Related papers: Operational versus umbral methods and the Borel tr…
In this contribution we first summarize how contour integration methods can be used to derive closed formulae for functional determinants of ordinary differential operators. We then generalize our considerations to partial differential…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…
In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…
A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded…
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type…
The differential transform method is used to find numerical approximation of solution to a class of certain nonlinear differential algebraic equations. The method is based on Taylor's theorem. Coefficients of the Taylor series are…
In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and…
We use the Laplace transform and the Gamma function to introduce a new integral transform and name it the Laplace-type transform possessing the property of mapping a function to a functional sequence, which cannot be achieved by the Laplace…
We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise…
We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.
We review a recently-discovered link between the functional relations approach to integrable quantum field theories and the properties of certain ordinary differential equations in the complex domain.
We introduce a new numerical method for the computation of the inverse nonlinear Fourier transform and compare its computational complexity and accuracy to those of other methods available in the literature. For a given accuracy, the…
Recently, D. S. Kim and T. Kim have studied applications of um- bral calculus associated with p-adic invariant integrals on Zp (see [6]). In this paper, we investigate some interesting properties arising from umbral calculus. These…
In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational…