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We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
We generate hierarchies of derivative nonlinear Schr\"odinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established…
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…
This work introduces a methodology for generating linear operators that approximately represent nonlinear systems of perturbed ordinary differential equations. This is done through the application of classical perturbation theory via the…
Simulating quantum systems with their environments often requires non-unitary operations, and mapping these to quantum devices often involves expensive dilations or prohibitive measurement costs to achieve desired precisions. Building on…
Using repeated Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we transform the coupled, integral-differential NLO singlet DGLAP equations first into coupled differential…
We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…
Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
In this note we extend the Differential Transfer Matrix Method (DTMM) for a second-order linear ordinary differential equation to the complex plane. This is achieved by separation of real and imaginary parts, and then forming a system of…
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions;…
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…
We construct a sheaf theoretic and derived geometric machinery to study nonlinear partial differential equations and their singular supports. We establish a notion of derived microlocalization for solution spaces of non-linear equations and…
Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in $\mathbb{R}^{m+1}$, whose solutions include polymonogenic and polyharmonic functions, among others. In this work,…
In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…