Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.Comment: 40 page

Correlation between Adomian and Partial Exponential Bell Polynomials

We obtain some recurrence relationships among the partition vectors of the partial exponential Bell polynomials. On using such results, the $n$-th Adomian polynomial for any nonlinear operator can be expressed explicitly in terms of the partial exponential Bell polynomials. Some new identities for the partial exponential Bell polynomials are obtained by solving certain ordinary differential equations using Adomian decomposition method

Decomposition of Differential Polynomials

We present an algorithm to decompose nonlinear differential polynomials in one variable and with rational functions as coefficients. The algorithm is implemented in Maple for the {em constant field} case. The program can be used to decompose differential polynomials with more than one thousand terms effectively

Exact Solution of Coupled Nonlinear PDEs Via Sumudu Decomposition Method

In this paper, we apply the Sumudu Decomposition Method on system of coupled nonlinear partial differential equations to calculate the analytical solutions in closed form. The nonlinear term can easily be handled with the help of He’s polynomials. The proposed technique is tested on four problems. Calculated results show the potential of the technique. Keyword: Nonlinear PDEs, He’s polynomials, Sumudu transform, Adomian decomposition metho

Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L^\ell_k$ in terms of the null spaces of mutually commuting second order factors.Comment: minor changes; we correct typos, add further explanation and clarify the treatment of the higher order operators in the case of even dimensions; results unchange

Spectral properties of operators using tridiagonalisation

A general scheme for tridiagonalising differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey-Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).Comment: 14 pages, corrections, to appear in Analysis and Application