"Consider the problem uz = uzr, 0 < x < 1, t > 0, with boundary condition u(0, t) = 0, u(1, t) = cos(9t), and initial condition u(x, 0) = x.Solve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Writeu = S+U, where S(x, t) = A(t)(1 – x) + B(t)x is (affine) linear in x, that isS(r, t)The initial boundary value problem for U becomesPDE: U= Ura + f(x,t)where f(x, t) =BC: U(0, t) = 0, U(1, t) = 0IC: U(x,0) =Now solve for U.The eigenfunctions to use areFirst decompose the inhomogeneity in the PDE as= (x)"Xf(x,t) = fn (t)X„(x),j=1where fn (t) =You find thatU(2,t) = ET,(t)X,(x),j=1whereTndsThen u(x, t)S(x, t) + U(x, t)

Given the problem ut-uxx=0, 0<x<1, t>0,---1 with boundary condition u0,t=0, u1,t=cos9t and…

"Consider the problem ut = Uzr, 0 < x < 1, t > 0, with boundary condition u(0, t) = 0, u(1, t) = cos(9t), and initial condition u(x,0) = x. Solve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write u = S +U, where S(x, t) = A(t)(1 – x) + B(t)x is (affine) linear in æ, that is S(x,t) = The initial boundary value problem for U becomes PDE: U̟ = Urz + f(x,t) where f(x, t) = BC: U(0,t) = 0, U(1, t) = 0 1C: U(x,0) = Now solve for U. The eigenfunctions to use are X„(x) = First decompose the inhomogeneity in the PDE as f(x,t) = E fn (t)X,„(x), j=1 where fn (t) = You find that U(x, t) = T,(t)X„(x) , j=1 where T, = ds

"Consider the problem ut = Uzr, 0 < x < 1, t > 0, with boundary condition u(0, t) = 0, u(1, t) = cos(9t), and initial condition u(x,0) = x. Solve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write u = S +U, where S(x, t) = A(t)(1 – x) + B(t)x is (affine) linear in æ, that is S(x,t) = The initial boundary value problem for U becomes PDE: U̟ = Urz + f(x,t) where f(x, t) = BC: U(0,t) = 0, U(1, t) = 0 1C: U(x,0) = Now solve for U. The eigenfunctions to use are X„(x) = First decompose the inhomogeneity in the PDE as f(x,t) = E fn (t)X,„(x), j=1 where fn (t) = You find that U(x, t) = T,(t)X„(x) , j=1 where T, = ds

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

"Consider the problem uz = uzr, 0 < x < 1, t > 0, with boundary condition u(0, t) = 0, u(1, t) = cos(9t), and initial condition u(x, 0) = x.
Solve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write
u = S+U, where S(x, t) = A(t)(1 – x) + B(t)x is (affine) linear in x, that is
S(r, t)
The initial boundary value problem for U becomes
PDE: U
= Ura + f(x,t)
where f(x, t) =
BC: U(0, t) = 0, U(1, t) = 0
IC: U(x,0) =
Now solve for U.
The eigenfunctions to use are
First decompose the inhomogeneity in the PDE as
= (x)"X
f(x,t) = fn (t)X„(x),
j=1
where fn (t) =
You find that
U(2,t) = ET,(t)X,(x),
j=1
where
Tn
ds
Then u(x, t)
S(x, t) + U(x, t)

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