Question

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1. Describe the basic idea behind Euler’s Method. Compare this with the Improved Euler’s Method - in what way is it an improvement? Finally, compare both these methods with the Runge-Kutta method - what is the difference, and why does Runge-Kutta give more accurate results?

2. Why would we choose to use one of the numerical methods, instead of solving the differential equation to obtain an exact answer?

3. What is the characteristic equation? What is the connection between the characteristic equation and the original differential equation?

4. What is the indicial equation? What is the connection between the indicial equation and the original different equation?

Answer 1

Basic idea behind euler method

euler method is a numerical method to solve first order first degree diffrential equation with a given intial value. it is the most basic explicit method for numerical integration of ordinary diffrential equations.

compare with improved euler method, in what way it is an improvement?

improved euler method eliminates the stability problems noted in euler method

this diffrers from euler method in that the function f is evaluated at the end point of the step instead of the starting point. euler method has yn+1 on both sides, so when applying it we have to solve an equation this makes implmentation more costly whereas improved euler method can achieve a higher order more accurately possibly to more function evaluations

Compare with runge kutta method

improved euler is two step integration method usong predictor and second order corrector where as runge kutta is based on simpson rule . it uses 3 predictors

error in euler is higher than runge kutta method because truncation error in higher order methods is less compared to euler method.

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