Nathan Summary and different of equations wave conditions for the heat boundary ⑨ Heat equation Equation 8eI=k8¥ : Separation ODES of X 't XX ' ) - and Tct ) General ucx.tl - - boundary for cases XCO ) this XK ) - XkT=0 - recent ) :[ ht an e- : O X>0 case conditions O solution n : T 't O UCO ,t)=uCL,t ) In t> 0 , two Usually (i - , ucx.tk/Cx)TCt ) variables Xcx ) for Axel , , X=Xn=(¥5 near : "Ttsin(n¥ ) , . . . . The constants condition an Cii ) Ux re an (x , (at ) this General ulxit ) - from the initial =fCx ) : -_&§fCx)sin(n¥)dx - Ux X' CO In O) come - - an={ X 'll ) vk.tl 9.2 . . . . Egan O ' X { CIT " O n , ) ht x " . Hoos ( NTI) )dx O , n - 9.2 - aid n=0 , : e- §fCx cost aqjofcxdx { - X >O solution ao > ( Litt O case t n , , . . . . . . . ② Wave Equation equation 8¥ 58¥ = : Separation ODES X " + Usually Ci ) - two recoil ) and Tct ) Xcx ) O - re t> 0 : T "tXET=0 , cases , ucx.tk/CxJTCt ) variables of for XX Axel , boundary for ( Lit ) - conditions : O XCO )=XC 4=0 In this General X> case solution 0,7=7*415 ul x.t ) , a- 9,2 . . . . : ulx.tl-%Cancost.IT/tbnsin(ntEtDsin(nII) The constants conditions an , UCX O) , bn from the initial we Cx O ) gcx ) come (x ) , , = { ( ii ) 2of( "¥)&× bn-ni-fgcxls.int#olx an re, x , Cat ) - rex ) sink a. 2 n, , . . . ( Lit 1=0 X' ( 01--1144=0 In this General case solution in> O ,x=Xn=f¥o? 2%62 ' vk.tl " " : ulxit -aotbott%2fancoslntftltbnsint.t #tHcoslnTT)&SofCx { q§fc×)dx ) an " cos (m)dx , n' O , n- 9,2 . . . . { •" § To 9 G) cos 9- guide 4¥ )d× , n , n- 9,2 . . . .