Partial Differential Equation Formula Sheet u671y

MAT 409 Partial Differential Equations Solutions to 2nd Order Differential Equations Two distinct real numbers for a. a1,  a2 1. y = Ae a1 x + Be a 2 x One real number for a. a1 = a2 2. y = Ae a1 x + Bxe a1 x pair for a. a = b  ci   3. Complex bx y = e ( A cos(cx ) + Bsin(cx ))  Full Fourier Series Expansion ∞ ⎛nπx   ⎞  ⎛ nπx ⎞  1 f (x) = a0 + ∑ an cos⎜  ⎟+   bn sin⎜  ⎟,  where − L ≤ x ≤ L ⎝  L ⎠  ⎝  L ⎠  2 n=1  ⎛nπx   ⎞  ⎛nπx   ⎞  1 L 1 L f (x)cos dx , b = f (x)sin⎜  ⎟dx ⎜   ⎟     ∫ ∫ n ⎝  L ⎠  ⎝  L ⎠  L −L L −L Fourier Cosine Series Expansion Fourier Sine Series ∞ Expansion ⎛nπx   ⎞  1 f (x) = a0 + ∑ an cos⎜  ⎟,  where 0 ≤ x ≤ L ∞ ⎛ nπx ⎞  ⎝   ⎠   2 L n=1 f (x) = ∑ bn sin⎜  ⎟,  where 0 ≤ x ≤ L ⎝  L ⎠  n=1 ⎛ nπx ⎞  2 L an = ∫ f (x)cos⎜  ⎟dx   ⎛ nπx ⎞  2 L ⎝  L ⎠  L 0 bn = ∫ f (x)sin⎜  ⎟dx   ⎝  L ⎠  L 0 an =



Double Summation Legendre Polynomials  The solution 0   k, 0  , 0    2  ∞ ∞ ⎛ nπx ⎞  ⎛ mπy ⎞   The solution u = ∑ ∑ An,m sin⎜  ⎟sin   ⎜  ⎟  ∞ ⎝  L ⎠  ⎝  K ⎠  n=1 m=1 u = ∑ An ρn ( x )  The initial condition with f(x) n=0 ∞ ∞ ⎛ nπx ⎞  ⎛ mπy ⎞   The initial condition with f(x) f (x, y) = ∑ ∑ An,m sin⎜  ⎟sin   ⎜  ⎟  ∞ ⎝  L ⎠  ⎝  K ⎠   n=1 m=1 f ( x) = ∑ An ρn ( x )  The constant  n=0 K L ⎛   ⎞   ⎛   ⎞    The constant 2 2 nπx mπy An,m = f (x)sin⎜  ⎟sin   ⎜  ⎟dxdy   ∫ ∫ 2n +1 1 ⎝  L ⎠  ⎝  K ⎠  LK 0 0  An k n = ∫ −1 f ( x)ρn ( x)dx 2  Double Summation & Bessel Function  The solution –L    L  ⎛aq   t ⎞  ⎛ q r ⎞  ∞ ∞ ⎡  ⎛aq   n,i t ⎞  ⎛ nπθ ⎞  ⎛nπθ   ⎞  ⎛ qn,i r ⎞⎤    1 ∞ z = ∑ cos⎜  0,i ⎟C  0,i J 0 ⎜  n,i ⎟+ ⎟+   Dn,i sin⎜  ⎟J  n ⎜  ⎟⎥      ∑ ∑⎢cos ⎟C   n,i cos⎜    ⎜  ⎝  L ⎠  ⎝  L ⎠  ⎝  K ⎠⎦    2 i=1 ⎝  k ⎠  ⎝  k ⎠  n=1 i=1 ⎣  ⎝  K ⎠   The initial condition with f(r,) ⎛aq   t ⎞  ⎛ q r ⎞  ∞ ∞ ⎡  ⎛aq   n,it ⎞  ⎛ nπθ ⎞  ⎛nπθ   ⎞  ⎛ qn,i r ⎞⎤    1 ∞ f (r,θ ) = ∑ cos⎜  0,i ⎟C  0,i J 0 ⎜  n,i ⎟+ C cos + D sin ⎜   ⎟   ⎜   ⎟J  ⎜  ⎟⎥      ∑ ∑⎢cos ⎜   ⎟     n,i ⎝  L ⎠  n,i ⎝  L ⎠  n ⎝  K ⎠⎦    2 i=1 ⎝  k ⎠  ⎝  k ⎠  n=1 i=1 ⎣  ⎝  K ⎠    The constants ⎛nπθ   ⎞  ⎛ qn,i r ⎞  ⎛nπθ   ⎞  ⎛ qn,i r ⎞  2 K L 2 K L rf r,θ cos J dθdr rf r,θ sin ⎜   ⎟   ⎜   ⎟J  ⎜  ⎟dθdr ( ) ( ) ⎜   ⎟     ∫ ∫ ∫ ∫ n 0 −L ⎝  L ⎠  ⎝  K ⎠  ⎝  L ⎠  n ⎝  K ⎠  π 0 −L  C = π Dn,i = n,i 2 2 K 2 J n+1 K 2 J n+1 (qn,i ) (qn,i )





