arXiv subjects

Daniele Mortari

Publications and source records attributed to Daniele Mortari.

21 records · Page 2Linked to original sources

arXiv2019

The Tensor Theory of Connections

This paper extends the univariate Theory of Connections, introduced in (Mortari,2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons,1984), by providing analytical expressions, called "constrained expressions," representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g(x,y), satisfy all constraints no matter what the g(x,y) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the "Tensor Theory of Connections," validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDEs and stochastic differential equations.

math.GM↗

arXiv2017

The Theory of Connections. Connecting Points

This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with embedded specific constraints. The paper first shows how to express the most general explicit function passing through a single point in three distinct ways: linear, additive, and rational. Then, functions with constraints on single, two, or multiple points are introduced as well as those satisfying relative constraints. This capability allows to obtain general expressions to solve linear differential equations with no need to satisfy constraints (the "subject to:" conditions) as the constraints are already embedded in the constrained expression. In particular, for expressions passing through a set of points, a generalization of the Waring's interpolation form, is introduced. The general form of additive constrained expressions is introduced as well as a procedure to derive its coefficient functions, requiring the inversion of a matrix with dimensions as the number of constraints.

math.OC↗

arXiv2017

Least-squares Solutions of Linear Differential Equations

This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second order differential equations. The proposed method has two steps. The first step consists of writing a constrained expression, introduced in Ref. \cite{Mortari}, that has embedded the differential equation constraints. These expressions are given in term of a new unknown function, $g (t)$, and they satisfy the constraints, no matter what $g (t)$ is. The second step consists of expressing $g (t)$ as a linear combination of $m$ independent known basis functions, $g (t) = \mathbfξ^T \mathbf{h} (t)$. Specifically, Chebyshev orthogonal polynomials of the first kind are adopted for the basis functions. This choice requires rewriting the differential equation and the constraints in term of a new independent variable, $x\in[-1, +1]$. The procedure leads to a set of linear equations in terms of the unknown coefficients vector, $\mathbfξ,$ that is then computed by least-squares. Numerical examples are provided to quantify the solutions accuracy for initial and boundary values problems as well as for a control-type problem, where the state is defined in one point and the costate in another point.

math.CA↗