Use the Laplace transform to solve the heat equationu_t=u_{xx} 0<x<1 text{ and } t>0{u}{left({x},{0}right)}= sin{{left(pi{x}right)}} {u}{left({0},{t}right)}={u}{left({1},{t}right)}={0}

Chesley 2021-02-21 Answered

Use the Laplace transform to solve the heat equation
ut=uxx0<x<1 and t>0

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Answered 2021-02-22 Author has 105 answers
Step 1
The given heat equation is ut=uxx0<x<1 and t>0
Also initial and boundary conditions are u(x,0)=sin(πx)  and  u(0,t)=u(1,t)=0
Taking the Laplace transform of heat equation on both sides
The Auxiliary equation of the homogeneous part is (D2s)U=0
So, the complementary solution of the differential equation is
For the particular solution
=sin(πx)π2s       [as  (π2s)!0]
Step 2
So, the complete solution of differential equation is
We have u(0,t)=u(1,t)=0
Taking Laplace transform we get
Substituting (2) in (1)
The solution of the above system will give c1=c2=0
Therefore, equation (1) reduces to the form
Step 3

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According to this example using the Laplace Transform, I am just curious to HOW we go from
More specific, I am curious to why we get an 4 and an 8. What formula or equation are being used to get those numbers?I know that the Laplace Transform calculates it for us, however, I want to calculate it "by hand" and show why we get the answer. Any feedback would be gladly appreciated!