Chapter 6 Differential Equations Chem 302 - Math 252 Chapter 6 Differential Equations

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

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

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

Picard Method – Example Continue to iterate until a desire level of accuracy is obtained in y

Picard Method – Example 2

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

Euler Method – Example

Taylor Method Based on Taylor expansion Euler method is Taylor method of order 1 Use chain rule

Taylor Method – Example

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)

Improved Euler Method – Example

Modified Euler Method Modified Euler Method Use derivative halfway between points i & i+1

Modified Euler Method – Example

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

Runge Methods

Kutta Methods

Runge-Kutta-Gill Methods

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

Systems of Equations – Example 1

Systems of Equations – Example 2

Systems of Equations – Example 3

Systems of Equations – Example 4

Systems of Equations – Example 5