Question
Asked Feb 16, 2020
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Partial Differential Equation

by separation of variables.  I have begun but am stuck

Plz & thank you!!

1. Solve Ut-Ux.
BCs şu(0t)=D0
lu1Bt)=3
IC u(x0)=4x-x² 0<X<3
0<x< 3
t70
Separation of variables
let ucx,t)= X (x)T(t)
A=0, San 15 trivial
A+0 T+ XZ T=0 >T4) = Aet
X"t A²X=0 => xu)= Bsin ArtACCoS(A
ulx.t) basy VA e-2t (D sin(Ax) tE ws (An))
4(0,E) =0 > Ee 12E -0>E-0
Sin d= 3
Sand=3. ()
SDAWAADM
Sind=l
%3D
u(3it) =3 =) De^t
myst
D=3
%3D
help_outline

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1. Solve Ut-Ux. BCs şu(0t)=D0 lu1Bt)=3 IC u(x0)=4x-x² 0<X<3 0<x< 3 t70 Separation of variables let ucx,t)= X (x)T(t) A=0, San 15 trivial A+0 T+ XZ T=0 >T4) = Aet X"t A²X=0 => xu)= Bsin ArtACCoS(A ulx.t) basy VA e-2t (D sin(Ax) tE ws (An)) 4(0,E) =0 > Ee 12E -0>E-0 Sin d= 3 Sand=3. () SDAWAADM Sind=l %3D u(3it) =3 =) De^t myst D=3 %3D

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Step 1 Given equation is a heat equation with non-homogeneous boundary conditions.

In case of non-homogeneous boundary conditions, we apply the transformation,

Advanced Math homework question answer, step 1, image 1

Step 2 Now, the new heat equation with homogeneous boundary conditions is:

Advanced Math homework question answer, step 2, image 1

Step 3 Solve for En by applying integral by parts to second integrand as well.

Advanced Math homework question answer, step 3, image 1

...

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