Partial Differential Equations
Partial Differential Equations MATH 467
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This 3 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 467 at Millersville University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/223531/math-467-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.
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Date Created: 10/15/15
1 Euler Equations A homogeneous Euler Equation is a second order linear ordinary differential equation of the form tzu add u 0 1 The expressions 04 and 8 are constants There are two popular ways to determine the solution to an Euler equation The rst is to assume that fundamental solutions have the form ut tf where r is a constant Differentiating this solution and substituting into eq 1 produces t2rr 71W atrtf l tT 0 rr 71tf artf tf 0 rr 71 ow 0 assuming tf 7 0 r20471r 0 Thus to solve an Euler equation we would solve this quadratic equation for r As an alternative we may perform a change of variables by letting t 61 In this case du du dz du 1 E 3 d271 d du d du 1 W amplt2 7 d du 1 dud 1 a g g 2 d2u1 du1 dz2 t2 dz t2 7 d271 du 1 7 dz2 dz t2 Substituting in eq 1 yields which is a second order linear constant coe icient ordinary di ferential equation lts characteristic equation is the same as in eq 2 above To solve the Euler equation we would once again solve the quadratic function for r and reverse the change of variables Heat Equation in 3D These notes will brie y outline the derivation of the heat equation in three dimensions Throughout these notes the following quantities will be referenced 0 speci c heat of material amount of heat per unit mass necessary to raise the temperature one degree p density of material mass per unit volume um y2 t temperature of the material at location z y2 at time t Qzy 2 t amount of heat energy generated per unit volume per unit time at location my2 at time t my2 heat energy ux at location zy 2 K0 thermal conductivity of the material the amount of power per unit length per degree the material can conduct k thermal diffusivity of the material We will also need Gauss s Theorem Divergence Theorem Gauss7s Theorem Suppose region R C R3 is bounded by the closed surface S and that nm y 2 denotes the unit outward normal to S If the components of the vector eld Fm y 2 have contin uous rst partial derivatives in R then SFndSR VFdV Suppose the material in question occupies the region in three dimensional space denoted by R and is bounded by the surface S Let the unit normal vector to the surface S be denoted n The heat energy contained in R at time t is the value of R Cpu W The heat energy leaving region B through its boundary S is given by the surface integral n dS S The heat energy generated per unit time at time t in region R can be written as R de Thus the conservation law of heat energy in region R can be expressed as RcpudV iS ndSR QdV 7 V dVQdV Gauss s Theorem R R wdv ltQv gtdv R R ltQvwgt w R Since R is an arbitrary threedimensional region then 0 i i 814 7 7 0 1 6p at V d Q In three dimensions Fourier s Law of Heat Conduction can be stated as 7K0Vu Substituting this into eq 1 yields 8 mg V 7K0Vu 7 Q 0 814 605 K0 V Vu Q KOVZu Q 814 i 2 Q a 7 kV u GP If there are no sources of heat in region R then the heat equation simpli es to 814 7k 2 at VU
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