In this assessment, we will employ the group problem-solving process. Given an inhomogeneous second-order partial differential equation: utt + utx - 2uxx = t₁ (1) then the PDE (1) can be solved using a change of variable technique. In this group activities, you and your team members will be guided to employ this approach from the beginning of the process i.e., proposing the new variables and equations to be used till the solving parts. a) By considering homogeneous equation below (i.e., taking RHS = 0): + utx - 2uxx = 0, propose the characteristic functions in term of x and t. b) From your answer in part (a), select new variables (e.g., n and k) in term of your characteristic equations and then employ change of variables and chain rules to transform the inhomogeneous PDE into a simpler equation: Utt -gunk = t. c) Hence, work out the general solution of PDE (1): u= A(x+t) + B(x-2t)-t(x+t)(x-2t). Hint: From the derived equation in (b), re-write t in terms of n and k and solve. (2) (3) (4)
In this assessment, we will employ the group problem-solving process. Given an inhomogeneous second-order partial differential equation: utt + utx - 2uxx = t₁ (1) then the PDE (1) can be solved using a change of variable technique. In this group activities, you and your team members will be guided to employ this approach from the beginning of the process i.e., proposing the new variables and equations to be used till the solving parts. a) By considering homogeneous equation below (i.e., taking RHS = 0): + utx - 2uxx = 0, propose the characteristic functions in term of x and t. b) From your answer in part (a), select new variables (e.g., n and k) in term of your characteristic equations and then employ change of variables and chain rules to transform the inhomogeneous PDE into a simpler equation: Utt -gunk = t. c) Hence, work out the general solution of PDE (1): u= A(x+t) + B(x-2t)-t(x+t)(x-2t). Hint: From the derived equation in (b), re-write t in terms of n and k and solve. (2) (3) (4)
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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Transcribed Image Text:In this assessment, we will employ the group problem-solving process. Given an inhomogeneous
second-order partial differential equation:
Utt utx - 2uxx = t,
(1)
then the PDE (1) can be solved using a change of variable technique. In this group activities, you and
your team members will be guided to employ this approach from the beginning of the process i.e.,
proposing the new variables and equations to be used till the solving parts.
a) By considering homogeneous equation below (i.e., taking RHS = 0):
utt + utx - 2uxx = 0,
propose the characteristic functions in term of x and t.
b) From your answer in part (a), select new variables (e.g., n and k) in term of your characteristic
equations and then employ change of variables and chain rules to transform the inhomogeneous PDE
into a simpler equation:
-gunk = t.
c) Hence, work out the general solution of PDE (1):
(2)
u= A(x+t) + B(x-2t)t(x+t)(x-2t).
Hint: From the derived equation in (b), re-write t in terms of n and k and solve.
(3)
(4)
d) Propose the solution of PDE (1) that satisfies initial conditions ut(0, x) = 0 and u(0, x) = 0.
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