Special Issue "Mathematics on Partial Differential Equations"

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

Deadline for manuscript submissions: closed (31 December 2013)

Special Issue Editors

Guest Editor
Prof. Dr. Andreas Ruffing (Website)

Technische Universität München, Fakultät für Mathematik, Lehr- und Forschungseinheit M6, "Mathematische Modellbildung", Boltzmannstraße 3, 85747 Garching
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
Guest Editor
Prof. Dr. Igor Pazanin (Website)

Faculty of Science, Department of Mathematics, University of Zagreb, Croatia
Interests: partial differential equations and its applications; mathematical modeling in fluid mechanics; perturbation methods; asymptotic analysis
Guest Editor
Prof. Dr. Martin Bohner (Website)

106 Rolla Building, Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, Missouri 65409-0020, USA
Fax: +(573) 341 4741
Interests: differential equations; difference equations; dynamic equations on time scales; applications in biology; economics and finance

Special Issue Information

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed Open Access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. For the first couple of issues the Article Processing Charge (APC) will be waived for well-prepared manuscripts. English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Published Papers (8 papers)

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Research

Open AccessArticle The Second-Order Shape Derivative of Kohn–Vogelius-Type Cost Functional Using the Boundary Differentiation Approach
Mathematics 2014, 2(4), 196-217; doi:10.3390/math2040196
Received: 31 December 2013 / Revised: 26 August 2014 / Accepted: 1 September 2014 / Published: 26 September 2014
Cited by 1 | PDF Full-text (958 KB) | HTML Full-text | XML Full-text
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 [...] 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)
Open AccessArticle A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter
Mathematics 2014, 2(3), 119-135; doi:10.3390/math2030119
Received: 18 January 2014 / Revised: 18 June 2014 / Accepted: 20 June 2014 / Published: 9 July 2014
Cited by 2 | PDF Full-text (404 KB) | HTML Full-text | XML Full-text
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 [...] 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)
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)
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
PDF Full-text (204 KB) | HTML Full-text | XML Full-text
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 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 Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval
Mathematics 2014, 2(1), 53-67; doi:10.3390/math2010053
Received: 22 December 2013 / Revised: 20 February 2014 / Accepted: 21 February 2014 / Published: 4 March 2014
PDF Full-text (264 KB) | HTML Full-text | XML Full-text
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 [...] 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)
Open AccessArticle Some New Integral Identities for Solenoidal Fields and Applications
Mathematics 2014, 2(1), 29-36; doi:10.3390/math2010029
Received: 31 December 2013 / Revised: 7 February 2014 / Accepted: 19 February 2014 / Published: 3 March 2014
PDF Full-text (171 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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)
Open AccessArticle One-Dimensional Nonlinear Stefan Problems in Storm’s Materials
Mathematics 2014, 2(1), 1-11; doi:10.3390/math2010001
Received: 17 October 2013 / Revised: 12 December 2013 / Accepted: 20 December 2013 / Published: 27 December 2013
Cited by 1 | PDF Full-text (224 KB) | HTML Full-text | XML Full-text
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 [...] 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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