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Mathematics, Volume 2, Issue 2 (June 2014), Pages 68-118

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Research

Open AccessArticle Numerical Construction of Viable Sets for Autonomous Conflict Control Systems
Mathematics 2014, 2(2), 68-82; doi:10.3390/math2020068
Received: 29 December 2013 / Revised: 6 March 2014 / Accepted: 2 April 2014 / Published: 11 April 2014
Cited by 1 | PDF Full-text (255 KB) | HTML Full-text | XML Full-text
Abstract
A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. [...] Read more.
A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. N. Krasovskii and A. I. Subbotin. The viability kernel is constructed as the limit of sets generated by a Pontryagin-like backward procedure. This method is implemented in the framework of a level set technique based on the computation of limiting viscosity solutions of an appropriate Hamilton–Jacobi equation. To fulfill this, the authors adapt their numerical methods formerly developed for solving time-dependent Hamilton–Jacobi equations arising from problems with state constraints. Examples of computing viability sets are given. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
Open AccessArticle Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models
Mathematics 2014, 2(2), 83-95; doi:10.3390/math2020083
Received: 31 December 2013 / Revised: 27 March 2014 / Accepted: 16 April 2014 / Published: 8 May 2014
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Abstract
In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The [...] Read more.
In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
Open AccessArticle The Riccati System and a Diffusion-Type Equation
Mathematics 2014, 2(2), 96-118; doi:10.3390/math2020096
Received: 31 December 2013 / Revised: 27 February 2014 / Accepted: 14 April 2014 / Published: 15 May 2014
Cited by 3 | PDF Full-text (289 KB) | HTML Full-text | XML Full-text
Abstract
We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation [...] Read more.
We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)

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