Heat, wave and Laplace equations, separation of variables, Fourier series and boundary value problems.
34 cards · basic cards · AI-written, checked twice. Edit anything.
- What is the heat equation in one dimension?
Show answer
u_t = alpha * u_xx, where u is temperature, t is time, x is position, and alpha is the thermal diffusivity constant- What is the wave equation in one dimension?
Show answer
u_tt = c^2 * u_xx, where u is displacement, c is wave speed, and subscripts denote partial derivatives with respect to time and position- What is Laplace's equation?
Show answer
nabla^2 u = 0, or in 2D: u_xx + u_yy = 0; describes steady-state and equilibrium problems- What is the separation of variables method?
Show answer
Assume solution u(x,t) = X(x)*T(t) as a product of single-variable functions, which reduces the PDE to two ODEs- What is a Fourier series?
Show answer
An infinite sum of sines and cosines that represents a periodic function: f(x) = a0/2 + sum of a_n*cos(nx) + b_n*sin(nx)- What is a boundary value problem?
Show answer
A differential equation with conditions specified at the boundaries of the domain, determining a unique solution- What role do initial conditions play in solving the heat equation?
Show answer
They specify the temperature distribution u(x,0) at time zero, allowing determination of all Fourier coefficients in the solution- What are eigenvalues in the context of PDEs?
Show answer
Special values of a parameter (lambda) for which a boundary value problem has non-trivial solutions- What are eigenfunctions in PDEs?
Show answer
Non-trivial solutions to a differential equation that correspond to specific eigenvalues- What does orthogonality of functions mean?
Show answer
Two functions are orthogonal if their inner product (integral of their product over the domain) equals zero- What are the Dirichlet conditions for Fourier series convergence?
Show answer
Function must be bounded, have finitely many discontinuities, and finitely many extrema in one period- What is periodic extension?
Show answer
The process of extending a function defined on an interval to a periodic function on the entire real line- What is a half-range Fourier series?
Show answer
A Fourier series that extends a function defined on (0,L) to either an even or odd function on (-L,L)- When should you use only a cosine series in Fourier representation?
Show answer
When the function is even, or when boundary conditions require an even extension of the domain- When should you use only a sine series in Fourier representation?
Show answer
When the function is odd, or when boundary conditions require an odd extension of the domain
FlashKeepers