Python代写｜Chapter 13 Numerical Methods for ODEs

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1. Show that if f(t,y)= g(t), the improved Euler method reduces to the simple trapezoidal rule quadrature and the classical fourth-oder RK method reduces to the simple Simpson’s rule quadrature.

2.Prove that all one-step methods are D-stable.

3. Consider the Leap Frog (also called midpoint) method

y”+1-y”-1=2△tf”.

(a) Show that it is D-stable and consistent, hence convergent.

(b) Show that the two roots of π（s，z）= p（ξ）- zo（≤）， Si and &must lie on the unit circle if z is in the A-stability region of this method and this occur only when z=ib, where b∈R and |b|<1. Therefore, the region of A-stability of the Leap Frog method is the open interval (一i,i). Hint: [Sill-a| =1 and each root is solution ofξ-1/5=2z.

13.1 Introduction

In this chapter we study numerical methods for the initial value problem(IVP)：

Here, f is a given function of the independent variable t and the unknown function y, and α is a constant. Often, t represents time but does not have

In (13.1)-(13.2), y and f may be vector-valued, in which case we have an IVP for a system of ordinary difffferential equations (ODEs).

Example 13.1.

y0 (t) = tsin y(t), 0 < t ≤ 2π,

y(0) = α.

Example 13.2.

y’1(t) = y1(t)y2(t) − y21(t),

y’2(t) = −y2(t) + t2 cosy1(t), 0 < t ≤ T,

y1(0) = α1, y2(0) = α2.

These two are examples of fifirst order ODEs. It is the type of IVP’s we will focus on. Higher order ODEs can be written as fifirst order systems by introducing new variables as we illustrate in the next two examples.

Example 13.3. The Harmonic Oscillator.

y ” (t) + k2y(t) = 0.

If we defifine

y1 = y,

y2 = y’,

we get

y10(t) = y2(t),

y’2(t) = −k2y1(t).

Example 13.4.

y 000 (t) + 2y(t)y 00 (t) + cos y0 (t) + et = 0.

Introducing the variables

y1 = y,

y2 = y‘ ,

y3 = y ” ,

we obtain the fifirst order system:

y1‘(t) = y2(t),

y’2(t) = y3(t),

y’3(t) = −2y1(t)y3(t) − cos y2(t) − et

If f does not depend explicitly on t we call the ODE (or the system of ODEs) autonomous. We can turn a non-autonomous system into an autonomous one by introducing t as a new variable.

Example 13.5. Consider the ODE

y‘(t) = sin t − y2(t).

If we defifine

y1 = y,

y2 = t,

we can write this ODE as the autonomous system

y10(t) = sin y2(t) − y21(t),

y‘2(t) = 1.

Continuity of f guarantees local existence of solutions but not uniqueness.We need an extra assumption on f to ensure uniqueness of solutions.