation (E) ˆ a(t) ˆ x ′′(t) ̃ γ ̃ ′ + q(t)f(x [τ (t)]) = 0. Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of (E). Obtained results essentially improve and complement earlier ones. Variou...

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show...

A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations

This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...

Fractional-order Legendre wavelets and their applications for solving fractional-order differential equations with initial/boundary conditions

In this manuscript a new method is introduced for solving fractional differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use fractional-order Legendre wavelets and operational matrix of fractional-order integration. First the fractional-order Legendre wavelets (FLWs) are presented. Then a family of piecewise functions is proposed, based on whi...

ON TE EXISTENCE OF PERIODIC SOLUTION FOR CERTAIN NONLINEAR THIRD ORDER DIFFERENTIAL EQUATIONS