Related papers: Transcendental numbers as solutions to arithmetic …
Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters…
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups.…
The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables…
The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the…
In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\mathbb{C}(\{t\})$ of meromorphic functions at $0$. We show that Klazar's result is an instance of…
We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be…
Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional…
In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New…
This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number…
The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The…
It is well known that value at a non-zero algebraic number of each of the functions $e^{x}, \ln x, \sin x, \cos x, \tan x, \csc x, \sec x, \cot x, \sinh x,$ $ \cosh x,$ $ \tanh x,$ and $\coth x$ is transcendental number (see Theorem 9.11 of…
The principal aim of this article is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions and derive their estimates developing the…
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…
In this note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.
Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit…
Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with…