Mathematics on Partial Differential Equations

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 December 2013) | Viewed by 39444

Special Issue Editors

"Mathematische Modellbildung", Lehr- und Forschungseinheit M6, Fakultät für Mathematik, Technische Universität München, Boltzmannstraße 3, 85747 Garching, Germany
Interests: difference equations and functional analysis; spectral properties of Schrödinger difference operators; basic partial difference schemes; numerical methods in discrete Schrödinger theory; special functions and orthogonal polynomials; moment problems / birth and death processes
Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia
Interests: partial differential equations and its applications; mathematical modeling in fluid mechanics; perturbation methods; asymptotic analysis
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Interests: Hamiltonian systems; Sturm–Liouville equations; boundary value problems; difference equations; variational analysis; control theory; optimization; dynamical systems; oscillation; fractional differentiation equations; positivity; matrix analysis; eigenvalue problems; computational mathematics; time scales
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Published Papers (8 papers)

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Research

958 KiB  
Article
The Second-Order Shape Derivative of Kohn–Vogelius-Type Cost Functional Using the Boundary Differentiation Approach
by Jerico B. Bacani and Gunther Peichl
Mathematics 2014, 2(4), 196-217; https://doi.org/10.3390/math2040196 - 26 Sep 2014
Cited by 101 | Viewed by 4528
Abstract
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape [...] Read more.
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape derivative for general domains is computedvia the boundary differentiation scheme. Additionally, the second-order shape derivative ofJ at the solution of the Bernoulli problem is computed using Tiihonen’s approach. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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404 KiB  
Article
A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter
by Marco A. Herrera-Valdez, Sergei K. Suslov and José M. Vega-Guzmán
Mathematics 2014, 2(3), 119-135; https://doi.org/10.3390/math2030119 - 09 Jul 2014
Cited by 44 | Viewed by 5548
Abstract
Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a [...] Read more.
Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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289 KiB  
Article
The Riccati System and a Diffusion-Type Equation
by Erwin Suazo, Sergei K. Suslov and José M. Vega-Guzmán
Mathematics 2014, 2(2), 96-118; https://doi.org/10.3390/math2020096 - 15 May 2014
Cited by 12 | Viewed by 5805
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 in [...] 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)
204 KiB  
Article
Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models
by Narcisa Apreutesei
Mathematics 2014, 2(2), 83-95; https://doi.org/10.3390/math2020083 - 08 May 2014
Cited by 14 | Viewed by 4474
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 existence [...] 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)
255 KiB  
Article
Numerical Construction of Viable Sets for Autonomous Conflict Control Systems
by Nikolai Botkin and Varvara Turova
Mathematics 2014, 2(2), 68-82; https://doi.org/10.3390/math2020068 - 11 Apr 2014
Cited by 9 | Viewed by 4778
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. 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)
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264 KiB  
Article
Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval
by Alexander Fedotov
Mathematics 2014, 2(1), 53-67; https://doi.org/10.3390/math2010053 - 04 Mar 2014
Cited by 33 | Viewed by 4059
Abstract
In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an [...] Read more.
In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
171 KiB  
Article
Some New Integral Identities for Solenoidal Fields and Applications
by Vladimir I. Semenov
Mathematics 2014, 2(1), 29-36; https://doi.org/10.3390/math2010029 - 03 Mar 2014
Cited by 16 | Viewed by 4094
Abstract
In spaces Rn, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
224 KiB  
Article
One-Dimensional Nonlinear Stefan Problems in Storm’s Materials
by Adriana C. Briozzo and María F. Natale
Mathematics 2014, 2(1), 1-11; https://doi.org/10.3390/math2010001 - 27 Dec 2013
Cited by 96 | Viewed by 4910
Abstract
We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature Tf : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a [...] Read more.
We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature Tf : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a heat flux boundary condition of the type q(t) = q 0 t , and in the second case, we assume a temperature boundary condition T = Ts < Tf at the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution. Full article
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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