Applications of Differential Equations and Dynamical Systems

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 September 2018)

Special Issue Editors

Departamento de Matemáticas, Universidad del Valle, Calle 13 No. 100-00, Cali 760032, Colombia
Interests: epidemic spreading in population networks; complex networks based on ODE systems; optimal control for ODE systems; numerical methods for optimization and optimal control, population and epidemiological modeling, applications of optimization techniques in biology, ecology, and epidemiology
Departamento de Matemáticas, Universidad Autónoma de Occidente, Calle 25 No. 115-85, Km 2 via Cali-Jamundí, Cali 760031, Colombia
Interests: optimal control and viable control for ODE systems; numerical methods for optimization; controlled ODE systems and viability theory; viable controls for deterministic and stochastic dynamical systems; population and epidemiological modeling, applications of optimization techniques in biology, ecology, and epidemiology
Departamento de Matemáticas, Universidad del Valle, Calle 13 No. 100-00, Cali 760032, Colombia
Interests: optimal control for ODE systems; numerical methods for optimization and optimal control, population and epidemiological modeling, applications of optimization techniques in economics, biology, ecology, and epidemiology; mathematical models and methods for sustainable management of natural resources

Special Issue Information

Dear Colleagues,

Ordinary differential equations and dynamical systems play a fundamental role in mathematical modeling due to their capacity to express the time evolution of various processes and phenomena that arise in the real world. The aim of this Special Issue is to present recent developments in the theory, qualitative analysis, numerical solutions, and applications of ODE systems with particular emphasis on natural and social science applications.

Dr. Heliana Arias-Castro
Dr. Lilian S. Sepúlveda-Salcedo
Dr. Olga Vasilieva
Guest Editors

Manuscript Submission Information

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Keywords

  • ordinary differential equations (ODE)
  • linear and nonlinear ODE systems
  • qualitative analysis of ODE systems, including bifurcations, limit cycles, periodic solutions, and chaotic behavior
  • deterministic and stochastic ODE systems
  • numerical methods for solution of ODE systems
  • controlled ODE systems
  • complex networks based on ODE
  • mathematical modeling involving ODE systems in natural and social sciences

Published Papers (3 papers)

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Research

24 pages, 332 KiB  
Article
Periodic Solution and Asymptotic Stability for the Magnetohydrodynamic Equations with Inhomogeneous Boundary Condition
by Igor Kondrashuk, Eduardo Alfonso Notte-Cuello, Mariano Poblete-Cantellano and Marko Antonio Rojas-Medar
Axioms 2019, 8(2), 44; https://doi.org/10.3390/axioms8020044 - 11 Apr 2019
Cited by 3 | Viewed by 4929
Abstract
We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In [...] Read more.
We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h ( x , t ) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier–Stokes equations with inhomogeneous boundary conditions. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)
11 pages, 341 KiB  
Article
A Two Dimensional Discrete Mollification Operator and the Numerical Solution of an Inverse Source Problem
by Manuel D. Echeverry and Carlos E. Mejía
Axioms 2018, 7(4), 89; https://doi.org/10.3390/axioms7040089 - 23 Nov 2018
Cited by 5 | Viewed by 2544
Abstract
We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional [...] Read more.
We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)
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15 pages, 293 KiB  
Article
On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem
by José Rodrigo González Granada, Joachim Gwinner and Victor A. Kovtunenko
Axioms 2018, 7(4), 76; https://doi.org/10.3390/axioms7040076 - 27 Oct 2018
Cited by 6 | Viewed by 2806
Abstract
This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the [...] Read more.
This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)
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