B.TECH BIOTECH

1. TEAM MEMBERS : 1.DN Vaisnavi (57) 3.Anitha .R(62)
2.Jenanisankari.C(59) 4.Faazna(63)

2. Founder of Taylor’s Series
Brook Taylor  Brook Taylor was an English mathematician who
is best known for his Taylors series and Taylors
theorem.
 Born : 18 august 1685, England.
 Died : 29 December 1731,London,England .
 He wrote a book called “Methodus
incrementorum directa et inversa”,added a new
branch to higher mathematics now called as
“calculus of finite difference”

3. Founder’s of RK Method of Fourth order
Runge Kutta  C.Runge and M.W.Kutta are the two German
mathematicians discovered Runge Kutta
method of fouth order ordinary differential
equations
 Runge was known for astronmoical
spectroscopy and kutta for aerodynamics.
 So both decided to work under same domain
called Numerical Analysis which resulted in
“Runge kutta Method”

4. General Applications of Taylor’s series
and RK method
 The Taylor series method and RK Method is an earliest analytic-
numeric algorithms used for is used for solving initial value problems
for ordinary differential equations.
 RK Method is generally used for analysing equations of motion for
multibody systems with flexible parts, which are fairly stiff, time-
dependent and non-linear functions.
 Taylors series is used to find sum of the series , to evaluate limits and
it is used to approximate polynomial function.

5. Applications in Biotechnology
 Runge kutta method is used to determine the BioKinetic parameters
in Environmental Biotechnology.
 Both Taylors series and RK method is used for
Numerical Analysis in Biomechanical Modelling.
 Numerical Methods of solving problem has been
Used to design Biomedical Instruments.

6. Definitions
A differential equation is any equation which contains
derivatives, either ordinary derivatives or partial derivatives.
 a Taylor series is a representation of a function as an infinite
sum of terms that are calculated from the values of the function's
derivatives at a single point.
 A method of numerically integrating ordinary differential
equations by using a trial step at the midpoint of an interval to
cancel out lower-order error terms is called RK Method.

7. Working Rule for Taylor’s series
𝑑𝑦
𝑑𝑥
= 𝑓 𝑥, 𝑦 , 𝑦( 𝑥0 = 𝑦0
Consider the first order differential equation with initial
conditions ,
𝑦 𝑥0 = 𝑦0
To find y(𝑥1) , y(𝑥2) ………………
𝑑𝑦
𝑑𝑥
= 𝑓 𝑥, 𝑦 , 𝑦 𝑥0 = 𝑦0

8. Working Rule For Taylors series
 Write the values of f , 𝑥0 , 𝑦0 , 𝑥1 , 𝑥2 from the given equation.
 Differentiating the given equation , to find the derivatives such as
𝑦′
, 𝑦′′
, 𝑦′′′
etc ………..
 Substituting the values of 𝑥0 , 𝑦0 in these derivatives.
 Then find 𝑦0
′
, 𝑦0
′′
, 𝑦0
′′′
…………….
 Find the value of ‘h’ by h = 𝑥1 − 𝑥0 or h = 𝑥2-𝑥1.

9. Working Rule for Taylors Series
 By Taylor’s method 𝑦1 is given by
substituting 𝑥1 and 𝑦1in the above equation for finding
𝑦1
′
𝑦1
′′
𝑒𝑡𝑐 … … … . . 𝑡ℎ𝑒𝑛 By Taylors method 𝑦2 is given by
𝑦1 = 𝑦0 +
ℎ
1!
𝑦0
′
+
ℎ2
2!
𝑦0
′′
+
ℎ3
3!
𝑦0
′′′
+ ⋯
𝑦2 = 𝑦1 +
ℎ
1!
𝑦1
′
+
ℎ2
2!
𝑦1
′′
+
ℎ3
3!
𝑦1
′′′
+ ⋯

10. Example 1 Taylors series Method
𝑑𝑦
𝑑𝑥
= 𝑥 + 𝑦 𝑤𝑖𝑡ℎ 𝑦 1 = 0 𝑎𝑛𝑑 𝑔𝑒𝑡 𝑦 1.1 𝑎𝑛𝑑 𝑦 1.2 𝑏𝑦 𝑇𝑎𝑦𝑙𝑜𝑟𝑠 𝑚𝑒𝑡ℎ𝑜𝑑
solution : To find y(1.1)
𝑥0 = 1 𝑦0 = 0 f = (x +y) 𝑥1 = 1.1 𝑥2 = 1.2 h = 𝑥1 - 𝑥0 , h = 0.1
𝑑𝑦
𝑑𝑥
= 𝑥 + 𝑦 , differentiating with respect to x , we get
𝑦′ = x+y 𝑦′′ = 1 +𝑦′
𝑦′′′ = 𝑦′′ sub (𝑥0, 𝑦0) = (0,1) in this eq ,we get 2
𝑦0
′
= 𝑥0 + 𝑦0
=1+0
= 1

11. Example sum
likewise we get 𝑦0
′′
= 2 , 𝑦0
′′′
= 2
BY Taylors method ,
𝑦1 = 0.1 +
0⋅1
1
1 +
0.1 2
2
2 +
0⋅1 3
6
2 + ⋯
= 0.1103
(𝑥1 , 𝑦1) = (1.1,0.1103)
To find y(1.2):
Substituting (𝑥1 , 𝑦1 ) in equation 2, we get 𝑦1
′
= 1.2103 , 𝑦1
′′
=2.2103 ,𝑦1
′′′
= 2.2103
By Taylors method ,
𝑦2 = 0.1103 +
0.1
1
1.2103 +
0 ⋅ 1 2
2
2.2103 +
0 ⋅ 1 3
0
2.2103 + ⋯
= 0.2464
(𝑥 , 𝑦 ) = (1.2 , 0.2464)

12. Working Rule for RK Method
To find 𝑦1
𝑑𝑦
𝑑𝑥
= 𝑓 𝑥 , 𝑦 when y(𝑥0) = 𝑦0
The fourth order RK Algorithm is given by,
 𝑘1 = ℎ𝑓(𝑥0, 𝑦0 )
 𝐾2 = ℎ𝑓 𝑥0 +
ℎ
2
, 𝑦0 +
𝑘1
2
 𝑘3 = ℎ𝑓 𝑥0 +
ℎ
2
, 𝑦0 +
𝑘2
2
 𝑘4 = ℎ𝑓(𝑥0 + ℎ , 𝑦0 + 𝑘3)
 ∆𝑦 =
1
6
[𝑘1 + 2𝐾2 + 2𝑘3 + 𝑘4]
 𝑦1 = 𝑦0 + ∆𝑦
To find 𝑦2
𝑑𝑦
𝑑𝑥
= 𝑓 𝑥 , 𝑦 when y(𝑥0) = 𝑦0
The fourth order RK Algorithm is given by,
 𝑘1 = ℎ𝑓(𝑥1, 𝑦1 )
 𝐾2 = ℎ𝑓 𝑥1 +
ℎ
2
, 𝑦1 +
𝑘1
2
 𝑘3 = ℎ𝑓 𝑥1 +
ℎ
2
, 𝑦1 +
𝑘2
2
 𝑘4 = ℎ𝑓(𝑥1 + h , 𝑦1+ 𝑘3)
 ∆𝑦 =
1
6
[𝑘1 + 2𝐾2 + 2𝑘3 + 𝑘4]
 𝑦2 = 𝑦1 + ∆𝑦

13. Example 2 – RK Method
Apply RK Method of fourth to find y(0.2) given 𝑦′
= x+y , y(0) = 1
Solution:
𝑥0 = 0 𝑦0 = 1 f = (x +y) 𝑥1 = 0.2 h = 𝑥1 - 𝑥0 , h = 0.2
To find y(0.2): (using the Algorithm)
 𝑘1 = 0.2 1
𝑘1 = 0.2
 𝑘2 = 0.2𝑓 0 +
0.2
2
, 1 +
0.2
2
 𝑘2 = 0.2 0.1 + 1.1
 𝑘2 = 0.24

14. Example 2 – RK Method
 𝑘3 = 0.2𝑓 0 + 0.1 , 1 + 0.12
 𝑘3 = 0.2 0.1 + 1.22
 𝑘3 = 0.244 likewise
 𝑘4 = 0.2888
 ∆𝑦 =
1
6
[𝑘1 + 2𝐾2 + 2𝑘3 + 𝑘4] , ∆𝑦 =
1
6
[0.2+2(0.24)+2(0.244)+0.2888]
 we get ∆𝑦 = 0.2428
 𝑦1 = 𝑦0 + ∆𝑦
 𝑦1 = 1+0.2428
 𝑦1 = 1.2428 , the solution is (0.2 , 1.2428 )

15. References
Taylor series revisited for numerical methods at Numerical
Methods for the STEM Undergraduate
 https://en.wikipedia.org/wiki/Taylor_series
 www.Mathsworld.com
 https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods