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**Chapter 6 Differential Equations**

Chem Math 252 Chapter 6 Differential Equations

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**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

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**Differential Equations**

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

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**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)

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**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

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**Picard Method – Example**

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

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**Picard Method – Example 2**

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**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

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Euler Method – Example

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**Taylor Method Based on Taylor expansion**

Euler method is Taylor method of order 1 Use chain rule

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**Taylor Method – Example**

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**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)

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**Improved Euler Method – Example**

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**Modified Euler Method Modified Euler Method**

Use derivative halfway between points i & i+1

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**Modified Euler Method – Example**

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

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Runge Methods

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Kutta Methods

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**Runge-Kutta-Gill Methods**

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

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**Systems of Equations – Example 1**

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**Systems of Equations – Example 2**

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**Systems of Equations – Example 3**

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**Systems of Equations – Example 4**

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**Systems of Equations – Example 5**

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