heat conduction vs. one-dimensional wave equation conduction and...microsoft word - heat conduction...
TRANSCRIPT
Heat Conduction and One-Dimensional Wave Equations โ๐ ๐ข!! = ๐ข! vs. ฮฑ๐๐ข!! = ๐ข!!
Heat Conduction: โ! ๐ข!! = ๐ข!
Boundary conditions: ๐ข(0, ๐ก) = 0,๐ข(๐ฟ, ๐ก) = 0
Case: Bar with both ends kept at 0 degree General Solution: ๐ข ๐ฅ, ๐ก = ๐ถ!!
!!! ๐!โ!!!!!!/!!๐ ๐๐ !"#
!
Steady State Solution: ๐ฃ(๐ฅ) = 0 Other info:
๐ถ! = ๐! =2๐ฟ
๐ ๐ฅ ๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
Heat Conduction: โ! ๐ข!! = ๐ข!
Boundary conditions: ๐ข!(0, ๐ก) = 0,๐ข!(๐ฟ, ๐ก) = 0
Case: Bar with both ends perfectly insulated General Solution: ๐ข ๐ฅ, ๐ก = ๐ถ! + ๐ถ!!
!!! ๐!โ!!!!!!/!!๐๐๐ !"#
!
Steady State Solution: ๐ฃ(๐ฅ) = ๐ถ! Other info: ๐ถ! =
!!!
๐ถ! = ๐! =2๐ฟ
๐ ๐ฅ ๐๐๐ ๐๐๐ฅ๐ฟ๐๐ฅ
!
!
Heat Conduction: โ! ๐ข!! = ๐ข!
Boundary conditions: ๐ข 0, ๐ก = ๐!,๐ข ๐ฟ, ๐ก = ๐!
Case: Bar with ๐! degrees at the left end, and ๐!degrees at the right end General Solution: ๐ข ๐ฅ, ๐ก = !!!!!
!๐ฅ + ๐! + ๐ถ!!
!!! ๐!โ!!!!!!/!!๐ ๐๐ !"#
!
Steady State Solution: ๐ฃ ๐ฅ = !!!!!
!๐ฅ + ๐!
Other info: ๐ฃ(๐ฅ) = ๐ด๐ฅ + ๐ต , and ๐ค ๐ฅ, 0 = ๐ ๐ฅ โ ๐ฃ(๐ฅ)
๐ถ! = ๐! =2๐ฟ
(๐ ๐ฅ โ ๐ฃ ๐ฅ )๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
One-Dimensional Wave Equations: ฮฑ!๐ข!! = ๐ข!!
Boundary conditions: ๐ข 0, ๐ก = 0,๐ข ๐ฟ, ๐ก = 0
Initial conditions: ๐ข(๐ฅ, 0) = ๐(๐ฅ),๐ข!(๐ฅ, 0) = ๐(๐ฅ)
Case: Undamped One-dimensional Wave Equation General Solution: ๐ข ๐ฅ, ๐ก = (๐ด!!
!!! ๐๐๐ !"#$!+ ๐ต!๐ ๐๐
!"#$!)๐ ๐๐ !"#
!
Other info: ** See Special Cases Below **
๐ด! = ๐! =2๐ฟ
๐ ๐ฅ ๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
๐ต! =๐ฟ๐๐๐
๐! =2๐๐๐
๐ ๐ฅ ๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
One-Dimensional Wave Equations: ฮฑ!๐ข!! = ๐ข!!
Boundary conditions: ๐ข 0, ๐ก = 0,๐ข ๐ฟ, ๐ก = 0
Initial conditions: ๐ข ๐ฅ, 0 = 0 ,๐ข!(๐ฅ, 0) = ๐(๐ฅ)
Case: Special Case of Undamped One-dimensional Wave Equation ๐๐๐๐ก๐๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐ก = 0 General Solution: ๐ข ๐ฅ, ๐ก = (๐ต!๐ ๐๐
!"#$!)๐ ๐๐ !"#
!
Other info: ๐ด! = 0
๐ต! =๐ฟ๐๐๐
๐! =2๐๐๐
๐ ๐ฅ ๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
One-Dimensional Wave Equations: ฮฑ!๐ข!! = ๐ข!!
Boundary conditions: ๐ข 0, ๐ก = 0,๐ข ๐ฟ, ๐ก = 0
Initial conditions: ๐ข(๐ฅ, 0) = ๐(๐ฅ),๐ข!(๐ฅ, 0) = 0
Case: Special Case of Undamped One-dimensional Wave Equation ๐๐๐๐ก๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ = 0 General Solution: ๐ข ๐ฅ, ๐ก = (๐ด!!
!!! ๐๐๐ !"#$!)๐ ๐๐ !"#
!
Other info: ๐ต! = 0
๐ด! = ๐! =2๐ฟ
๐ ๐ฅ ๐ ๐๐๐๐๐ฅ๐ฟ๐๐ฅ
!
!
๐ต! = 0