# "Initial-boundary value problems arise in practice" by Dr. Hongqiu Chen

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Dear all,

Mathematical Soft Matter Unit (Fried Unit) would like to invite you to a Seminar by Dr. Hongqiu Chen.

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Date: Wednesday, May 20, 2015

Time: 13:30-14:30

Venue: C015, LevelC, Lab 1

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Speaker:

Dr. Hongqiu Chen

Associate Professor, Department of Mathematical Sciences,

The University of Memphis.

Title: Initial-boundary value problems arise in practice

Abstract:

The full equations for waves on the surface of an ideal fluid were

derived by Euler and Bernoulli in the 18th century. This system is not

always convenient in practice, however, model equations that formally

approximate solutions of the full problem have been developed from the

19th century up to today. It is natural to run controlled laboratory

experiments to see how well these model equations work.

Much of the mathematical theory developed for model equations is

concerned with the pure initial-value problem where the waveform is

assumed to be known everywhere at a given instant of time. In fact, it

is very difficult to determine even a fairly simple waveform at all points

in some spatial domain at one time. Much easier and far more accurate

is the determination of the wave at fixed spatial points for an interval

of time. Mathematically, this leads to initial-boundary-value problems

rather than pure initial-value problems.

Thus, in the wave tank pictured below, a flap-type wavemaker is

turned on at time t = 0 and waves are generated which propagate down

the channel. If the frequency and throw of the wavemaker are adjusted

appropriately, the waves generated are in the formal range of validity of

various model equations. The wave amplitudes are monitored at several

stations along the channel as a function of elapsed time. These measurements

are then used in the context of initial-boundary-value problems to

check the model's predictive power.

This program throws up several interesting mathematical issues. First,

one needs to understand whether or not the relevant boundary-value

problems are well posed in the classical sense of having existence and

uniqueness of solutions. The solutions should also vary only slightly

with small variations of the imposed boundary conditions - i.e. the model

should be robust. The relevant model equations are nonlinear and exact

solutions are not available even for very simple boundary data. Thus,

when such models are used in practice, a numerical scheme for approximating

its solutions must be implemented. This requires the spatial

domain to be cut off, and that in turn raises the thorny issue of whether

or not this will have an unwanted, articial eect on the solutions. Theory

relating to this issue will also be discussed.

Finally, we will discuss briefly the use of a modal decomposition to

approximate solutions of the model, a practice followed by some coastal

engineers and oceanographers.

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