Related papers: A New Operator Theory of Linear Partial Differenti…
Solving partial differential equations (PDEs) efficiently is essential for analyzing complex physical systems. Recent advancements in leveraging deep learning for solving PDE have shown significant promise. However, machine learning…
Theory of differential operators on associative algebras is not extended to the non-associative ones in a straightforward way. We consider differential operators on Lie algebras. A key point is that multiplication in a Lie algebra is its…
We introduce an abstract framework for elliptic boundary value problems in a variational form. Given a non-negative quadratic form in a Hilbert space, a boundary pair consists of a bounded operator, the boundary operator, and an auxiliary…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
This paper describes a general formalism for obtaining localized solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems. This class includes the important cases of Schr\"odinger's…
We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality…
The present paper plans to examine the existence, uniqueness and data dependence of the solution of the fractional functional differential equation with the abstract operator of Volterra, in the context of the Picard operators. We present…
We introduce a general algebraic setting for describing linear boundary problems in a symbolic computation context, with emphasis on the case of partial differential equations. The general setting is then applied to the Cauchy problem for…
This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and…
The boundary-value problem on semi-axis for one class operator-differential equations of the fourth order, the main part of which has the multiple characteristic is investigated in this paper in Sobolev type weighted space. Correctness and…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
This paper presents a novel method that allows to generalise the use of the Adam-Bashforth to Partial Differential Equations with local and non local operator. The Method derives a two step Adam-Bashforth numerical scheme in Laplace space…
The objective in this paper is to demonstrate that four of the most used techniques in applied mathematics, viz., Fourier series, Fourier transform, Laplace transform and the Fourier-Laplace transform can be introduced using eigenvalue…
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
In this paper, we derive certain formulas giving the Laplace transforms of two generalized fractional integral operators introduced recently in [Fract. Calc. Appl. Anal. 20 (2) (2017), 422--446]. The main results provide generalizations to…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
Relations have been derived which establish connection between a scalar or a vector functions and the integral of Laplace operator of these functions (the integral property of Laplace operator). The integral property of Laplace operator was…