Transcription of Euler’s Method, Taylor Series Method, Runge Kutta …
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Euler’s Method, Taylor Series Method, Runge Kutta …
Euler s Method, Taylor Series Method, RungeKutta methods , Multi-Step methods and :We start with the differential equationdy(t)dt=f(t,y(t))( )y(0) =y0 This equation can be nonlinear, or even a system of nonlinear equations (in which caseyisa vector andfis a vector ofndifferent functions).Numerical Solution of an ODE:The idea behind numerical solutions of aDifferentialEquationis to replace differentiation by differencing. A computer cannot differentiate but itcan easily do a difference. (Differentiation is a continuous process. Differencing is a discreteprocess.) Now we introduce the most important tool that will be used in this section. By thetime you ve mastered this section, you ll be able to doTaylor Expansionsin your sleep.(I am already doing Taylor expansions in your sleep, right?!) Taylor Series Expansion:You ll recall (?) from your calculus class that if a functiony(t)behaves nicely enough, then its Taylor Series expansion converges:y(t+ t)=y(t)+ ty (t)+12 t2y (t)+13!
Kutta Methods, Multi-Step Methods and Stability. REVIEW: We start with the differential equation dy(t) dt = f (t,y(t)) (1.1) y(0) = y0 This equation can be nonlinear, or even a system of nonlinear equations (in which case y is a vector and f is a vector of n different functions).
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