Second Order Linear Partial Differential Equations Part IV

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2008, 2012 Zachary S Tseng E 4 1 Second Order Linear Partial Differential Equations Part IV One dimensional undamped wave equation; D Alembert solution of the wave equation; damped wave equation and the general wave equation; two dimensional Laplace equation The Second type of Second Order Linear Partial Differential Equations in 2 independent variables is the one dimensional wave equation. Together with the heat conduction equation, they are sometimes referred to as the evolution Equations because their solutions evolve , or change, with passing time. The simplest instance of the one dimensional wave equation problem can be illustrated by the equation that describes the standing wave exhibited by the motion of a piece of undamped vibrating elastic string.

Eigenvalues: 2 2 2 L n π λ= , n = 1, 2, 3, … Eigenfunctions: L n x X n π =sin , n = 1, 2, 3, … Next, substitute the eigenvalues found above into the second equation to find T(t). After putting eigenvalues λ into it, the equation of T becomes 0 2 2 2 ′′+ 2 T= L n T a π. It is a second order homogeneous linear equation with constant ...

  Linear, Equations, 2 2 2