Related papers: Eikonal 2.0: General Solutions for Modified Eikona…
We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are…
Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} \chi(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) \chi (n) f(n)$ have been obtained in a unified manner, where $(\chi (n))$ is a periodic…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise,…
In present paper, we establish sufficient conditions for existence and stability of solutions for system of nonlinear implicit fractional differential equations. The main techniques are based on method of successive approximations. Finally,…
We investigate the solutions of the second-order difference equation $u_{n+2}=(au_n)/(1+bu_nu_{n+1})$ using a group of transformations (Lie symmetries) that leaves the solutions invariant.
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
In this work we introduce two different generalizations of the Fokker-Planck equation in (1+1) dimensions by replacing the spatial derivatives in terms of generalized Dunkl-type derivatives involving reflection operators. As applications of…
In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. The new method is an iterative two-scale method that uses a parareal-like update scheme in combination with…
We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We demonstrate…
The existence and stability results for a class of fractional differential equations involving generalized Katugampola derivative are presented herein. Some fixed point theorems are used and enlightening examples of obtained result are also…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity…
We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless they share with BV…
In this paper, some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worthy mentioning that when $\alpha=1$ the initial value problem (1.1) reduces to a classical…
We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients based on the explicit formula for Kostka-Foulkes polynomials is given.
The bright soliton solutions and singular solutions are constructed for space-time fractional EW and modified EW equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and…
Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…