Transcription of The 1-D Heat Equation - MIT OpenCourseWare
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The 1-D Heat Equation - MIT OpenCourseWare
The 1-D Heat Equation Linear Partial Differential equations Matthew J. Hancock Fall2006 1 The 1-D Heat Equation Physical derivation Reference:Guenther&Lee ,Myint-U&Debnath and [Sept. 8, 2006] Inametalrod with non-uniformtemperature,heat(thermal energy)istransferred from regions of higher temperature to regions of lower temperature. Three physical principles are used here. 1. Heat(orthermal) energy of abody with uniformproperties: Heat energy = cmu, where m is the body mass, u is the temperature, c isthe specificheat, units[c]= L2T 2U 1 (basicunitsareM mass, Llength, T time, U temperature). c istheenergy required to raise a unit mass of the substance 1 unit in temperature. 2. Fourier s law of heat transfer: rate of heat transfer proportional to negative temperaturegradient, Rate of heat transfer u = (1) K0 area x where K0 isthethermal conductivity,units[K0]= MLT 3U 1.
This is an example of a Sturm-Liouville problem (from your ODEs class). There are 3 cases: λ > 0, λ < 0 and λ = 0. (i) λ < 0. Let λ = −k2 < 0. Then the solution to (14) is X = Aekx + Be−kx 5 for integration constants A, B found from imposing the BCs (15),
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