CP08_ode - Differential Equations Most fundamental and...

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1 1 Differential Equations Differential Equations Most fundamental and basic equations in physics as well as frequently occurring problems appear as differential equations. 2 Examples: ) ( ) ( 2 2 t kx dt t x d m = Simple harmonic oscillator Schrödinger equation (example for 1D) ) , ( ) ( ) , ( 2 1 ) , ( 2 2 t x x V x t x m dt t x d i ψ + = h Part 1 classification 4 Differential Equations 5 is initial value problem for the second order ordinary linear homogeneous differential equation 0 0 2 2 ) 0 ( ) 0 ( ) ( ) ( v t dt dx x t x t kx dt t x d m = = = = = Simple Harmonic Oscillator 6 ODE or PDE The ordinary differential equations (ODE) – have functions of one only independent variable Example: stationary Schrödinger equation The partial differential equations (PDE) – have functions of several independent variables Example: time dependent Schrödinger equation for ) , ( t r Ψ
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2 7 ODE: Linear or Nonlinear A linear differential equation – all of the derivatives appear in linear form and none of the coefficient depends on the dependent variable example: A nonlinear differential equation – if the coefficients depend on the dependent variable, or the derivatives appear in a nonlinear form Examples: ) ( ) ( 2 2 t kx dt t x d m = 0 ) ( ) ( 0 ) ( ) ( ) ( 2 2 2 2 2 2 = = t x dt t x d t t x dt t dx dt t x d 8 Order of ODE The order n of an ordinary differential equation is the order of the highest derivative appearing in the differential equation Examples: second order third order 0 ) ( ) ( 0 ) ( ) ( 3 3 2 2 2 = = dt t dx dt t x d t t x dt t x d t 9 General or partial solution Example: General solution: Partial solutions: 0 ) ( ) ( = t x dt t dx t Ce t x = ) ( t t e t x e t x 8 . 4 ) ( 0 . 2 ) ( = = 10 Homogeneous and nonhomogeneous ODE A homogeneous equation: the each term contains either the function or its derivative, but no other functions of independent variables A nonhomogeneous equation: contains additional term (source terms, forcing functions) which do not involve the dependent variable 0 ) ( ) ( 2 2 = t kx dt t x d m ) cos( ) ( ) ( 0 2 2 t F t kx dt t x d m ω = 11 Three major categories of ODE Initial-value problems – involve time-dependent equations with given initial conditions: Boundary-value problems – involve differential equations with specified boundary conditions: Eigenvalue problems – involve solutions for selected parameters in the equations In reality, a problem may have more then just one of the categories above 0 0 2 2 ) 0 ( , ) 0 ( , 0 ) ( ) ( v t dt dx x t x t kx dt t x d m = = = = = b a y b x y y a x y x y dx x y d = = = = = ) ( , ) ( , 0 ) ( ) ( 2 2 α 12 Three general classifications in physics 9 Propagation problems - are initial value problems in open domains where the initial values are marched forward in time (or space) . The order may be one or greater. The number of initial values must be equal to the order of the differential equation. 9 Equilibrium problems – are boundary-value problems in closed domains where boundary values are specified at boundaries of the solution domain. The order of ODE must be at least two.
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This note was uploaded on 09/29/2009 for the course PHYSICS 811 taught by Professor Godunov during the Fall '09 term at Old Dominion.

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CP08_ode - Differential Equations Most fundamental and...

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