Partial differential equation
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PDE Utt = Uxx+sin(3x) 0<x<1 0<t<
BC's U(0,t)=0
U(1,t)= 0
IC's U(x,0)= 2sin(2x)
Ut(x,0)= 0
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Solution Summary
This provides an example of solving a partial differential equation. The functions of partial differential equations are examined.
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PDE Utt = Uxx+sin(3x) 0<x<1 0<t<
BC's U(0,t)=0
U(1,t)= 0
IC's U(x,0)= 2sin(2x)
Ut(x,0)= 0
We first must consider the homogenous equation:
We start as always with separating the wave equation:
Plugging it back into the homogenous equation we obtain:
Each side is independent of the other (since each side is a function of a single independent variable).
This can occur if an only if both sides equal the same constant number.
Hence:
The spatial equation becomes:
Case 1:
The general solution of the equation is:
Using boundary conditions we have:
Hence for we get only the trivial solution.
Case 2:
The general solution of the equation is:
Using boundary conditions we have:
Hence for we get only the trivial solution.
So we are left with case 3:
The general solution of the equation is:
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