Question

Find product solutions, \(u(x, t)=b(t) \phi(x)\), to the wave equation satisfying the boundary conditions \(u(0, t)=0\) and \(u_{x}(1, t)=0\). Use these solutions to find a general solution of the heat equation satisfying these boundary conditions.

Short answer

The general solutions for both the wave and the heat equations given the boundary conditions are \(u(x, t) = \sum (Acos(kc) + Bsin(kc))sin(kx)\) and \(u(x, t) = \sum A \ exp(-k^2 \ \alpha^2 \ t) \ sin(kx)\) respectively.

Step-by-step solution

Find the product solution

The wave equation is typically written as \(u_{tt} = c^2 u_{xx}\), where \(c\) is the speed of wave propagation. If \(u(x, t) = b(t) \phi(x)\), then upon substituting this into the wave equation, we get \(b''(t)\phi(x) = c^2 b(t)\phi''(x)\). This can be rearranged to \(\frac{b''(t)}{b(t)} = c^2 \frac{\phi''(x)}{\phi(x)}\). The left-hand side is a function of \(t\) only while the right-hand side is a function of \(x\) only, hence, both sides must equal to the same constant, say \(-k^2\). This gives two ordinary differential equations \(b''(t) + k^2b(t) = 0\) and \(\phi''(x) - \frac{k^2}{c^2}\phi(x) = 0\).

Apply the boundary conditions

Apply the boundary conditions to solve the system of equations. For \(\phi(x)\), the given boundary conditions are \(\phi(0) = 0\) and \(\phi'(1) = 0\). Solving the second ordinary differential equation with these conditions will give the solution for \(\phi(x)\) as \(\phi(x) = Asin(kx)\) where \(A\) is a constant.

Find the general solution of the wave equation

The general solution of the wave equation can be found using the method of superposition. The general solution is given by \(u(x, t) = \sum (Acos(kc) + Bsin(kc))sin(kx)\). This solution satisfies both the wave equation and the given boundary conditions.

Find a general solution of the heat equation

The heat equation is typically written as \(u_{t} = \alpha^2 u_{xx}\), where \(\alpha\) is the thermal diffusivity. For the heat equation, the process is similar to the wave equation. The method of separation of variables gives us two ordinary differential equations \(b'(t) - k^2\alpha^2b(t) = 0\) and \(\phi''(x) - k^2\phi(x) = 0\) . Thus, the general solution of the heat equation satisfying the given boundary conditions is \(u(x, t) = \sum A \ exp(-k^2 \ \alpha^2 \ t) \ sin(kx)\), which satisfies the heat equation and the given boundary conditions.

Key concepts

Boundary Conditions

Boundary conditions are essential in solving differential equations because they provide the constraints that a solution must satisfy. In the context of the wave and heat equations, they specify what happens at the edges — or boundaries — of the spatial domain where the phenomena are occurring.

For the given wave equation problem, the boundary conditions are:

• At the boundary where the position, x, is zero (i.e., the start of the domain), the solution for every moment in time, t, must satisfy \(u(0, t) = 0\). This means there is no displacement at the starting point.
• At the boundary where x equals 1, the spatial derivative with respect to x must satisfy \(u_{x}(1, t) = 0\). This implies that the rate of change of wave amplitude in space at this point is zero, resembling a stationary or "clamped" end.

These conditions are used to find specific solutions to the differential equations that model the wave and heat phenomena. The boundary conditions are crucial because they directly influence the specific form of the functions \(b(t)\) and \(\phi(x)\). It ensures that the solutions are viable and physically meaningful.

Separation of Variables

Separation of variables is a powerful mathematical technique used to solve partial differential equations (PDEs). It involves breaking down a PDE into simpler, one-dimensional problems called ordinary differential equations (ODEs). This is done by assuming a solution can be written as a product of functions, each dependent on a single variable.

• For instance, in the wave equation, we assume a solution in the form \(u(x, t) = b(t) \phi(x)\).
• Plugging this form back into the equation, it allows us to separate the variables such that each side of the resulting equation depends only on one independent variable.

In the wave equation problem, by equating the time-dependent part and the space-dependent part to a constant, like \(-k^2\), we get two separate ODEs: one in terms of time \(t\) and the other in terms of space \(x\).

This method provides a systematic way to derive sets of simpler equations, which can be solved with conventional mathematical techniques. The solutions to these ODEs are then combined to form the general solution of the original PDE.

Heat Equation

The heat equation is a fundamental PDE describing how heat diffuses through a given region over time. It is a model used to study thermal conduction. Mathematically, it is expressed as \(u_{t} = \alpha^2 u_{xx}\), where \(\alpha\) is the thermal diffusivity. This determines how quickly heat spreads through the material.

In solving the heat equation using separation of variables:

• We separate the temperature function as \(u(x, t) = b(t) \phi(x)\), much like in the wave equation.
• By substituting this product into the heat equation, it results in two ODEs after variable separation.

The spatial component \(\phi(x)\) is similar, satisfying the same boundary conditions previously derived in the wave equation problem: \(\phi(0) = 0\) and \(\phi'(1) = 0\). This leads us to conclude that \(\phi(x) = A\sin(kx)\), where \(A\) is a constant.

The time-dependent part is described by an exponential decay \(e^{-k^2 \alpha^2 t}\), which indicates how heat dissipates over time. The general solution of the heat equation, respecting the same boundary conditions, therefore takes the form \(u(x, t) = \sum A \exp(-k^2 \alpha^2 t) \sin(kx)\). This reflects how heat redistributes in the medium, ensuring adherence to the constraints given by the boundary conditions.