Related papers: Symbolic analysis of second-order ordinary differe…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
A new analytical operator method is discussed which solves linear ordinary differential equations with regular singularities. Solutions are obtained in analytic series form and also in Mellin-Barnes-type contour integral form. Exact series…
The first order nonlinear ODE \dot \phi(t) + \sin\phi(t)=q(t),q(t)=B+A\cos\omega t, where A,B,\omega are real constants, is considered, the transformation converting it to a second order linear homogeneous ODE with polynoimial coefficients…
Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful…
We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…
We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our…
We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also…
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…