Related papers: The Peano-Baker series
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
We propose a sufficient condition of the convergence of a power-log series that formally satisfies an algebraic ordinary differential equation (ODE) of arbitrary order. A general form and properties of the functional coefficients of such a…
In this expository style of writing I will give an introduction of Gr\"{o}bner bases and compute it for some algebras and then show how to use it to compute Hilbert series for algebras from chains.
Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…
We present a dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
A semi-Peano algebra is an algebra for which each operation is injective, and the images of the operations are pairwise disjoint. The most straightforward non-trivial kind of finitely presented semi-Peano algebra are algebras with a single…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
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…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
S. Banach \cite{Banach} proved that good differential properties of function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). On the other hand it is very well…
There is a recent interest for the verification of monadic programs using proof assistants. This line of research raises the question of the integration of monad transformers, a standard technique to combine monads. In this paper, we extend…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
The general term of the Poincare normalizing series is explicitly constructed for non-resonant systems of ODE's in a large class of equations. In the resonant case, a non-local transformation is found, which exactly linearizes the ODE's and…
In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential…
A generalization of the reduction technique for ODEs recently introduced by Gao and Liu is given. It is shown that the use of algebraic methods allows the extension of the procedure to much more general flows, as well as the derivation of…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
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…
For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B--series and corresponding growth functions are constructed. From these, convergence results based on the order of the…