Differentiate between ODE & PDE?

In this post, you will learn that what is the difference between ordinary differential equations (ODE) & partial differential equations (PDE).

First, we will start with dependent and independent variables.

Dependent & Independent Variable:

Take any equation which gives the relationship between two variables i.e., x & y. 

\mathrm{y=2x^2}

• Compute the value of y by taking x as 1, 2 & 3.

\mathrm{y=2\left(1\right)^2=2;\;\;y=2\left(2\right)^2=8;\;y=2\left(3\right)^2=18}

• Here, the value of y depends on the value of x.
• So, we can say that variable x is independent while variable y is dependent.  

\mathrm{y=2x^2\;\;\Rightarrow y=f\left(x\right)}

• Independent variables can be more than one.

\mathrm{Z=2x^2+2y^2\;\;\Rightarrow Z=f\left(x,\;y\right)}

• These variables are important factors in any chemical process.
• Consider a plug flow reactor in which reactant A of concentration C_A0 is reacting.

• The concentration of reactant A decreases along the length of PFR.
• It means the concentration is a function of spatial coordinates along the length of the reactor.

• In the above example, we are more interested to develop the relationship between concentration (C_A) & spatial coordinate (x) and their rate change (dC_A/dx).
• A differential equation is needed to observe how physical values like concentration, temperature, and so on vary with independent variables such as space and time.

Differential Equation:

These equations are used to express many general laws of nature.

• It provides a relationship between dependent and independent variables.
• Depending on the no. of independent variables, the differential equation can be categorized as Ordinary differential derivative (ODE) & Partial differential derivative (PDE). 

ODE:

It contains only one independent variable. So, the derivative of the dependent variable with respect to the independent variable is the ordinary derivative.

\mathrm{\frac{dy}{dx}-ax=0;\;\;\;\;\frac{d^3y}{{dx}^3}+\frac yx-b=0}

Rate equation,

\mathrm{-\frac{dC_A}{dt}=k\left(C_A\right)^n}

• It involves differentials with respect to only one independent variable.
• For example, in the above equation, the independent variable is x and the solution is y(x).

PDE:

It contains several independent variables. So, the derivative of the dependent variable with respect to one independent variable while holding other independent variables constant is a partial derivative.

\mathrm{\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=0;\;\;\;\;\frac{\partial^2z}{\partial x\partial y}+\frac{\partial z}{\partial x}+z=0}

Heat Conduction equation,

\mathrm{\frac{\partial^2T}{\partial^2x}+\frac{\partial^2T}{\partial^2y}+\frac{\partial^2T}{\partial^2z}=\frac1\alpha\frac{\partial T}{\partial t}}

Fick’s 2nd law equation,

\mathrm{\frac{\partial^2C}{\partial^2x}+\frac{\partial^2C}{\partial^2y}+\frac{\partial^2C}{\partial^2z}=\frac1D\frac{\partial T}{\partial t}}

• It involves differentials with respect to many independent variables.
• For example, in the above equation, the independent variables are x, y, and the solution is y(x, y).
• PDEs have a more complex solution than ODEs.