Skip to main content

Applied Mathematics

Peter Staab

Section 7.2 Modeling with PDEs

In this section, we show where two important differential equations arise based on modeling the physical situation. The first will be the heat transport or heat equation and the second will be the wave equation by modeling the motion of a taut string. In both situations we will only show this for motion in one-dimension although similar PDEs model more complex behavior.
🔗

Subsection 7.2.1 The Heat Equation

In this section, we will derive the transport of heat in a solid bar. In short, heat moves in a substance from a warmer region to a cooler region and if we know the direction of motion and the rate then we can understand heat flow.
🔗
We will consider a bar of length \(L\) with fixed density \(\rho\text{,}\) constant cross sectional area \(A\) and insulated sides. This derivation will result in flow only in the direction of the length of the bar. The shape of the cross section and material in the bar does not matter but we will show a cylindrical bar.
🔗
Figure 7.2.1. A cylindrical bar which conducts heat.
🔗
The small segment of rod has heat flux \(Q(x,t)\) entering in the left side and has heat flux \(Q(x+\Delta x,t)\) exiting the right side. The temperature is \(T(x,t)\) on the left and \(T(x+\Delta x,t)\) on the right.
🔗
Figure 7.2.2. The heat flux through a slice of thickness \(\Delta x\) of the bar.
🔗
Next, we consider the amount of heat in the segment. The heat is proportional to the mass of the segment (\(\rho \Delta x\)) and the temperature. The proportionality constant, \(c\) is called the specific heat. The amount of heat in this small segment is:
🔗
\begin{equation*} c \rho T \Delta x \end{equation*}
🔗
The temporal change in heat of this element is the derivative with respect to \(T\) and the only quantity that depends on time is the temperature \(T\text{:}\)
🔗
\begin{equation*} c \rho \frac{\partial T}{\partial t} \Delta x \end{equation*}
🔗
and is equal to the total amount of heat entering from the two ends or
🔗
\begin{equation*} c \rho \frac{\partial T}{\partial t} \Delta x = Q(x,t) - Q(x+\Delta x,t) \end{equation*}
🔗
Divide both sides of this equation by \(\Delta x\) and take the limit \(\Delta x \rightarrow 0\text{,}\)
🔗
\begin{equation} c \rho \frac{\partial T}{\partial t} = \lim_{\Delta x \rightarrow 0} \frac{Q(x,t) - Q(x+\Delta x,t)}{\Delta x} = -\frac{\partial Q}{\partial x}\tag{7.2.1} \end{equation}
🔗
The relationship between the heat flux, \(Q\) and the temperature is called Fourier’s Law, and states that the heat transferred across unit area is proportional to the temperature gradient or
🔗
\begin{equation*} Q = -k \frac{\partial T}{\partial x} \end{equation*}
🔗
where \(k\) is the proportionality constant called the thermal conductivity and the negative sign is due to the fact that heat flows from hot to cold (in a negative direction). Substituting Fourier’s Law into (7.2.1) results in
🔗
\begin{equation*} c \rho \frac{\partial T}{\partial t} = -\frac{\partial }{\partial x} \biggl( -k \frac{\partial T}{\partial x} \biggr) \end{equation*}
🔗
and assuming that \(k\) is a constant,
🔗
\begin{equation*} c \rho \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial {x}^2} \end{equation*}
🔗
rearranging we get
🔗
\begin{equation*} \frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial {x}^2} \end{equation*}
🔗
where \(\kappa = k/c \rho\) is called the thermal diffusivity.
🔗

Remark 7.2.3.

The one-dimensional heat equation is
🔗
\begin{equation*} \frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial {x}^2} \end{equation*}
🔗
and is typically solved for a given initial condition \(T(x,0)=f(x)\) and with boundary conditions. We will see some examples of this in Section 7.5.
🔗
🔗
🔗

Subsection 7.2.2 Wave Equation: Modeling a Taut String

Next, we turn to modeling the motion of a taut string fixed on both ends. A common example of this is a guitar string. We look at a small part of the string that has been displaced vertically from rest:
🔗
Figure 7.2.4. A small section of a taut string. The vectors \(\vec{T}\) and \(\vec{T}'\) are the force vectors on the string, the variable \(u\) is the vertical displacement of the string from rest position.
🔗
where \(u(x,t)\) be the vertical position of the string at position \(x\) and time \(t\) and the tension at \(x\) and \(x+\Delta x\) are the force vectors \(\vec{T}\) and \(\vec{T}'\text{.}\) To determine an equation that describes the motion of the string, we need to balance the horizontal a vertical forces. We can write the two force vectors \(\vec{T}\) and \(\vec{T}'\) in terms of the horizontal and vertical components where the subscripts in this case are the components as shown in the figure below.
🔗
Figure 7.2.5. The force vectors from Figure 7.2.4 are written in terms of the component vectors.
🔗
And recall that using Newton’s 2nd law of motion the sum of the forces on a object is equal to the mass of the object times its acceleration in that direction or \(\sum F = ma\text{.}\) There is no horizontal motion of the string, so the horizontal acceleration is zero and therefore:
🔗
\begin{align*} -\vec{T}_x +\vec{T}'_x \amp = 0 \\ -||\vec{T}|| \cos \alpha + || \vec{T}'|| \cos \beta \amp = 0 \end{align*}
🔗
and solving for \(||\vec{T}'||\)
🔗
\begin{equation} ||\vec{T}'|| = ||\vec{T}|| \frac{\cos \alpha}{\cos \beta}\tag{7.2.2} \end{equation}
🔗
Next, we examine the vertical forces. The main difference is that there is a vertical acceleration which is the 2nd derivative of the position \(u\) with respect to \(t\text{.}\)
🔗
\begin{align*} \sum F \amp = ma \\ -\vec{T}_y + \vec{T}_y' \amp = ma \\ -||\vec{T}||\sin \alpha + ||\vec{T}'|| \sin \beta \amp = \rho \Delta x \frac{\partial^2 u}{\partial {t}^2} \end{align*}
🔗
where the mass of the short piece is \(\rho\) the mass density of the string and \(\Delta x\) is the horizontal length of the piece. Substituting (7.2.2) into the equation above,
🔗
\begin{equation*} -||\vec{T}||\sin \alpha + ||\vec{T}|| \frac{\cos \alpha}{\cos \beta} \sin \beta = \rho \Delta x \frac{\partial^2 u}{\partial {t}^2} \end{equation*}
🔗
and dividing through by \(||\vec{T}|| \cos \alpha\)
🔗
\begin{align*} -\frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta} \amp = \frac{\rho \Delta x}{||\vec{T}|| \cos \alpha} \frac{\partial^2 u}{\partial {t}^2} \\ -\tan \alpha + \tan \beta \amp = \frac{\rho \Delta x}{||\vec{T}|| \cos \alpha} \frac{\partial^2 u}{\partial {t}^2} \end{align*}
🔗
The tangent of the angles are the slopes of the function \(u(x,t)\) at \(x+\Delta x\) and \(x \) respectively, or \(\frac{\partial u}{\partial x}\) at \(x+\Delta x\) and \(x\) therefore the above can be written
🔗
\begin{align*} \frac{\partial u}{\partial x} \biggr\vert_{x+\Delta x} - \frac{\partial u}{\partial x} \biggr\vert_x \amp = \frac{\rho \Delta x}{||\vec{T}|| \cos \alpha} \frac{\partial^2 u}{\partial {t}^2} \end{align*}
🔗
Divide by \(\Delta x\) and let \(T=||\vec{T}|| \cos \alpha\) be the horizontal tension in the string
🔗
\begin{align*} \frac{1}{\Delta x} \biggl( \frac{\partial u}{\partial x} \biggr\vert_{x+\Delta x} - \frac{\partial u}{\partial x} \biggr\vert_x \biggr) \amp = \frac{\rho}{T} \frac{\partial^2 u}{\partial {t}^2} \end{align*}
🔗
take the limit as \(\Delta x \rightarrow 0\)
🔗
\begin{align*} \frac{\partial^2 u}{\partial {x}^2} \amp = \frac{\rho}{T} \frac{\partial^2 u}{\partial {t}^2} \end{align*}
🔗
The parameter \(\rho/T\) plays an important role in the solution, and we’ll define \(c=\sqrt{T/\rho}\text{,}\) so the equation can be written:
\begin{equation} \frac{\partial^2 u}{\partial {t}^2} = c^2 \frac{\partial^2 u}{\partial {x}^2}\tag{7.2.3} \end{equation}
🔗
Typically, (7.2.3) is called the wave equation or since there is once one spatial variable, the one-dimensional wave equation.
🔗
🔗

Subsection 7.2.3 Important Second-Order PDEs

The following is a list of other important second-order partial differential equations. Some of these are two-dimensions versions of the wave and heat equations derived above.
🔗
\begin{align*} \frac{\partial^2 u}{\partial {t}^2} \amp = c^2 \frac{\partial^2 u}{\partial {x}^2} \amp \amp \text{1D wave equation} \\ \frac{\partial u}{\partial t}\amp = c^2 \frac{\partial^2 u}{\partial {x}^2} \amp \amp \text{1D heat equation} \\ \frac{\partial^2 u}{\partial {x}^2} + \frac{\partial^2 u}{\partial {y}^2} \amp = 0 \amp\amp \text{2D Laplace's equation} \\ \frac{\partial^2 u}{\partial {t}^2} \amp = c^2 \biggl( \frac{\partial^2 u}{\partial {x}^2}+ \frac{\partial^2 u}{\partial {y}^2}\biggr) \amp \amp \text{2D wave equation} \\ \frac{\partial u}{\partial t}\amp = c^2 \biggl( \frac{\partial^2 u}{\partial {x}^2}+ \frac{\partial^2 u}{\partial {y}^2}\biggr) \amp \amp \text{2D heat equation} \end{align*}
🔗
The remainder of this chapter covers solution techniques of these and other partial differential equations.
🔗
🔗
🔗
PrevTopNext