pde_notes01 - Notes for TUT course MAT-51316 Robert Pich e...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 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 b a u ( x, t ) A dx = φ ( a, t ) A - φ ( b, t ) A + b a f ( x, t, u ) A dx, which can be rearranged to give b a ( u t + φ x - f ) dx = 0 . Because [ a, b ] is arbitrary, this implies the conservation equation u t + φ x = f.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern