1 Summary table

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







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












u (0) = 0  u (L ) = 0           {
            x  0 < x < L2
u (x,0) =  L − xL2-< x < L  No λn = (nπ)2,n = 1,2,3,⋅⋅⋅
Xn  = BLnsin(√λnx )        ∑ ∞        (√ ---) −kλnt
         n=1 Bnsin   λnx e






u (0) = 0  u (L ) = 0  u (x,0) = 100  No λ  = (nπ)2,n = 1,2,3,⋅⋅⋅
Xn  = BLsin(√λ--x)
  n    n       n        ∑ ∞        (√ ---)
         n=1 Bnsin   λnx e−kλnt






u (0) = T0  u (L ) = 0  u (x,0) = x  No      (nπ)2
λn =   L  ,n(√=-1,2),3,⋅⋅⋅
Xn  = Bnsin  λnx                  ∑          (√ ---)
       T0 − T0L x +  ∞n=1Bn sin   λnx e− kλnt






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






∂u(0)= 0
 ∂x  u (L ) = T
        0  u (x,0) = 0  No      (  )
λn =  n2πL 2,n(√=-1,3),5,⋅⋅⋅
Xn  = Ancos   λnx        T + ∑ ∞       A  cos(√ λ-x)e−kλnt
        0    n=1,3,5,⋅⋅⋅ n       n






∂u(0)
 ∂x = 0  u (L ) = 0  u (x,0) = f (x)  No λn = (n2πL)2,n = 1,3,5,⋅⋅⋅
Xn  = Ancos(√ λnx)        ∑ ∞            (√ ---) −kλnt
         n=1,3,5⋅⋅⋅An cos   λnx e






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






u (0) = 0  u (L )+ ∂u∂(Lx)-= 0  u (x,0) = f (x)  λ = 0
X0 = A
 0    0  tan(√ λ-L) = − λ
X  = B nsin(√λ-nx)
  λ    λ      n             ∑         (√ ---)
       A0 +   ∞n=1Bn sin   λnx e−kλnt






Heat PDE        2
∂∂ut = α ∂∂ux2-− βu  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(0,t)
-∂x--= 0  ∂u(π,t)
-∂x-- = 0  u (x,0) = x  λ0 = 0
X0 = A0  λn = n2,n = 1,2,3,⋅⋅⋅
X (x) = A0 + ∑ ∞  Ancos(nx)
              n=1  π    ( −βt   )   2∑ ∞  ((−1)n−1)        −(n2α+β)t
2 + c0 e  − 1 + π   n=1---n2---cos(nx)e






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

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

              ◜--------------------λ>◞0◟---------------------◝
        λ◜=◞0◟◝  ∞∑        (∘ ---)        ∑∞       (∘ ---)
u(x,t) = a0 +    An cos   λnx e−kλnt +   Bn sin    λnx e− kλnt
              n=1                     n=1