# Numerical Solution for Initial Value Problem Numerical Analysis.

## Presentation on theme: "Numerical Solution for Initial Value Problem Numerical Analysis."— Presentation transcript:

Numerical Solution for Initial Value Problem Numerical Analysis

Introduction x0x0 x1x1 x2x2 x3x3 xnxn

Well-posed Problem

Well-posed Formula: Example

Classes of Methods

Euler Method

Euler Method: Formula

Euler Method: Geometry Meaning

Euler Method: Example

Taylor order n

Taylor order n: Formula

Taylor order n: Example

Interpolate other values

Runge-Kutta Method (RK) Runge-Kutta 25

Taylor Theory in 2 Variables 26

Runge-Kutta order 2 27

Runge-Kutta order 2 28

Runge-Kutta order 2 29

Runge-Kutta order 2 30

RK2: Midpoint Formula 31

RK2: Modified Euler Formula 32

RK2: Heuns Formula 33

RK2: Example 34

RK2: Example 35

RK2: Example 36

RK2: Example 37

RK2: Example 38

Runge-Kutta order 4 39

RK4: Example 40

RK4: Example 41

RK4: Example 42

RK4: Example 43

RK4: Example 44

RK4 vs Other Methods 45

RK4 vs Other Methods 46

RK4 vs Other Methods 47

Multi-Step Method 48

Multi-Step Method 49

Multi-Step Method: Explicit case 50

Multi-Step Method: Explicit case 52

Multi-Step Method: Explicit case 53

Multi-Step Method: Implicit case 54

Multi-Step Method: Implicit case 55

Predictor-Corrector Method 2 (Predictor Formula) (Corrector Formula) (inner iteration) 56

Predictor-Corrector Method 57

Predictor-Corrector Method 58

Predictor-Corrector Method: Example 59

Predictor-Corrector Method: Example 60

Predictor-Corrector Method: Example 61

Predictor-Corrector Method: Example 62

Predictor-Corrector Method: Example 63

High ODE and System of ODE

RK4 in vector form

ABM4 in vector form

High ODE and System of ODE: Example

Divided Different Method for BVP

Differential Equation to Divided Different Form

Divided Different Formula for Linear BVP

Existing of Solution

Divided Different for BVP: Example