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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 1 st 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 e x 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 1 st (calculated) point User selects x 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 Use chain rule Euler method is Taylor method of order 1

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Taylor Method – Example

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

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Improved Euler Method – Example

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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 –2 nd order Runge-Kutta 4 th 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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