b)
prove:
xβxβzβββyβzβ2yβ=2x
Differentiating f(xy2,zβ2x)=0 , we get:
y2f1β+(βxβzββ2)f2β=0
2xyf1β+βyβzβf2β=0
where f1 and f2 are partial derivatives of f with respect to its first input and second input. When we multiply the first equality above by 2x and the second by y:
2x(βxβzββ2)=yβyβzββΉxβxβzβββyβzβ2yβ=2x
a)
solution of two dimensional heat equation βx2β2uβ+βy2β2uβ=0 is
u(x,y)=(AcosΞ»x+BsinΞ»x)(CeΞ»y+DeβΞ»y)
boundary conditions:
u(x,β)=0
u(x,0)=50
u(0,y)=0
u(1,y)=0
0β€xβ€1,0β€y<β
then:
u(0,y)=A(CeΞ»y+DeβΞ»y)=0βΉA=0
u(x,y)=BsinΞ»x(CeΞ»y+DeβΞ»y)
u(1,y)=BsinΞ»(CeΞ»y+DeβΞ»y)=0βΉΞ»=nΟ
then:
u(x,y)=Bsin(Οnx)(CeΟny+DeβΟny)
u(x,β)=Bsin(Οnx)CeΟny=0βΉC=0
u(x,y)=B1βsin(Οnx)eβΟny
where B1 = BD
the most general solution:
u(x,y)=βBnβsin(Οnx)eβΟny
where
Bnβ=β«01βu(x,0)sin(Οnx)=50β«01βsin(Οnx)=
=βΟn50βcos(nΟx)β£01β=Οn50β(1βcos(nΟ))
for odd n:
u(x,y)=βΟn100βsin(Οnx)eβΟny
for even n:
u(x,y)=0
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