The aim of this work is to study some noncommutative differential operators. We take an algorithmic approach as well as further developing the mathematics, and design and analyse algorithms to solve fundamental problems. We also give applications to differential equations. First we consider how to factor skew polynomials. These are polynomials in a differential or difference operator. Using the...

in this paper, boundary value problems of fractional order are converted into an optimal control problems. then an approximate solution is constructed from translations and dilations of a b-spline function such that the exact boundary conditions are satisfied. the fractional differential operators are taken in the riemann-liouville and caputo sense. several example are given and the optimal err...

on utilizing the spectral representation of selfadjoint operators in hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in hilbert spaces via a taylor's type expansion are given.

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

In the course of writing the book [9] and various papers [10, 11, 12, 13, 14, 15, 16] we encountered many q-differential equations but were frustrated by a lack of understanding about natural forms for such equations. One has operators of the type qKP or qKdV for example but even there, expressing the resulting equations (even via Hirota type equations or in bilinear form) seemed curiously diff...

Following Weyl’s account in The Classical Groups we develop an analogue of the (first and second) Fundamental Theorems of Invariant Theory for rings of differential operators: when V is a k-dimensional complex vector space with the standard SLkC action, we give a presentation of the ring of invariant differential operators D(C[V n])SLkC and a description of the ring of differential operators on...

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford’s spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein’s conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on ...

The quantization of a pair of commuting differential operators is a pair of non-commuting differential operators. Both at the classical and quantum level the flows of the KP hierarchy are defined and further one can consider switching, up to a sign, the ordering of the operators. We discuss the interaction of these operations with the quantization.

1. Elliptic Differential Operators 1 1.1. Partial differential operators 1 1.2. Sobolev spaces and Hölder spaces 10 1.3. Apriori estimates and elliptic regularity 20 1.4. Elliptic operators on compact manifolds 28 2. Non-linear Elliptic Equations 35 2.1. Banach manifolds and Fredholm operators 35 2.2. Moduli space of nonlinear elliptic equations 39 2.3. The Sard-Smale theorem and transversality...

A family (Dλ)λ∈C of differential operators on the sphere S n is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of S which preserve the smaller sphere Sn−1 ⊂ S. The family of conformally covariant differential operators from S to Sn−1 introduced by A. Juhl is obtained by composing these operators on S and taking restrictions to Sn−1.