Prof. Muhammad Saeed ( Ordinary Differential Equations )

2M.Sc. Physics 1.Solution of ODEs 1.Initial Value Problems ODEs of the form: Eulers Method It can be derived from Taylor Series a)Single-Step Mwthods: a)Single-Step Mwthods:

M.Sc. Physics3 Runge-Kutta Methods i)2 nd Order R-K ii)3rd Order R-K Modified Eulers Method

M.Sc. Physics4 iii)4th Order R-K

M.Sc. Physics5 iii)5th Order R-K

M.Sc. Physics6 iii)Runge-Kutta-Fehlberg

M.Sc. Physics7 b)Multistep Methods: Adams Method Milnes Method

M.Sc. Physics8 Adam-Moulton Method

M.Sc. Physics9 2.Higher Order Initial Value Problems Transform the equation into the first order equations. Apply any previous method on the equations simultaneously. 3.System of ODEs Lower Order Conversion Finite Difference Method Replace all derivatives by difference formulas and form a matrix for function (y) values.

M.Sc. Physics10 4.Boundary Value Problems Shooting Method Finite Difference Method 5.Partial Differential Equations Finite Difference Method Guess derivatives initial value and compare the boundary values. Solution of Laplace Equation.

M.Sc. Physics11