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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…
A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…
We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on Bour-Mayer method to determine compatibility conditions via Jacobi-Mayer brackets. We solve…
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
We present a new approach for estimating parameters in rational ODE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a…
We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider i) classes of optimization problems of…
A new method to find first integrals of nonlinear differential equations in Jacobi-type form is presented. The basic idea of our approach is to use one-parameter perturbed motions to find well-conceived nonlocal constants that are conserved…
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…
In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use…
In [1,2] we have developed a method (we call it the S-function method) that is successful in treating certain classes of rational second order ordinary differential equations (rational 2ODEs) that are particularly `resistant' to canonical…
Link between the Painleve property and the first integrals of nonlinear ordinary differential equations in polynomial form is discussed. The form of the first integrals of the nonlinear differential equations is shown to determine by the…
The Numerov method for linear second-order differential equations is generalized to include equations containing a first derivative term. The method presented has the same degree of accuracy as the ordinary Numerov sixth-order method. A…
In this paper we use the comparison method for investigation of first order polynomial differential equations. We prove two comparison criteria for these equations. The proved criteria we use to obtain some global solvability criteria for…
Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and tau functions…
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 introduce a formalism to analyze partially defined functions between ordered sets. We show that our construction provides a uniform and conceptual approach to all the main definitions encountered in elementary real analysis including…
Parameter estimation for ordinary differential equations (ODEs) plays a fundamental role in the analysis of dynamical systems. Generally lacking closed-form solutions, ODEs are traditionally approximated using deterministic solvers.…
Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs…
In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…