First order PDEs

• While studying necessary condition for differentiability of analytic function $f(z)=u(x,y)+\imath v(x,y)$, we had came across first order PDEs Cauchy-Riemann equations
  $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}~$$and$$~\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$$

• Later we also saw that when $f(z)=u(x,y)+\imath v(x,y)$ is an analytic function, then $u$ and $v$ are harmonic functions and satisfy Laplace's equation:
  

  $$u_{xx}+u_{yy} =$$ 0

  $$v_{xx}+v_{yy} = 0.$$

  
  

  This is second order PDE.

• Whatever a physical quantity under investigation is a function of more than one independent variable, its (differential calculus based) model involve partial differential equations. eg. many physical quantities like electric field, magnetic field, position, velocity, acceleration, wave function in quantum mechanics are functions of $3D$ space and time:
  $$u=u(\bar{r},t)=u(x,y,z,t).$$
  Then the partial derivative of $u$ w.r.t. $x$ is defined as

  $$\frac{\partial u}{\partial x}= \lim_{\Delta x \to 0} \frac{u(x+\Delta x,y,z,t)-u(x,y,z,t)}{\Delta x},$$ (1.1)

  change in $u$ w.r.t. $x$ keeping other variables constant and so on.

• In this part of the course, we are going to study analytical and numerical techniques of solving partial differential equations. In a generic setting, we assume that $u$ is dependent variable and $x_1, x_2,..., x_n$ are independent variables (known). i.e.
  $$u=u(x_1, x_2,..., x_n).$$
  In typical models of science and engineering, the independent variables have physical meaning like spatial co-ordinate in a particular direction, temporal variable, etc. In that case, we will be dealing with a situation like $u=u(x,y,z,t).$

• Quite often, we make our life simple by simply considering only two independent variables $x$ and $y$ and assume $u$ is a dependent variable.

• While for the sake of simplicity, we will be dealing with PDEs involving $(x,y,u,u_x,u_y)$, much of the theory can be extend to generic multi-variable case.
  Definition 1 (Order of PDE)   The order of a PDE is defined to be the order of the highest partial derivative occurring in the equation. eg.

  $$\begin{cases} u_{t}+au_{x}=0 & \text{first order}\\ u_{xx}+u... ...cond order}\\ u_t+ku_{xx}=0 & \mbox{second order.} \end{cases}$$ (1.2)

  Definition 2 (Degree of PDE)   The degree of a PDE is defined to be the degree of the highest order derivative occurring in the equation, after the equation has been rationalized. eg. all above PDEs are of first degree.