Examples Of Differntial Equations-Numerical Analysis-Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Examples ofExamples ofDifferentialDifferentialEquationsEquations

We considered thedifferential equation of firstorder with the initialcondition

y ( t^0

) =^ y

. (^0) ( , )

dy

f t y dt^



We obtained the solutionof the given differentialequation in the form of arecurrence relation

1

(^ ,

m^

m^

m^

m

y^

y^

hf t

y

Euler’s algorithmLet us try to approximate thesolution of the given IVP at(N+1) equally spaced numbersin the interval [a ,b]

( ,^

), ,

( )

y^

f t y a^

t^ b

y a



  ^

^



INPUT

endpoints a, b; integer

N, initial condition (alpha) OUTPUT

approximate w to y

at the (N+1) values of t

Step 3 Set^ w = w + h f (t , w); (compute w

).i

t = a + i h (compute t

)i

Step 4

OUTPUT

(t , w)

Step 5

STOP

ExampleUse Euler’s method toapproximate the solution ofIVP^ y’= y - t

,^ 0 < t < 2,

y ( 0 ) = 0.5 with N = 10.

Recall

Runge-Kutta(Order Four)METHOD

The fourth-order R-Kmethod was described as^1

1

2

3 4

1 (^

2

2

)

6 n^

n y^

y^

k^

k^

k^ k

^ 

^

^

^

