3. Introduction Chapter 1
• Any equation including independent variable x, Dependent variable y,
and derivatives of dependent variable y in different orders Known as
differential equation.
• We Have two major type:
1. Ordinary Differential Equations
2. Partial Differential Equations.
4. Introduction Chapter 1
Ordinary Differential Equations (ODE)
• They are equations that they include ordinary Derivatives.
• They are as:
• First order ODEs
• Second order ODEs
• ….
• …..
nth order ODEs
6. Introduction Chapter 1
Order of differential Equation
• The highest derivative in differential equation is the order of a differential
equation.
7. Introduction Chapter 1
Degree of a Differential Equations
Degree of a differential equation is the exponent
of the highest derivative in a differential equation.
8. Introduction Chapter 1
How important are differential equations in computer science?
If you develop a computer program to predict
future values, especially on engineering
quantities, you will almost certainly use
differential equations as part of your predictive
algorithms in your tool kit.
9. Introduction Chapter 1
Physics is largely governed by differential equations, more specifically partial
differential equations. They appear in electromagnetism through Maxwell’s
equations, thermodynamics through heat equation, and semiconductors and
quantum mechanism through Schrodinger’s equation. So there won’t be
computers, let alone computer science, if we didn’t understand differential
equations in the first place.
Recently I had an opportunity to use ordinary differential equations for work related
to hashing function, an active topic in compute algorithm research. They arise as
Euler Lagrange equations for some constrained optimization. The technical detail
can be found
10. Introduction Chapter 1
Generally it depend what in CS are You going to do?Eg.
You do not know what mathematical/statistical tool would
be helpful in case of data mining/data science in near
future. Also very helpful could be in subjects as
knowledge discovery, visual data processing, or neural
nets etc.
Also current state of CS and IT in commercial env. is that
You are in financial, insurance or telecommunication
company where knowledge of other parts of math (eg.
financial math. or econometrics )could be desired.
11. Introduction Chapter 1
Differential equation modelling has given us tool to predict the future and analyze the
past but it is only possible due to computer revolution.
1) Differential equations model the complex behavior of physical world.
2) Climate modelling, Big bang theory modelling. prediction of building failures all the
natural phenomenon (heat equations, fluid flow equations, hooks law etc) can be
modelled using differential equations.
3) A large part of todays super computing powers is used to do modelling of differential
equations and help us understand and predict the behavior of Nature.
......even the heating and cooling problem of computer is modelled using the differential
equation and solved by computer itself.
12. Introduction Chapter 1
1.Differential equation are greatly used in game development
For example
In a simple video game involving a jumping motion, a differential equation is
used to model the velocity of a character after the command is given to return
them to the ground in a simulated gravitational field.
Process control and rendering graphics are two particularly
common applications.
13. Introduction Chapter 1
“Differential equations in terms of computer science is
numerical analysis which is the performance of algorithms in
computers and their numerical stability. This opens a broad
range of areas. I don’t think any comp sci graduate should
graduate without it.
Usually problems in computer science are interested with
problems that may be formulated in terms of linear
programming.”
14. Introduction Chapter 1
Physics in general uses a lot of computer science. Computer
applications are involved in several aspects such as modeling (TIM
the incredible machine) underlying logic (Chess or Go) or complex
fluid flow, machine learning or financial analysis.
Differential equation may be used in computer science to model
complex integration or non linear phenomena.
15. Introduction Chapter 1
We see PDEs everywhere. More or less every phenomena be it physical,
chemical or even biological can be represented in terms PDEs. Be it the
behavior of waves, be it the flow of fluids or be it even a chemical reaction
almost everything can be represented in terms of PDEs.
Well what we as researchers do to find out the solution of a particular
problem is try and find a PDE which would represent the behavior of the
system that we are concerned with and then try and solve the PDE in
whatever way we can and then from the solution we infer what further
implications can that phenomena have.
21. Introduction Chapter 1
• Actual Solution
The actual solution to a differential equation is the
specific solution that not only satisfies the differential
equation, but also satisfies the given initial
condition(s).
