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Unformatted text preview: Notes for TUT course MAT-51316 Robert Pich´e 2.9.2007 1 Transport Equation Initial Value Problem • how to derive the one dimensional transport equation • how to solve initial value problems for this equation using the method of characteristics • how to compute and plot solutions using Maple function PDEplot 1.1 Transport Equation Let u ( x, t ) denote the density (units [quantity] · [volume]- 1 ) of a substance (e.g. mass, energy, ...) as a function of position x and time t . (This is a one dimensional model, it is assumed that there are no density variations in the y and z directions.) The amount of substance in an interval a ≤ x ≤ b of a tube-shaped region of constant cross section A is R b a u ( x, t ) A dx . A b a x Let φ ( x, t ) be the flux (units [quantity] · [time]- 1 · [area]- 1 ). The net flux into the tubular interval is φ ( a, t ) A- φ ( b, t ) A . Let f ( x, t, u ) be the rate (units [quantity] · [time]- 1 · [volume]- 1 ) at which substance density increases by processes other than flux, for example chemical reaction. This is called a source term. The rate of increase of the total amount of substance in the interval is d dt Z b a u ( x, t ) A dx = φ ( a, t ) A- φ ( b, t ) A + Z b a f ( x, t, u ) A dx, which can be rearranged to give Z b a ( u t + φ x- f ) dx = 0 ....
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This note was uploaded on 03/29/2008 for the course MATH 442 taught by Professor Howard during the Spring '08 term at Texas A&M.
- Spring '08