1 Summary table

Heat PDE ∂u   ∂2u
∂t = k∂x2   in 1D  (in a rod)






Left side Right side initial condition u(x,0)  λ = 0  λ > 0










u (0) = 0  u(L) = 0  {             L-
   x  0L< x < 2
  L− x 2 < x < L  No λn = (nLπ)2,n = 1,2,3,⋅⋅⋅
Xn = Bn sin(√ λnx)
u(x,t) = ∑ ∞n=1Bn sin (√ λnx) e− kλnt





u (0) = 0  u(L) = 0  100  No λ =  (nπ)2,n = 1,2,3,⋅⋅⋅
Xn = BL sin(√ λ-x)
un(x,t)n = ∑ ∞  Bn sin (√ λ-x) e− kλnt
          n=1  n      n





u (0) = T0  u(L) = 0  x  No      (nπ)2
λn =  L   ,(n√ =-1,2),3,⋅⋅⋅
Xn = Bn sin  Tλ0nx  ∑ ∞       (√ ---) −kλnt
u(x,t) = T0 − L x + n=1Bn sin   λnx e





∂u(0)
-∂x-= 0  ∂u(L)
-∂x--= 0  x  λ0 = 0
X0 = A0       (  )2
λn =  nLπ  ,n(√ =-1,2),3,⋅⋅⋅
Xn = An cos ∑λn∞x       (√ ---)
u(x,t) = A0 + n=1 An cos   λnx e−kλnt





∂u(∂0x)= 0  u(L) = T0  0  No      (  )
λn =  n2Lπ 2,n(√ =-1,3),5,⋅⋅⋅
Xn = An cos ∑λnx           (√ ---)
u(x,t) = T0 + ∞n=1,3,5,⋅⋅⋅Ancos   λnx e−kλnt





∂u(0)= 0
 ∂x  u(L) = 0  f (x)  No λn = (nπ)2,n = 1,3,5,⋅⋅⋅
Xn = A2Ln cos(√ λnx)
u(x,t) = ∑ ∞     An cos(√ λnx)e− kλnt
          n=1,3,5⋅⋅⋅





u (0) = 0  ∂u(L)-
 ∂x  = 0  f (x)  No      (nπ)2
λn =  2L   ,(n√ =-1,3),5,⋅⋅⋅
Xn = Bn s∑in∞  λnx      (√ ---) −kλnt
u(x,t) =  n=1,3,5⋅⋅⋅Bn sin   λnx e





u (0) = 0         ∂u(L)
u(L)+  -∂x-= 0  f (x)  λ0 = 0
X0 = A0     (√--- )
tan  λnL  =(√−-λn)
Xn = Bn sin  ∑λn∞x       (√ ---) −kλ t
u(x,t) = A0 + n=1 Bn sin   λnx e   n





u (− 1) = 0  u(1) = 0  f (x)  No √---  nπ
 λn = -2   (√n-=-1),2,3,⋅(⋅√⋅-- )
Xn = An c∑os∞  λnx  ,sin (λ√nx- ) −λ t  ∑ ∞          (√ ---) −λ t
u(x,t) =  n=1,3,⋅⋅⋅An cos  λnx  e  n +   n=2,4,⋅⋅⋅Bn sin   λnx e  n





Heat PDE ∂u= α ∂2u2-− βu
∂t    ∂x  in 1D  (in a rod) with α,β > 0  for 0 < x < π







Left side Right side initial condition λ = 0  λ > 0  analytical solution           u (x,t)












∂u(∂0x,t)= 0  ∂u∂(πx,t) = 0  u (x,0) = x  λ0 = 0
X0 = A0        2
λn = n ,n = 1∑,2,∞3,⋅⋅⋅
X (x) = A0 +  n=1 Ancos(nx)       (       )             n            (     )
π2 + c0 e−βt − 1 + π2∑ ∞n=1 ((−1n)2−1)cos(nx)e− n2α+βt






(TO DO) Heat PDE for periodic conditions u (− L) = u(L)  and ∂u(∂−xL)-= ∂u∂(Lx)-

     (   )
λ  =  n-π 2,n = 1,2,3,⋅⋅⋅
 n     L

               --------------------λ>0----------------------
        λ=0   ◜∞∑        (∘ ---)      ◞◟∑∞       (∘ ---)     ◝
u(x,t) = ◜◞a◟0◝+   An cos   λnx e−kλnt +   Bn sin    λnx e− kλnt
              n=1                     n=1