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Differential Equations2D

Partial Differential Equations - Theory & Concepts

Introduction to PDEs, classification, Separation of Variables, and solving the Heat, Wave, and Laplace equations.

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1D Wave Equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Vibrating string fixed at both ends ( $x=0$ , $x=L$ ). The solution is a superposition of normal modes (standing waves).

Normal Mode (

n

)1

Determines the spatial frequency $\sin(n\pi x/L)$ .

Wave Speed (

c

)1.0

u(x,t)

u_n(x,t) = \sin\left(\frac{1\pi x}{L}\right) \cos\left(\frac{1\pi (1.0) t}{L}\right)

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