Transcription of Numerical Methods for Differential Equations with Python
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Numerical Methods for Differential Equations with Python
JohnS ButlerNumerical Methods forDifferential Equations withPythonC O N T E N T Si initial value problems61 Numerical solutions to initial value approximation of of Forward Euler for one about Ordinary Differential s Sheet222 higher order order Taylor Methods233 runge kutta of Second Order Runge Kutta second order: Midpoint 2nd Order Runge Kuttaa0= : Heun s Order Runge Kutta Kutta fourth s 4th Order Runge choice of method and Sheet2 374 multi-step of a explicit multistep Derivation of a explicit method three step four step of the implicit multi-step of Adam s step-size multi-step Sheet3 535 consistency,convergence and Step Sheet4 Value Problem Review Questions642 Contents3ii Numerical solutions to boundary value prob-lems696 boundary value of order Value theorems about boundary value Shooting Shooting method for non-linear Difference method80iii Numerical solutions to partial Differential equa-tions847 partial Differential Operators
The equation can be approximated using the forward Euler as w i+1 w i h = sin(x i). Rearranging the equation gives the discrete difference equation with the unknowns on the left and the know values of the right w i+1 = w i +hsin(x i). The Python code bellow implements this difference equation. The output of the code is shown in Figure 1.1.1.
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