Numerical Methods

1
Numerical Methods

• Chapter 5.2 Taylor Methods

2
Introduction

• Differential equations - used to model problems
  involving a change of some variable with respect
  to another.
• Problems require solution to an initial-value
  problem (i.e. Diff Eq that satisfies a given
  initial condition)
• Diff Eq often too complicated to solve exactly,
  so 1 of 2 approaches taken to approximate
  solutions
• Simplify the differential equation to one that
  can be solved exactly and use solution of
  simplified equation to approximate the original
• Find methods for directly approximating the
  solution of the original problem. (common
  approach)
• More accurate results
• Realistic error information can be obtained

3
Introduction

• Methods considered in this chapter do not produce
  continuous approximations to solution of the
  initial-value problem.
• Approximations found at certain specified (often
  equally spaced) points.
• Some method of interpolation (often Hermite) used
  if intermediate values are required.

4
Introduction

• Begin by considering approximation of solution
  y(t) to problem of form dy/dt f(t,y), for a lt
  t lt b subject to initial condition y(a) a
• Later, extend the methods to a system of
  first-order differential equations in the form
• dy1 /dt f1(t, y1, y2,, yn)
• dy2 /dt f2(t, y1, y2,, yn)
• dyn /dt fn(t, y1, y2,, yn)
• For a lt t lt b subject to initial conditions
• y1(a) a1, y2(a) a2 , ... , yn(a)
  an

5
Introduction

• Also examine relationship of a system of the type
  described on the previous slide to the general
  nth-order initial value problem of the form
• y(n) f(t, y, y, y,, y (n-1) )
• For a lt t lt b subject to the multiple initial
  conditions
• y(a) a0, y(a) a1 , ... , y (n-1)
  (a) an-1
• Well-Posed Condition Suppose that f and fy ,
  its first partial derivative with respect to y,
  are continuous for t in a,b and for all y. Then
  the initial-value problem
• y f(t,y), for a lt t lt b with y(a) a, has
  a unique solution y(t) for a lt t lt b, and the
  problem is well-posed

6
Taylor Methods - Motivation

• Function we need to expand in a Taylor polynomial
  is the unknown solution to the problem y(t).
• In its most elementary form this leads to Eulers
  Method
• Seldom used in practice
• Simplicity of derivation illustrative of
  technique used for more advanced procedures

7
Eulers Method -objective
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Eulers Method -objective
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Eulers Method
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Eulers Method - Example
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Eulers Method Error Bound
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Eulers Method Error Bound Example
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Taylors Method of order n
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Taylors Method of order n