$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$

The aim of this is to introduce and motivate partial differential equations (PDE). The section also places the scope of studies in APM346 within the vast universe of mathematics.

A *partial differential equation (PDE)* is an *equation* involving *partial derivatives*. This is not so informative so let’s break it down a bit.

An *ordinary differential equation (ODE)* is an equation for a function which depends on one independent variable which involves the independent variable, the function, and derivatives of the function:
\begin{equation*}
F( t, u(t), u'(t), u^{(2)}(t), u^{(3)}(t),
\ldots, u^{(m)}(t)) = 0.
\end{equation*}
This is an example of an ODE of *order* $m$ where $m$ is a highest order of the derivative in the equation. Solving an equation like this on an interval $t\in [0,T]$ would mean finding a function $t \mapsto u(t) \in \mathbb{R}$ with the property that $u$ and its derivatives satisfy this equation for all $t \in [0,T]$. The problem can be enlarged by replacing the real-valued $u$ by a vector-valued one $\mathbf{u}(t)= (u_1 (t), u_2 (t), \dots, u_N (t))$. In this case we usually talk about *system of ODEs*.

Even in this situation, the challenge is to find functions depending upon exactly one variable which, together with their derivatives, satisfy the equation.

When you have function that depends upon several variables, you can differentiate with respect to either variable while holding the other variable constant. This spawns the idea of *partial derivatives*. As an example, consider a function depending upon two real variables taking values in the reals:
\begin{equation*}u:
\mathbb{R} ^n \to \mathbb{R}.
\end{equation*}
As $n=2$ we sometimes visualize a function like this by considering its *graph* viewed as a surface in $\mathbb{R}^3$ given by the collection of points
\begin{equation*}
{ (x,y,z) \in {\mathbb{R}^3}: z = u(x,y) }.
\end{equation*}
We can calculate the derivative with respect to $x$ while holding $y$ fixed. This leads to $u_x$, also expressed as $\partial_x u$,
$\frac{\partial u}{\partial x}$, and $\frac{\partial\ }{\partial x}u$. Similarly, we can hold $x$ fixed and differentiate with respect to $y$.

A *partial differential equation* is an equation for a function which depends on more than one independent variable which involves the independent variables, the function, and partial derivatives of the function:
\begin{equation*}
F(x,y, u(x,y), u_x (x,y), u_y (x,y), u_{xx} (x,y), u_{xy}
(x,y), u_{yx} (x,y), u_{yy} (x,y)) = 0.
\end{equation*}
This is an example of a PDE of order 2. Solving an equation like this
would mean finding a function $(x,y) \to u(x,y)$ with the property that $u$ and is partial derivatives satisfy this equation is true for all admissible arguments.

Similarly to ODE case this problem can be enlarged by replacing the real-valued $u$ by a vector-valued one $\mathbf{u}(t)= (u_1 (x,y), u_2 (x,y), \dots, u_N (x,y))$. In this case we usually talk about *system of PDEs*.

PDEs are often referred as *Equations of Mathematical Physics* (or *Mathematical Physics* but it is incorrect as Mathematical Physics is now a separate field of mathematics) because many of PDEs are coming from different domains of physics (acoustics, optics, elasticity, hydro and aerodynamics, electromagnetism, quantum mechanics, seismology etc).

However PDEs appear in other fields of science as well (like quantum chemistry, chemical kinetics); some PDEs are coming from economics and financial mathematics, or computer science.

Many PDEs are originated in other fields of mathematics.

(Some are actually systems)

- Simplest First Order Equation \begin{equation*} u_x = 0. \end{equation*}
- Transport Equation \begin{equation*} u_t + c u_x = 0. \end{equation*}
- $\bar{\partial}$ Equation (complex variables)
\begin{equation*}
\bar{\partial}f:=\frac{1}{2}(f_x+ if_y) =0,
\end{equation*}
($\bar{\partial}$ is known as "di-bar" or Wirtinger derivatives), or as $f= u+iv$
\begin{equation*}
\left\{\begin{aligned}
u_x-v_y=0,\\
u_y+v_x=0.\end{aligned}\right.
\end{equation*}
Those who took
*Complex variables*know that those are*Cauchy-Riemann equations*. - Laplace’s Equation (in 2D) \begin{equation*} \Delta u:=u_{xx} + u_{yy} = 0 \end{equation*} or similarly in the higher dimensions.
- Heat Equation \begin{equation*} u_t = k\Delta u; \end{equation*}

(The expression $\Delta$ is called the *Laplacian* (*Laplace operator*) and is defined as $\partial_{x}^2 + \partial_{y}^2+\partial_{z}^2$ on $\mathbb{R}^3$).

- Schrödinger Equation (quantum mechanics) \begin{equation*} i\hbar \partial_t\psi =\bigl(-\frac{\hbar^2}{2m}\Delta +V\bigr)\psi. \end{equation*} Here $\psi$ is a complex-valued function.
- Schrödinger-Pauli Equation (quantum mechanics) see in Section 14.3.
- Dirac Equations (relativistic quantum mechanics).
\begin{equation*}
i\hbar\partial_t\psi =
\bigl(\beta mc^2-
\sum_{1\le k\le 3}ic\alpha_k\hbar \partial_{x_k}\bigr)\psi
\end{equation*}
with
*Dirac $4\times 4$-matrices*$\alpha_1,\alpha_2,\alpha_3,\beta$. See also in Section 14.3. Elasticity Equation (homogeneous and isotropic) \begin{equation*} \mathbf{u}_{tt}=\lambda \Delta\mathbf{u}+ \mu \nabla(\nabla\cdot\mathbf{u}). \end{equation*}

*homogeneous*means "the same in all places" (an opposite is called*inhomogeneous*) and*isotropic*means "the same in all directions" (an opposite is called*anisotropic*).Wave Equation \begin{equation*} u_{tt} - c^2\Delta u = 0; \end{equation*} sometimes $\square:=c^{-2}\partial_t^2-\Delta$ is called (

*d'Alembertian*or*d'Alembert operator*). It appears in elasticity, acoustics, electromagnetism and so on.One-dimentsional wave equation \begin{gather*} u_{tt} - c^2u_{xx} = 0 \end{gather*} often is called

*string equation*and describes f.e. a vibrating string.Equation of oscillating rod (with one spatial variable) \begin{gather*} u_{tt} + Ku_{xxxx} = 0 \end{gather*} or plate (with two spatial variables) \begin{equation*} u_{tt} + K\Delta^2 u = 0; \end{equation*}

Maxwell Equation (electromagnetism) in vacuum \begin{equation*} \left\{\begin{aligned} &\mathbf{E}_{t} - c\nabla \times \mathbf{H} = 0,\\ &\mathbf{H}_{t} + c\nabla \times \mathbf{E} = 0,\\ &\nabla\cdot\mathbf{E}=\nabla\cdot\mathbf{H}=0. \end{aligned}\right. \end{equation*} Here $\mathbf{E}$ and $\mathbf{H}$ are vectors of electric and magnetic intensities, so the first two lines are actually $6\times 6$ system. The third line means two more equations, and we have $8\times 6$ system. Such systems are called

*overdetermined*. See also Section 14.2.Dirac Equations (relativistic quantum mechanics)} \begin{gather*} i\hbar\partial_t\psi = \bigl(\beta mc^2- \sum_{1\le k\le 3}ic\hbar\alpha_k \partial_{x_k}\bigr)\psi \end{gather*} with

*Dirac $4\times 4$-matrices*$\alpha_1,\alpha_2,\alpha_3,\beta$. Here $\psi$ is a complex $4$-vector, so in fact we have $4\times 4$ system.Navier-Stokes Equation (hydrodynamics for incompressible liquid)\begin{equation*} \left\{\begin{aligned} &\rho\mathbf{v}_{t} + (\mathbf{v}\cdot \nabla ) \rho\mathbf{v} -\nu \Delta \mathbf{v}=-\nabla p,\\ &\nabla \cdot \mathbf{v} = 0, \end{aligned}\right. \end{equation*} where $\mathbf{v}$ is a velocity and $p$ is the pressure; when viscosity $\nu=0$ we get Euler equation \begin{equation*} \left\{\begin{aligned} &\rho\mathbf{v}_{t} + (\mathbf{v}\cdot \nabla ) \rho\mathbf{v} =-\nabla p,\\ &\nabla \cdot \mathbf{v} = 0. \end{aligned}\right. \end{equation*} Both of them are $4\times 4$ systems.

Yang-Mills Equation (elementary particles theory) \begin{align*} &\partial_{x_j} F_{jk} + [A_j,F_{jk}]=0,\\ &F_{jk}:= \partial_{x_j} A_k-\partial_{x_k} A_j +[A_j,A_k], \end{align*} where $A_k$ are traceless skew-Hermitian matrices. Their matrix elements are unknown functions, so we have 2nd order system of PDEs. This is a $2$-nd order system.

Einstein Equation for General Relativity \begin{gather*} G_{\mu\nu}+ \Lambda g_{\mu\nu}=\kappa T_{\mu\nu}, \end{gather*} where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}$ is the

*Einstein tensor*, $g_{\mu\nu}$ is the*metric tensor*, $R_{\mu\nu}$ is the*Ricci curvature tensor*, and $R$ is the*scalar curvature*, $T_{\mu\nu}$ is the*stress–energy tensor*, $\Lambda$ is the*cosmological constant*and $\kappa$ is the*Einstein gravitational constant*. Components of Ricci curvature tensor are expressed through the components of the metric tensor, their first and second derivatives. So we have 2nd order system of PDEs. This is a $2$-nd order system.Black-Scholes equation (Financial mathematics) is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives: \begin{gather*} {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0, \end{gather*} where $V$ is the price of the option as a function of stock price $S$ and time $t$, $r$ is the risk-free interest rate, and $\sigma$ is the volatility of the stock.

and so on...

**Remark 1.**

- Some of these examples are actually not single PDEs but the systems of PDEs.
- In all this examples there are
*spatial variables*$x,y,z$ and often*time variable*$t$ but it is not necessarily so in all PDEs. Equations, not including time, are called*stationary*(an opposite is called*nonstationary*). - Equations could be of different order with respect to different variables and it is important. However if not specified the order of equation is the highest order of the derivatives invoked.
- In the Textbook we will deal mainly with the wave equation, heat equation and Laplace equation in their simplest forms.