Non-Local Partial Differential Equations for Engineering and Biology

Mathematical Modeling and Analysis

verfasst von: Dr. Nikos I. Kavallaris, Prof. Dr. Takashi Suzuki

Verlag: Springer International Publishing

Buchreihe : Mathematics for Industry

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena.
This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.

Inhaltsverzeichnis

Frontmatter

Applications in Engineering

Frontmatter

Chapter 1. Micro-Electro-Mechanical-Systems (MEMS)

Abstract

In the current chapter we first present the construction of some non-local models describing the operation of an idealized MEMS (Micro-electro-mechanical system). In particular, the MEMS device is considered to be part of an electrical circuit and using elastic and electric theories two different non-local models are derived: a parabolic and a hyperbolic one. In the first place, the investigation of the structure of the corresponding non-local elliptic steady state problem is undertaken and some estimates of the pull-in voltage are obtained. Next, we focus on the mathematical analysis of the derived evolutionary non-local equations. Notably, the circumstances under which finite-time quenching occurs for both of evolutionary problems are investigated, so then some useful conclusions regarding the possible destruction of the MEMS device or the invalidity of the used models can be derived. Since, maximum principle is not available for both of the inspected non-local models, and thus comparison methods are not applicable, finally energy methods are called forth to investigate their long-time behavior.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 2. Ohmic Heating Phenomena

Abstract

The current chapter considers two main applications associated with Ohmic heating phenomena. Initially we deal with an application from food industry, building up two one-dimensional non-local problems illustrating the evolution of the temperature of the sterilized food. The former model consists of a diffusion-convection equation while the latter of a convection equation with non-local convection velocity. Both of these non-local models are investigated in terms of their stability and the occurrence of finite-time blow-up, where the latter in the current context indicates food burning. Different approaches should be followed though depending on the monotonicity of the nonlinearity appearing in the non-local term, since no maximum principle is available for the non-local parabolic problem when this nonlinearity is increasing. The second part of the chapter is devoted to the study of a non-local parabolic model illustrating the operation of the thermistor device. Notably, conditions under which finite-time blow-up, which here indicates the destruction of the thermistor device, occurs are investigated by using both energy and comparison methods.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 3. Linear Friction Welding

Abstract

The current chapter discusses an application arising in the process of linear friction welding applied in metallurgy. In the first place a one-dimensional non-local model defined in the half-line is constructed in order to describe the evolution of the temperature within the welding region. In this study we mainly consider two cases: the soft-material which is modeled by an exponential nonlinearity and the hard-material case when a power-law nonlinearity is regarded. In the former case the non-local problem has variational structure, and so can be treated as a gradient flow, which is used to derive appropriate a priori estimates for the solution. Thus parabolic regularity theory can be used to prove global-in-time existence and finally prove the convergence of its solution towards the unique steady state. On the other hand, the power-law case lacks such a variational structure and thus we have to appeal to a numerical scheme of Crank–Nicolson type in order to presume the long-tume behavior in this case as well as to confirm the analytical results derived in the exponential case.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 4. Resistance Spot Welding

Abstract

In the current chapter we debate a joining process called resistance spot welding which finds many applications ranging from the automobile industry to robotics. In the first part of the chapter we present the construction of a non-local mathematical model illustrating the phase transition occurs during this joining process. Since the derived model consists of a degenerate non-local parabolic equation its analytical study is rather hard due to many arising technicalities and so we appeal to a numerical approach. We then consider a time discretization scheme for solving the resulting non-local moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction which takes into account the nonlinearity. The stability and error estimates of the developed scheme are also investigated. Finally, some numerical experiments are presented which verify the efficiency of the developed numerical algorithm, as well as demonstrate the emergent interfaces due to the degeneracy of the problem.

Nikos I. Kavallaris, Takashi Suzuki

Emotion in Music

Frontmatter

Chapter 5. Gierer–Meinhardt System

Abstract

The purpose of the current chapter is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer–Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a diffusion driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 6. A Non-local Model Illustrating Replicator Dynamics

Abstract

The current chapter discusses a utilization from the field of evolutionary game dynamics and in particular from its subarea called replicator dynamics. Considering an infinite continuous strategy space, which for example might be considered as the sampling space of a continuously varying trait of a biological population, as well as payoff functions of Gaussian type we build up a non-local degenerate parabolic problem. As it is appropriate for degenerate problems, a regularized approximation is constructed and then some a priori estimates for its solutions are obtained. Using the derived estimates, we prove that solutions converge to the trivial solution if the initial population is small, whereas they undergo a blow-up in finite time if the initial population is large. In particular, in the latter case, it is shown that the blow-up set coincides with the whole strategy space, i.e. the finite-time blow-up is global.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 7. A Non-local Model Arising in Chemotaxis

Abstract

The current chapter deals with the biological phenomenon of chemotaxis. In the first place, a parabolic-parabolic Keller–Segel system is considered which describes the movement of some cell population towards a chemo-attractant produced by the population itself. Next, this version of Keller–Segel system is reduced to a non-local parabolic problem for the concentration of chemo-attractant in the case the chemo-attractant diffuses much faster than the cell population. Using the variational structure of the derived non-local parabolic problem we obtain some appropriate a priori estimates permitting us to derive global-in-time solutions when the total cell population is below the threshold \(8 \pi .\) It is also proven that the global-in-time solution converges to the unique steady state solution in the radial symmetric case. When the cell population exceeds the threshold \(8 \pi \) then all the radially symmetric solutions exhibit finite-time blow-up on the origin of the considered sphere, i.e. single-point blow-up occurs.

Nikos I. Kavallaris, Takashi Suzuki

Chapter 8. A Non-local Reaction-Diffusion System Illustrating Cell Dynamics

Abstract

Initially a reaction-diffusion system with non-local reaction terms is build up as a mathematical model to illustrate the evolution of protein dimers within human cells. The derived system inspects the situation when chemical reactions occur when the two chemicals within a cell are in distance R, where such a distance is the reaction radius. Next, the long-time behavior of the solutions of the preceding non-local system is investigated as well as the phase separation phenomenon occurring when the reaction takes place very fast is also examined. It is actually shown that a two-phase Stefan problem is derived in the limit of infinite chemical reaction rate. Next the convergence of the global-in-time solution to the preceding system towards the unique stationary solution is derived. The chapter closes with some results on the determination of the decay rate of the above convergence towards the unique stationary solution.

Nikos I. Kavallaris, Takashi Suzuki

Backmatter

Titel: Non-Local Partial Differential Equations for Engineering and Biology
verfasst von: Dr. Nikos I. Kavallaris
Prof. Dr. Takashi Suzuki
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-67944-0
Print ISBN: 978-3-319-67942-6
DOI: https://doi.org/10.1007/978-3-319-67944-0