Related papers: Solutions of DEs and PDEs as Potential Maps Using …
Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad…
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…
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincar\'e-Einstein manifolds, and other conformally singular geometries. These "generalised geodesics" agree with…
As a sequel to our previous work [C. Ma, Q. Zhang and W. Zheng, SIAM J. Numer. Anal., 60 (2022)], [C. Ma and W. Zheng, J. Comput. Phys. 469 (2022)], this paper presents a generic framework of arbitrary Lagrangian-Eulerian unfitted finite…
Coupled partial differential equations defined on domains with different dimensionality are usually called mixed dimensional PDEs. We address mixed dimensional PDEs on three-dimensional (3D) and one-dimensional domains, giving rise to a…
We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been…
The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of Carnot group type. As such, $J^k$ admits a submetry (\sR submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are…
This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides a detailed analysis. More precisely,…
This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from…
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical…
In this work we use Lie group theoretic methods and the theory of prolonged group actions to study two fully nonlinear partial differential equations (PDEs). First we consider a third order PDE in two spatial dimensions that arises as the…
Certain solutions of autonomous PDEs without any boundary conditions describing the spatiotemporal evolution of a dependent variable in an unbounded spatial domain can be characterised as a travelling wave moving with constant speed. In the…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine…
We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Fr\"olicher-Nijenhuis theory on the first jet bundle, $J^1\pi$. We prove…
Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems…
One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the…
In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via…
We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian…
Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to…