One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by continuous generalizations of Fibonacci sequence recently discussed Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with Friedmann in FLRW homogeneous cosmology. The method allows introduction sequences, hence Darboux-deformed (noninteger) numbers. Consi...

This paper aims to design a generalized cylinder with geodesic base curve according the Darboux frame in Euclidean 3-space. A is special ruled surface that constructed by continuous fixed motion of generator line called ruling along given curve. The necessary and sufficient conditions for be are studied. main results show an osculating whose helical geodesic, rulings directed unit vector.

n this paper, we apply picard’s iteration method followed by adomian decomposition method to solve a nonlinear singular cauchy problem of euler- poisson- darboux equation. the solution of the problem is much simplified and shorter to arriving at the solution as compared to the technique applied by carroll and showalter (1976)in the solution of singular cauchy problem.

1. The Fundamental Theorem of Calculus 1 2. Building the Definite Integral 5 2.1. Upper and Lower Sums 5 2.2. Darboux Integrability 7 2.3. Verification of the Axioms 10 2.4. An Inductive Proof of the Integrability of Continuous Functions 12 3. Further Results on Integration 13 3.1. The oscillation 13 3.2. Discontinuities of Darboux Integrable Functions 14 3.3. A supplement to the Fundamental Th...

It has been recently discovered that exceptional families of SturmLiouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an examp...

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all isothermic surfaces: they can be characterized by the fact that their parallel c...