Related papers: Eikonal 2.0: General Solutions for Modified Eikona…
A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update…
Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter $N$ or continuous parameter $x$. The analysis of these equations reveals to which order they…
As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a $T-$periodic solution to the second order differential equation \begin{equation*} u"=\frac{h(t)}{u^{\lambda}} \end{equation*} are…
The purpose of this paper is to give some explicit formulas involving M\"obius functions, which may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums…
We consider a singular fractional differential equation involving generalized Katugampola derivative and obtain the existence and uniqueness of its solution. A scheme for uniformly approximating solution is constructed by using Picard…
We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale…
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…
We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
This paper is devoted to ordinary differential equations of the form $$y''=a^3(x,y)y'^3+a^2(x,y)y'^2+a^1(x,y)y'+a^0(x,y)$$ The algebra of all differential invariants of point transformations is constructed for these equations in general…
In this article we consider a system of eikonal equations with a Dirichlet boundary condition. We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient.
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the…
A general fusion method to find solutions to the reflection equation in higher spin representations starting from the fundamental one is shown. The method is illustrated by applying it to obtaining the $K$ diagonal boundary matrices in an…
The paper is devoted to the multiple chordal Loewner differential equation with different driving functions on two time intervals. We obtain exact implicit or explicit solutions to the Loewner equations with piecewise constant driving…
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter.
We use a particular fractional generalization of the ordinary differential equations that we apply to the Riccati equation of constant coefficients. By this means the latter is transformed into a modified Riccati equation with the free term…
In this work, we give the general solution sequential linear conformable fractional differential equations in the case of constant coefficients for {\alpha}(\in)(0,1]. In homogeneous case, we use a fractional exponential function which…
In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of…