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u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0

u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.3: Euler's Method
Problem 1YT: Use Eulers method to approximate the solution of dydtx2y2=1, with y(0)=2, for [0,1]. Use h=0.2.
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We will solve the heat equation =2ur 2 Usc 0<r<6, t> 0 with boundary/initial conditions: (0, t) = 0, 2(6, t) = 0, 2, 0<x<3 and u(x, 0) = { 0, 3<x < 6 This models temperature in a thin rod of length L = 6 with thermal diffusivity a- 2 where the temperature at the ends is fixed at 0 and the initial temperature distribution is u(x, 0). For extra practice we will solve this problem from scratch.
Transcribed Image Text:We will solve the heat equation =2ur 2 Usc 0<r<6, t> 0 with boundary/initial conditions: (0, t) = 0, 2(6, t) = 0, 2, 0<x<3 and u(x, 0) = { 0, 3<x < 6 This models temperature in a thin rod of length L = 6 with thermal diffusivity a- 2 where the temperature at the ends is fixed at 0 and the initial temperature distribution is u(x, 0). For extra practice we will solve this problem from scratch.
Find Eigenfunctions for ). The problem splits into cases based on the sign of A (Notation: For the cases below, use constants a and b) • Case 1: A = 0 X(x) Plugging the boundary values into this formula gives 0 = X(0) %3D 0 = X(6) So X(x) = which means u(x, t) = We can ignore this case. • Case 2: A =-y² <0 (In your answers below use gamma instead of lambda) X(x) = Plugging the boundary values into this formula gives 0 – X(0) 0 = X(6) = So X(x) = which means u(x, t) = We can ingore this case. • Case 3: A= > X(x) = (In your answers below use gamma instead of lambda) Plugging in the boundary values into this formula gives 0 = X(0) = %3D 0 = X(6) Which leads us to the eigenvalues A, = Yn where Yn = and eigenfunctions X,(z) = (Notation: Eigenfunctions should not include any constants a or b.)
Transcribed Image Text:Find Eigenfunctions for ). The problem splits into cases based on the sign of A (Notation: For the cases below, use constants a and b) • Case 1: A = 0 X(x) Plugging the boundary values into this formula gives 0 = X(0) %3D 0 = X(6) So X(x) = which means u(x, t) = We can ignore this case. • Case 2: A =-y² <0 (In your answers below use gamma instead of lambda) X(x) = Plugging the boundary values into this formula gives 0 – X(0) 0 = X(6) = So X(x) = which means u(x, t) = We can ingore this case. • Case 3: A= > X(x) = (In your answers below use gamma instead of lambda) Plugging in the boundary values into this formula gives 0 = X(0) = %3D 0 = X(6) Which leads us to the eigenvalues A, = Yn where Yn = and eigenfunctions X,(z) = (Notation: Eigenfunctions should not include any constants a or b.)
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