相关论文: Differential Equations for $F_q$-Linear Functions,…

In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that…

New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…

We consider a recently proposed nonlinear Schroedinger equation exhibiting soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function which naturally emerges within nonextensive thermostatistics…

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized…

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the…

In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Fract. 6 (3) (2022)] on an Erd\'{e}lyi-type integral for Saran's three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an…

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…

When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…

We review recent contributions on nonlinear Dirichlet forms. Then, we specialise to the case of 2-homogeneous and local forms. Inspired by the theory of Finsler manifolds and metric measure spaces, we establish new properties of such…

Linearly independent Dirichlet L-functions satisfying the same Riemann-type of functional equation have been supposed for long time to possess off critical line non trivial zeros. We are taking a closer look into this problem and into its…

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher…

We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we…

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class…

Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb{K}(z)$. This paper is dedicated to proving the following…

There is no unified method to solve the fractional differential equation. The type of derivative here used in this paper is of Jumarie formulation, for the several differential equations studied. Here we develop an algorithm to solve the…

We prove that the Dirichlet $L$-functions associated with Dirichlet characters in $\mathbb{F}_{q}[x]$ are universal. That is, given a modulus of high enough degree, $L$-functions with characters to this modulus can be found that approximate…

This study is on Cauchy's function $f(z)$ and its integral, $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a closed simple contour $C$, in regard to their comprehensive properties over the entire $z=x+iy$ plane consisted of…

We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions F(p) cos p z + F(q) cos q z in the short-time asymptotic…