Theory and Applications of Conservation Laws

This thesis examines the properties, applications and usefulness of the different conservation lawsin everyday life from our homes to the industry.We investigate some mathematical derivations of linear and non-linear partial differentialequations which are used as models to solve problems for instance in the applications of oilrecovery process were we produce a model to find the amount of water that passes through theproduction well. Also investigations are made on derivations of the shallow water wave equationsin one-dimension in which case emphasis is placed on important assumptions which are used toproduce a simplified model which can be solved analytically.An overview of the different conservation laws are used in order to get the right models orequations. Emphasis was also placed on the different techniques used to solve the characteristicequations depending on the nature and direction of its characteristic speed. There are differentways of finding the solutions of the conservation differential equations but this thesis specificallyis concentrated on the two types of solutions, that is, the shock and rarefaction solutions whichare obtain at different characteristic speeds. Entropy conditions were also studied in order to geta phyically admissible weak solutions which allows a shock profile.A numerical scheme was employed to find an approximate solution to the shallow water waveequations. The numerical approach used in the thesis is based on the Lax Friedrichs schemewhich is built on differential equations by difference methods.Also further work will be needed for instance to look at two or three dimesional shallowwater equation as well as it can be extended to look at applications of conservation in a twodimensional or network systems of cars