OCE301 PDE: Separation of Variables

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OCE301 PDE Separation of Variables
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Reading Assignment

• Textbook Kreyszig, 8th edition,
• pp. 585-595.

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One-Dimensional Wave Equation
Governing differential equation
two independent variables
boundary conditions
(2nd order in x)
initial conditions
initial deflection
(2nd order in t)
initial velocity
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Method of Separating Variables
product method
original GDE
two ordinary differential equations
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Solving Boundary Value Problem in x
boundary conditions
ordinary differential equation
two boundary conditions
three scenarios to consider for k
a positive number
a negative number
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Boundary Value Problem in x (contd.)
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Solution in Time Domain
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Expression of un(x,t)
(eigenfunctions)
(eigenvalues)
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Solution of the Entire Problem
given initial displacement
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Compute Bn
Fourier sine series of f (x)
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Compute Bn
Fourier coefficients
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Summary
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Summary of Solution Procedure
A linear PDE with linear boundary(initial)
conditions
Apply separation of variables
Consider three scenarios for one ODE