Definitions Flashcards
What is a partial differential equation (PDE)?
A partial differential equation is an equation that involves partial derivatives of a function with respect to multiple variables.
True or False: A PDE can only have one independent variable.
False
Fill in the blank: A function that satisfies a partial differential equation is called a _______.
solution
What is the order of a partial differential equation?
The order of a PDE is the highest order of derivative present in the equation.
Provide an example of a second-order partial differential equation.
The wave equation: ∂²u/∂t² = c²∂²u/∂x².
What are the three main types of PDEs?
Elliptic, parabolic, and hyperbolic.
True or False: Elliptic PDEs typically describe steady-state phenomena.
True
What is the Laplace equation?
The Laplace equation is a second-order elliptic PDE given by ∇²u = 0.
Define boundary conditions in the context of PDEs.
Boundary conditions are constraints necessary to find a unique solution to a PDE, specified on the boundary of the domain.
What is the heat equation?
The heat equation is a parabolic PDE given by ∂u/∂t = α∇²u, where α is the thermal diffusivity.
Fill in the blank: The method of characteristics is used primarily for solving _______ PDEs.
hyperbolic
What does the term ‘initial conditions’ refer to in PDEs?
Initial conditions specify the state of the system at the beginning of the observation period.
True or False: The Poisson equation is a type of elliptic PDE.
True
What does the term ‘separation of variables’ refer to?
Separation of variables is a technique used to solve PDEs by reducing them to simpler ordinary differential equations.
What is a homogeneous PDE?
A homogeneous PDE is one in which all terms involve the unknown function or its derivatives, and there are no constant terms.
Name a common application of PDEs in physics.
The modeling of fluid dynamics using the Navier-Stokes equations.
True or False: The wave equation is an example of a parabolic PDE.
False
What is the significance of the Fourier series in relation to PDEs?
Fourier series are used to express solutions to PDEs, particularly in problems with periodic boundary conditions.
What is meant by ‘linear’ in the context of PDEs?
A linear PDE is one where the unknown function and its derivatives appear to the first power and are not multiplied together.
Fill in the blank: The term _______ refers to the qualitative behavior of solutions to PDEs under small perturbations.
stability
What is the primary purpose of a Fourier Series?
To represent a periodic function as a sum of sines and cosines.
True or False: A Fourier Series can only represent functions that are continuous.
False
Fill in the blank: The coefficients of a Fourier Series are calculated using the formulas a_n = (1/T) * ∫_0^T f(t) cos(nω_0 t) dt and b_n = (1/T) * ∫_0^T f(t) sin(nω_0 t) dt, where T is the period and ω_0 is the angular frequency.
Fourier coefficients
What is the general form of a Fourier Series for a function f(t) with period T?
f(t) = a_0/2 + Σ(a_n cos(nω_0 t) + b_n sin(nω_0 t)) from n=1 to ∞
Multiple Choice: Which of the following is NOT a component of a Fourier Series? A) Sine terms B) Cosine terms C) Exponential terms D) Polynomial terms
D) Polynomial terms
