$$\newcommand{\D}{\displaystyle} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})}$$

# 5.3 Second order partial differenatial equations

A seond order PDE for two independent variables $$x$$ and $$y$$ can be written as: $$$$A \frac{\partial^2\phi}{\partial x^2}+B \frac{\partial^2\phi}{\partial x \partial y}+C \frac{\partial^2\phi}{\partial y^2}+f=0 \tag{5.19}$$$$ If $$A, B, C$$ and $$f$$ are functions of $$x, y,\phi,\frac{\partial\phi}{\partial x}$$ and $$\frac{\partial\phi}{\partial y}$$, (5.19) is said to be quasi-linear. If $$A, B$$ and $$C$$ are only functions of $$x$$ and $$y$$, (5.19) is semi-linear. If $$f$$ is only function of $$x$$ and $$y$$, (5.19) is linear.

## 5.3.1 Example 3:

\begin{align} \left(\frac{\partial\phi}{\partial x}\right)\frac{\partial^2\phi}{\partial x^2}-\frac{\partial\phi}{\partial x}-e^{xy}\sin\phi& =0 \: \ \text{quasi-linear} \tag{5.20}\\ x \frac{\partial^2\phi}{\partial x^2}-\frac{\partial\phi}{\partial x}-e^{xy}\sin\phi& =0 \: \ \text{semi-linear} \tag{5.21}\\ \left(\frac{\partial^2\phi}{\partial x^2}\right)^2\frac{\partial^2\phi}{\partial y^2} &=0 \: \ \text{non-linear} \tag{5.22} \end{align}

A PDE that can be written in the following form: $$$$A \frac{\partial^2\phi}{\partial x^2}+B \frac{\partial^2\phi}{\partial x \partial y}+C \frac{\partial^2\phi}{\partial x^2}+D \frac{\partial\phi}{\partial x}+E \frac{\partial\phi}{\partial y}+F\phi+G=0 \tag{5.23}$$$$ where $$A, B, C, D, E, F$$, and $$G$$ are only functions of $$x$$ and $$y$$, is a 2nd order linear PDE. (5.23) is consequently a special case of (5.19). Notice that usually we will use the term non-linear as opposed to linear. Let us now investigate whether (5.19) has characteristics or not.

Consider a function $$\phi$$ such that $$u=\frac{\partial\phi}{\partial x}$$ and $$v=\frac{\partial\phi}{\partial y}$$, which in turn implies that $$\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$$.

(5.19) can then be written as a system of two 1st order PDEs: \begin{align} A \frac{\partial u}{\partial x}+B \frac{\partial u}{\partial y}+C \frac{\partial v}{\partial y}+f &=0 \tag{5.24} \\ \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x} & =0 \tag{5.25} \end{align}

It can be shown that a higher order (quasi-linear) PDE can always be written as a system of 1st order PDEs. Moreover, the resulting first order system can have many forms. If, for example, (5.19) is the potential equation in gas dynamics, we can denote $$\phi$$ as the potential speed. (5.25) is the condition of irrotational flow.

We will now try to write (5.24) and (5.25) as a total differential (see (5.13)(5.15)). Multiply (5.25) with any scalar $$\sigma$$ and add to (5.24): $$$$A\left[\frac{\partial u}{\partial x}+\left(\frac{B+\sigma}{A}\right)\frac{\partial u}{\partial y}\right]-\sigma\left[\frac{\partial v}{\partial x}-\frac{C}{\sigma}\frac{\partial v}{\partial y}\right]+f=0 \tag{5.26}$$$$ Now we have: $$\begin{equation*} \frac{du}{dx}=\frac{\partial u}{\partial x}+\frac{dy}{dx}\frac{\partial u}{\partial y},\ \frac{dv}{dx}=\frac{\partial v}{\partial x}+\frac{dy}{dx} \frac{\partial v}{\partial y} \end{equation*}$$

Hence: $$$$\frac{du}{dx}=\frac{\partial u}{\partial x}+\lambda\frac{\partial u}{\partial y},\ \frac{dv}{dx}=\frac{\partial v}{\partial x}+\lambda \frac{\partial v}{\partial y} \tag{5.27}$$$$ where we have defined $$\lambda$$ as: $$$$\lambda=\frac{dy}{dx} \tag{5.28}$$$$

By comparing (5.26) and (5.27): $$$$\frac{dy}{dx}=\lambda=\frac{B+\sigma}{A}=-\frac{C}{\sigma} \tag{5.29}$$$$

(5.29) inserted in (5.26) gives the compatibility equation $$$$A \frac{du}{dx}-\sigma \frac{dv}{dx}+f=0 \tag{5.30}$$$$ If the characteristics are real, i.e. $$\lambda$$ is real, we have transformed the original PDE into an ODE given by (5.30) along the directions defined by (5.28).

From (5.29): $$$$\sigma=-\frac{C}{\lambda} \tag{5.31}$$$$ which also gives: $$$$\lambda=\frac{B-\frac{C}{\lambda}}{A} \tag{5.32}$$$$

The following 2nd order equation can be obtained from (5.32) to determine $$\lambda$$: $$$$A\lambda^2-B\lambda+C=0 \tag{5.33}$$$$ or by using (5.28): $$$$A\cdot(dy)^2-B\cdot dy\cdot dx+C\cdot(dx)^2=0 \tag{5.34}$$$$

After $$\lambda$$ is found from (5.33) and (5.34), $$\sigma$$ can be found from (5.31) and (5.32) so that the compatibility equation in (5.30) can be determined. Instead of using (5.30), we can insert $$\sigma$$ from (5.29) in (5.30) so that we get the following compatibility equation: $$$$A \frac{du}{dx}+\frac{C}{\lambda}\frac{dv}{dx}+f=0 \tag{5.35}$$$$ 2nd degree equations (5.33) and (5.34) have the roots $$\lambda_1$$ and $$\lambda_2$$ $$$$\lambda_{1,2}=\frac{B\pm \sqrt{B^2-4AC}}{2A} \tag{5.36}$$$$

We have three possibilities for the roots in (5.36):

• $$B^2-4AC > 0 \Rightarrow\lambda_1$$ and $$\lambda_2$$ are real
• $$B^2-4AC < 0 \Rightarrow\lambda_1$$ and $$\lambda_2$$ are complex
• $$B^2-4AC = 0 \Rightarrow\lambda_1=\lambda_2$$ and real
The quasi-linear PDE (5.19) is called
• Hyperbolic if $$B^2-4AC > 0$$: $$\lambda_1$$ and $$\lambda$$ are real
• Elliptic if $$B^2-4AC < 0$$: $$\lambda_1$$ and $$\lambda$$ are complex
• Parabolic if $$B^2-4AC = 0$$: $$\lambda_1=\lambda_2$$ are real
The above terms come from the analogy with the cone cross-section equation of the expression $$Ax^2+Bxy+Cy^2+Dx+F=0$$. For example, $$x^2-y^2=1$$ represents a hyperbola if $$B^2-4AC > 0$$.

The roots $$\lambda_1$$ and $$\lambda_2$$ are characteristics. Hence:

• Hyperbolic: Two real characteristics.
• Elliptic: No real characteristics.
• Parabolic: One real characteristic.

## 5.3.2 Example 4: Examples of classification of various PDEs

The wave equation $$\dfrac{\partial^2u}{\partial x^2}-\dfrac{\partial^2u}{\partial y^2}=0$$ is hyperbolic since $$B^2-4AC=4 > 0$$ Characteristics are given by $$\lambda^2=1\to \dfrac{dy}{dx}=\pm 1$$

Laplace equation $$\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0$$ is elliptic since $$B^2-4AC=-4 < 0$$

The diffusion equation $$\dfrac{\partial u}{\partial y}=\dfrac{\partial^2u}{\partial x^2}$$ is parabolic since $$B^2-4AC=0$$

Linearisert potensial-ligning for kompressibel strømning: $$$$(1-M^2)\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}=0,\ M=\text{Mach-number.} \tag{5.37}$$$$

• $$M=1$$: Parabolic (degenerate).
• $$M < 1$$: Elliptic. Subsonic flow.
• $$M > 1$$: Hyperbolic. Supersonic flow.