Partial differential equations (PDEs) are mathematical models describing relationships between functions and their partial derivatives. They're crucial in physics, engineering, and applied math, representing phenomena like heat transfer, wave propagation, and fluid dynamics. PDEs come in various types, including elliptic, parabolic, and hyperbolic, each with unique properties and applications. Solving techniques range from analytical methods like separation of variables to numerical approaches like finite difference and finite element methods.
Solve the one-dimensional heat equation with initial condition and boundary conditions using the separation of variables method.
Find the steady-state temperature distribution in a rectangular plate with dimensions and , given the boundary conditions:
Solve the one-dimensional wave equation with initial conditions , , and boundary conditions using the separation of variables method.
Find the electric potential in a square region with the following boundary conditions:
Solve the advection equation with initial condition using the method of characteristics.