Boundary Value Problems/Lesson 6

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1D Wave Equation
Derivation of the wave equation using string model.

General form for boundary conditions.
$$ \alpha_{11} u(a,t) + \alpha_{12} u_x (a,t) = \gamma_{1} $$ $$ \alpha_{21} u(b,t) + \alpha_{22} u_x (b,t) = \gamma_{2} $$

Wave equation with Dirichlet Homogeneous Boundary conditions.
$$ \displaystyle \alpha_{11} u(a,t) = 0 $$ $$ \displaystyle \alpha_{21} u(b,t) = 0 $$ In the homogeneous problem $$ \displaystyle u_{xx}-\frac{1}{c^2} u_{tt} =0 $$ with $$ \displaystyle u(0,t) =0 $$, $$  \displaystyle u(L,t) =0 $$

Finding a solution: u(x,t)
Let $$ u(x,t)= X(x)T(t) $$ then substitute this into the PDE. $$ X T = \frac{1}{c^2} X T $$ $$ \frac{X}{X} = \frac{1}{c^2} \frac{T}{T} = \mu$$ Where $$ \mu $$ is a constant that can be positive, zero or negative. We need to check each case for a solution.

Wave Equation with nonhomogeneous Dirichlet Boundary Conditions
In the homogeneous problem $$ u_{xx}-\frac{1}{c^2} u_{tt} =0 $$ $$ \displaystyle \alpha_{11} u(x,t) = \gamma_{1}(t) $$ $$ \displaystyle \alpha_{21} u(x,t) = \gamma_{2} (t) $$

Wave Equation with resistive damping
In the homogeneous problem $$ u_{xx}=\frac{1}{c^2} u_{tt} + ku_t $$