Theory and Applications of Conservation Laws
This thesis examines the properties, applications and usefulness of the different conservation laws
in everyday life from our homes to the industry.
We investigate some mathematical derivations of linear and non-linear partial differential
equations which are used as models to solve problems for instance in the applications of oil
recovery process were we produce a model to find the amount of water that passes through the
production well. Also investigations are made on derivations of the shallow water wave equations
in one-dimension in which case emphasis is placed on important assumptions which are used to
produce a simplified model which can be solved analytically.
An overview of the different conservation laws are used in order to get the right models or
equations. Emphasis was also placed on the different techniques used to solve the characteristic
equations depending on the nature and direction of its characteristic speed. There are different
ways of finding the solutions of the conservation differential equations but this thesis specifically
is concentrated on the two types of solutions, that is, the shock and rarefaction solutions which
are obtain at different characteristic speeds. Entropy conditions were also studied in order to get
a phyically admissible weak solutions which allows a shock profile.
A numerical scheme was employed to find an approximate solution to the shallow water wave
equations. The numerical approach used in the thesis is based on the LaxFriedrichs scheme
which is built on differential equations by difference methods.
Also further work will be needed for instance to look at two or three dimesional shallow
water equation as well as it can be extended to look at applications of conservation in a two
dimensional or network systems of cars.
Place, publisher, year, edition, pages
Institutt for matematiske fag , 2014. , 71 p.
IdentifiersURN: urn:nbn:no:ntnu:diva-26800Local ID: ntnudaim:10418OAI: oai:DiVA.org:ntnu-26800DiVA: diva2:751708
Hanche-Olsen, Harald, Førsteamanuensis