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Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…
Two novel robust nonlinear stochastic full pose (i.e, attitude and position) estimators on the Special Euclidean Group SE(3) are proposed using the available uncertain measurements. The resulting estimators utilize the basic structure of…
A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear…
Solving semiparametric models can be computationally challenging because the dimension of parameter space may grow large with increasing sample size. Classical Newton's method becomes quite slow and unstable with intensive calculation of…
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional…
The single-step explicit time integration methods have long been valuable for solving large-scale nonlinear structural dynamic problems, classified into single-solve and multi-sub-step approaches. However, no existing explicit single-solve…
We compare three approaches to posing the index 3 set of differential algebraic equations (DAEs) associated with the constrained multibody dynamics problem formulated in absolute coordinates. The first approach works directly with the…
We present two approaches to system identification, i.e. the identification of partial differential equations (PDEs) from measurement data. The first is a regression-based Variational System Identification procedure that is advantageous in…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
The principle of finding an integrating factor for a none exact differential equations is extended to a class of third order differential equations. If the third order equation is not exact, under certain conditions, an integrating factor…
Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems…
In this paper we consider the numerical approximation of a semilinear reaction-diffusion model problem (PDEs) by means of reduced order methods (ROMs) based on proper orthogonal decomposition (POD). We focus on the time integration of the…
Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the…
An Exactly-Solvable (ES) potential on the sphere $S^n$ is reviewed and the related Quasi-Exactly-Solvable (QES) potential is found and studied. Mapping the sphere to a simplex it is found that the metric (of constant curvature) is in…
In this paper, the simultaneous identification of damping or anti-damping coefficient and initial value for some PDEs is considered. An identification algorithm is proposed based on the fact that the output of system happens to be…
Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities…
We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional {\em quadratic} first integrals, thus constructing a large…
Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter's invariance…
We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…