| Zeit: Mittwochs 11-13 |
| Ort: INF 294, SR -104 |
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Abstract: Real systems can be modeled at various levels of resolution. Information about the regime in which the system operates is coded in the parameters of the equations and, ideally, by changing these parameters we should be able to move smoothly between microscopic, mesoscopic and macroscopic level depending on whether we are interested in the accuracy of the description and disregard the cost, or conversely. Unfortunately, in most cases when we change the parameters to move from one regime to another, the type of the equation changes as well and the solutions in different regimes are often dramatically different. Problems of his type are called singularly perturbed. The lecture will be devoted to the mathematical methods of multiscale and asymptotic analysis with application to differential equations. The aim is to provide a systematic way of deriving the limit equations the coefficients of which encapsulate relevant information from the original regime. We will prove that solutions to the original equations in the micro regime converge to the solution of the derived equation in the macro regime when the relevant parameter converges to its critical value. We will consider regular and singular perturbation for the ordinary differential equations with applications (van der Pol oscillator, Duffing oscillator, Volterra-Lotka equation): quasi-stationary approximations (small parameter methods, Tichonov Theorem), matched asymptotic expansions, singular and regular limits for reaction-diffusion equations with examples from mathematical biology (such as shadow systems, SLEP method) and some techniques of homogenization (such as two-scale convergence, unfolding method) with applications to reactive flows).
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| Bibliography: |
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[1] J. Banasiak and M.Lachowicz, Methods of small parameter in mathematical biology and other applied sciences, chapters from the book in preparation, 2012. |
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[2] A. Marciniak-Czochra and A.Mikelic, Multiscale Methods with Applications in Sciences, chapters from the book in preparation, 2012. |
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[3] I. Takagi, Mathematical Analysis of Biological Pattern Formation - Singular Perturbation Methods, Script, 2011. |