Download presentation

1
**Chapter 6 Differential Equations**

Chem Math 252 Chapter 6 Differential Equations

2
**Differential Equations**

Many problems in physical chemistry (eg. kinetics, dynamics, theoretical chemistry) require solution to a differential equation Many can not be solved analytically Deal only with first order ODE Higher order equations can be reduced to a system of 1st order DE

3
**Differential Equations**

Simplest form Can integrate analytically or numerically (using techniques of Chapter 4)

4
**Differential Equations**

General case Many simpler problems can be solved analytically Many involve ex However, in chemistry (physics & engineering) many problems have to be solved numerically (or approximately)

5
**Picard Method Can not integrate exactly because integrand involves y**

Approximate iteratively by using approximations for y Continue to iterate until a desire level of accuracy is obtained in y Often gives a power series solution

6
**Picard Method – Example**

Continue to iterate until a desire level of accuracy is obtained in y

7
**Picard Method – Example 2**

8
**Euler Method Assume linear between 2 consecutive points**

Between initial point and 1st (calculated) point User selects Dx Need to be careful - too big or too small can cause problems

9
Euler Method – Example

10
**Taylor Method Based on Taylor expansion**

Euler method is Taylor method of order 1 Use chain rule

11
**Taylor Method – Example**

12
**Improved Euler (Heun’s) Method**

Euler Method Use constant derivative between points i & i+1 calculated at xi Better to use average derivative across the interval yi+1 is not known Predict – Correct (can repeat)

13
**Improved Euler Method – Example**

14
**Modified Euler Method Modified Euler Method**

Use derivative halfway between points i & i+1

15
**Modified Euler Method – Example**

16
Runge-Kutta Methods Improved and Modified Euler Methods are special cases 2nd order Runge-Kutta 4th order Runge-Kutta Runge Kutta Runge-Kutta-Gill

17
Runge Methods

18
Kutta Methods

19
**Runge-Kutta-Gill Methods**

20
Systems of Equations All the previous methods can be applied to systems of differential equations Only illustrate the Runge method

21
**Systems of Equations – Example 1**

22
**Systems of Equations – Example 2**

23
**Systems of Equations – Example 3**

24
**Systems of Equations – Example 4**

25
**Systems of Equations – Example 5**

Similar presentations

OK

1/14 5.2 Euler’s Method Compute the approximations of y(t) at a set of ( usually equally-spaced ) mesh points a = t 0 < t 1 <…< t n = b. That is, to.

1/14 5.2 Euler’s Method Compute the approximations of y(t) at a set of ( usually equally-spaced ) mesh points a = t 0 < t 1 <…< t n = b. That is, to.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on world book day images Ppt on review writing resources Ppt on standing order template Ppt on green revolution in india Ppt on tribals of india Ppt on water scarcity in uganda Ppt on chromosomes and genes are actually replicated Appt on falls of neuse Ppt on power system security Ppt on job satisfaction and morale