In this work, we construct commutative rings of two variable matrix differential operators that are isomorphic to a ring of meromorphic functions on a rational manifold obtained from the CP 1×CP 1 by identification of two lines with the pole on a certain rational curve. The commutation condition for differential operators is equivalent to a system of non-linear equations in the operators’ coeff...

Received: January 10, 2011 Accepted: February 10, 2011 doi:10.5539/jmr.v3n3p32 Abstract In this paper we obtained the formula for the common solution of the linear differential equation of the second order with the variable coefficients in the more common case. We also obtained the formula for the solution of the Cauchy problem.

We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the exis...

A new dual-Petrov-Galerkin method is proposed, analyzed and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the “dual” boundary conditions. The method leads to linear systems which are sparse for problems with cons...