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In a recent paper (A. Mitsopoulos and M. Tsamparlis, J. Geom. Phys. 170, 104383, 2021), a general theorem is given which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of…
We consider the generic quadratic first integral (QFI) of the form $I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ and require the condition $dI/dt=0$. The latter results in a system of partial differential equations…
An autonomous dynamical system is described by a system of second order differential equations whose solution gives the trajectories of the system. The solution is facilitated by the use of first integrals (FIs) that are used to reduce the…
We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent…
In general, a system of differential equations is integrable if there exist `sufficiently many' first integrals (FIs) so that its solution can be found by means of quadratures. Therefore, the determination of the FIs is an important issue…
The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects…
We consider the time-dependent dynamical system $\ddot{q}^{a}= -\Gamma_{bc}^{a}\dot{q}^{b}\dot{q}^{c}-\omega(t)Q^{a}(q)$ where $\omega(t)$ is a non-zero arbitrary function and the connection coefficients $\Gamma^{a}_{bc}$ are computed from…
In [1], we have presented the theoretical background for finding the Elementary Invariants for a 3D system of first order rational differential equations (1ODEs). We have also provided an algorithm to find such Invariants. Here we introduce…
A theorem is derived which determines higher order first integrals of autonomous holonomic dynamical systems in a general space, provided the collineations and the Killing tensors -- up to the order of the first integral -- of the kinetic…
We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only…
In this thesis we introduce the concept of a guided dynamical system, and exploit this idea to solve various problems in functional equations and PDE's. Our main results are 1) a necessary and sufficient condition for unique-solvability of…
We consider autonomous holonomic dynamical systems defined by equations of the form $\ddot{q}^{a}=-\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$, where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly…
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet…
Integrable partial differential equation (PDE) systems are of great interest in natural science, but are exceedingly rare and difficult to discover. To solve this, we introduce OptPDE, a first-of-its-kind machine learning approach that…
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results…
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of…
We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…