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Week 4

6/28 Week 3: Systems I

Ahat, P-System, Elastic String

With the tools in hand to study systems of partial differential equations, we familiarized ourselves with many classical systems. The process consisted of writing down the system, finding the appropriate flux matrix and taking the eigensystem of said matrix. As this was computationally intensive, we used Mathematica and Maple to speed calculations. What we did for each system below is contained in a mathematica notebook.

The “Ahat” system consists of a decoupled scalar equation and a system of two other equations which depend on the solution to the first. It appears in one form or another in the works of Professor Young and in several papers we’ve read. It is mainly of theoretical interest.

1) Ahat system: a, b, c, d, and F are functions of W

Ut + (aU+bV)x = 0
Vt + (cU+dV)x = 0
Wt + (F)x = 0

The “P” System consists of two coupled equations. Under certain conditions, many other systems of conservation laws reduce to the P-system. It models Isentropic gas dynamics and the one dimensional elastic string.

2)P System: U is velocity. F (pressure) is a function of V (reciprocal of density)

Ut + (F)x = 0
Vt - (U)x = 0

The “Elastic String” system consists of six coupled equations. It models the motion of a string in three dimensional space.

3) Elastic String: The Fi are functions of U1, U2, U3.

U1t + (V1)x = 0
U2t + (V2)x = 0
U3t + (V3)x = 0
V1t + (F1)x = 0
V2t + (F2)x = 0
V3t + (F3)x = 0


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