a. Domains of the heat equation
For each of the following heat equation scenarios, describe the domain of the PDE, including its dimension. If the solutions are time-dependent, identify the spatial slices of the domain and the initial spatial slice.
b. Domains of the wave equation
For each of the following wave equation scenarios, describe the domain of the PDE, including its dimension. If the solutions are time-dependent, identify the spatial slices of the domain and the initial spatial slice.
c. Domains of Laplace’s equation
For each of the following Laplace’s equation scenarios, describe the domain of the PDE, including its dimension. If the solutions are time-dependent, identify the spatial slices of the domain and the initial spatial slice.
Recall (from the exercises!) that a linear differential operator \(L\) is something of the form \[ L(u) = au + \sum_{i=1}^n b_i \frac{\partial u}{\partial x_i} + \sum_{i,j=1}^n c_{ij}\frac{\partial^2 u}{\partial x_i \partial x_j} + \cdots, \] where the sum only includes finitely many terms, and the coefficients \(a, b_i, c_{ij}, \ldots\) are functions only of the independent variables.
For example, the Laplacian is such an operator: \[ L(u) = \nabla^2 u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}. \] If we set \(L(u)=0\), we get Laplace’s equation.
For another example, the operator \[ L(u) = \frac{\partial u}{\partial t} - k \nabla^2 u, \] is linear. If we set \(L(u)=0\), we get the heat equation.
For a third example, the operator \[ L(u) = \frac{\partial^2 u}{\partial t^2} - a^2 \nabla^2 u, \] is linear. If we set \(L(u)=0\), we get the wave equation.
Linear PDEs
A linear partial differential equation consists of the following:
An equation \(L(u) = f\), where \(L\) is a linear differential operator and \(f\) is some function of the independent variables (but not of \(u\) or its derivatives).
A specification of the domain \(D\) over which the equation is considered, i.e., the domains of the solution \(u\) and the function \(f\).
Theorem (Superposition principle for homogeneous linear PDEs).
If \(u_1, u_2, \dots, u_n\) are solutions to a homogeneous linear PDE, then any linear combination \[ u = a_1 u_1 + a_2 u_2 + \dots + a_n u_n \] is also a solution, where \(a_1, a_2, \dots, a_n\) are constants.
Let \(L(u)=f\) be a non-homogeneous linear PDE, and let \(u_p\) be a particular solution to this PDE (i.e., \(L(u_p)=f\)).
Let \(u_h\) be a solution to the corresponding homogeneous PDE (i.e., \(L(u_h)=0\)). Show that \(u = u_p + u_h\) is also a solution to the non-homogeneous PDE.
Show that if \(u\) is any solution to the non-homogeneous PDE, then \(u - u_p\) is a solution to the homogeneous PDE.
Conclude that any solution \(u\) to the non-homogeneous PDE can be expressed as \[ u = u_p + u_h, \] where \(u_h\) is a solution to the homogeneous PDE.
Boundary and initial conditions
Suppose first \(u\) is time-independent:
If \(u\) is time-dependent:
A boundary condition is a specification of the values of the solution \(u\) (and possibly its derivatives) on the boundary of the spatial slices of \(D\).
An initial condition is a specification of the values of the solution \(u\) (and possibly its derivatives) on the initial spatial slice of \(D\).
Types of boundary conditions
Suppose first \(u\) is time-independent:
Dirichlet boundary condition: specifies the value of \(u\) on the boundary of \(D\).
Neumann boundary condition: specifies the value of the normal derivative of \(u\) on the boundary of \(D\).
If \(u\) is time-dependent:
Dirichlet boundary condition: specifies the value of \(u\) on the boundary of the spatial slices of \(D\).
Neumann boundary condition: specifies the value of the normal derivative of \(u\) on the boundary of the spatial slices of \(D\).
Boundary and initial value problems
A boundary value problem is a linear PDE together with a specification of boundary conditions.
An initial value problem is a linear PDE together with a specification of initial conditions.
For each of the following scenarios, identify the type of PDE (e.g., heat equation, wave equation, Laplace’s equation), whether the solution are time-dependent or time-independent, the domain \(D\) of the problem, and describe the boundary and/or initial conditions that would be appropriate for the scenario.
The temperature \(u\) at a certain position and time of a metal rod of length \(L\) that is insulated at both ends and is known to have an initial temperature profile.
The displacement \(u\) at a certain position and time of a vibrating string of length \(L\) that is fixed at both ends, given that the string is initially at rest but has an initial displacement profile.
The steady-state temperature \(u\) at a certain position of a metal plate of width \(x_0\) and height \(y_0\) that is held at a constant temperature along its boundary and has an initial temperature distribution.