As time goes to infinite the height of liquid will be in same level being flat

As time goes to infinite the height of liquid will be

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2. Present H(x, t) in terms of Fourier coefficients
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of variables 3. Solve the resulting ODE for F(x) 4. Solve the resulting ODE for G(t) 5. Full solution First step (define parameters): Initial condition has been given in Eq.1 Boundary conditions have been given in Eq.3 Governing equation is shown in Eq.2 Second step (separation of variables); (4) H ( x ,t ) = F ( x ) G ( t ) According to the partial derivative, we could both get the first derivative function response to t and the second derivative function response to x. Then we could get two equations below: (5) H t = FG ( t ) H xx = G F xx By substituting these two equations back to Eq.2, we will get a new relation: FG t = DG F xx G t DG = F xx F =− P 2 (let it be a constant) Third step (ODE for F(x)) F xx F =− P 2 F xx + P 2 F = 0 F ( x ) = acos ( px ) + bsin ( px ) Then apply boundary condition (Eq.3 ):
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