Simulation and Modelling - Old Questions

12. Write short notes on: (2 × 2.5 = 5)

    a. Differential equation

    b. Markov Chain 

a. Differential equation

The equation that consists of the higher order derivatives of the dependent variable is known as differential equations.
- The differential equation is said to be linear if any of the dependent variables and its derivatives have power of one and are multiplied by the constant.
E.g. M x’’ + D x’ + K x = K F(t)
where, M, D and K are constants; F(t) is the input to the system depending upon the independent variable t; x’’ and x’ are second and first order derivatives of dependent variable x.
- The differential equation is said to be non-linear if the dependent variable or any of its derivatives are raised to a power or are combined in other way like multiplication.

The differential equation is said to be partial if more than one independent variables occur in a differential equation.
-E.g. Equation of flow of heat in three dimensional body. It consists of four independent variables ( three dimensions and time ) and one dependent variable ( temperature ).

b. Markov Chain

If the future states of a process are independent of the past and depend only on the present , the process is called a Markov process. A discrete state Markov process is called a Markov chain. A Markov Chain is a random process with the property that the next state depends only on the current state.

Markov chains are used to analyze trends and predict the future. (Weather, stock market, genetics, product success, etc).