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Keywords = Stieltjes ordinary differential equation

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30 pages, 1693 KB  
Article
Numerical Solution of Stieltjes Differential Equations
by Francisco J. Fernández and F. Adrián F. Tojo
Mathematics 2020, 8(9), 1571; https://doi.org/10.3390/math8091571 - 11 Sep 2020
Cited by 9 | Viewed by 3245
Abstract
This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations [...] Read more.
This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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30 pages, 414 KB  
Article
Displacement Calculus
by Ignacio Márquez Albés and F. Adrián F. Tojo
Mathematics 2020, 8(3), 419; https://doi.org/10.3390/math8030419 - 14 Mar 2020
Cited by 6 | Viewed by 3643
Abstract
In this work, we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation [...] Read more.
In this work, we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation and integration. We find interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equations. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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