Simulation and modeling - Old Questions
5. Describe the importance of differential/Partial differential equations in simulation.
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 ).
Necessity of differential equations:
1. Most physical and chemical process occurring in the nature involves rate of change, which requires differential equations to provide mathematical model.
2. It can be used to understand general effects of growth trends as differential equations can represent a growth rate.