2 Differential Equations Many problems in physical chemistry (eg. kinetics, dynamics, theoretical chemistry) require solution to a differential equationMany can not be solved analyticallyDeal only with first order ODEHigher order equations can be reduced to a system of 1st order DE
3 Differential Equations Simplest formCan integrate analytically or numerically (using techniques of Chapter 4)
4 Differential Equations General caseMany simpler problems can be solved analyticallyMany involve exHowever, 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 yContinue to iterate until a desire level of accuracy is obtained in yOften gives a power series solution
6 Picard Method – Example Continue to iterate until a desire level of accuracy is obtained in y
12 Improved Euler (Heun’s) Method Euler MethodUse constant derivative between points i & i+1calculated at xiBetter to use average derivative across the intervalyi+1 is not knownPredict – Correct (can repeat)
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