Related papers: A Novel Solution to the Frenet-Serret Equations
The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural…
This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow is, to leading order, shearless and viscosity, negligible. The Froude number is large, so that…
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…
In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are…
Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of…
This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the euclidean space but…
In this thesis four separate problems in general relativity are considered, divided into two separate themes: coordinate conditions and perfect fluid spheres. Regarding coordinate conditions we present a pedagogical discussion of how the…
We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by…
By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be…
We study stationary solutions to the Keller--Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and…
This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
We consider a linear inhomogeneous fractional evolution equation which is obtained from a Cauchy problem by replacing its first-order time derivative with Caputo's fractional derivative. The operator in the fractional evolution equation is…
We propose a numerical scheme to solve the time dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element discretization in space. Since the…
We present a generic solution to the fundamental problem of how to connect two points in a plane by a smooth curve that goes through these points with a given slope. The smoothness of any curve depends both on its curvature and its length.…
We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders…
The cable equation describes the voltage in a straight cylindrical cable, this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some…
We study the properties of $\text{CAT}(\kappa)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(\kappa)$ condition locally. The main facts about…
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2{\alpha}.
The Bethe-Salpeter equation plays a crucial role in understanding the physics of correlated fermions, relating to optical excitations in solids as well as resonances in high-energy physics. Yet, it is notoriously difficult to control…