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**CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36**

KFUPM Read , 26-2, 27-1 CISE301_Topic8L4&5 KFUPM

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**Outline of Topic 8 Lesson 1: Introduction to ODEs**

Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODEs Lesson 7: Multiple step Methods Lesson 8-9: Boundary value Problems CISE301_Topic8L4&5 KFUPM

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**Lecture 31 Lesson 4: Runge-Kutta Methods**

CISE301_Topic8L4&5 KFUPM

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**Learning Objectives of Lesson 4**

To understand the motivation for using Runge-Kutta (RK) method and the basic idea used in deriving them. To get familiar with Taylor series for functions of two variables. To use RK method of order 2 to solve ODEs. CISE301_Topic8L4&5 KFUPM

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Motivation We seek accurate methods to solve ODEs that do not require calculating high order derivatives. The approach is to use a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion as possible. CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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**Taylor Series in Two Variables**

The Taylor Series discussed in Chapter 4 is extended to the 2-independent variable case. This is used to prove RK formula. CISE301_Topic8L4&5 KFUPM

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**Taylor Series in One Variable**

Approximation Error CISE301_Topic8L4&5 KFUPM

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**Taylor Series in One Variable - Another Look -**

CISE301_Topic8L4&5 KFUPM

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Definitions CISE301_Topic8L4&5 KFUPM

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**Taylor Series Expansion**

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**Taylor Series in Two Variables**

y+k y x x+h CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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**Runge-Kutta Method Alternative Formula**

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**Runge-Kutta Method Alternative Formula**

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**Runge-Kutta Method Alternative Formulas**

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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**Second order Runge-Kutta Method Example**

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**Second order Runge-Kutta Method Example**

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**Second order Runge-Kutta Method Example**

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CISE301_Topic8L4&5 KFUPM

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Summary RK methods generate an accurate solution without the need to calculate high order derivatives. Second order RK have local truncation error of order O(h3). Fourth order RK have local truncation error of order O(h5). N function evaluations are needed in the Nth order RK method. CISE301_Topic8L4&5 KFUPM

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**Lecture 32 Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs**

CISE301_Topic8L4&5 KFUPM

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**Learning Objectives of Lesson 5**

Use Runge-Kutta methods of different orders to solve first order ODEs. CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Methods RK2 CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Methods RK3 CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Methods RK4 CISE301_Topic8L4&5 KFUPM

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**Runge-Kutta Methods Higher order Runge-Kutta methods are available.**

Higher order methods are more accurate but require more calculations. Fourth order is a good choice. It offers good accuracy with a reasonable calculation effort. CISE301_Topic8L4&5 KFUPM

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**Fifth Order Runge-Kutta Methods**

CISE301_Topic8L4&5 KFUPM

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**Second Order Runge-Kutta Method**

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**Second Order Runge-Kutta Method**

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**Second Order Runge-Kutta Method**

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**Example 1 Second Order Runge-Kutta Method**

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**Example 1 Second Order Runge-Kutta Method**

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**Example 1 Second Order Runge-Kutta Method**

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**Example 1 Second Order Runge-Kutta Method**

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**Example 1 Second Order Runge-Kutta Method**

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**Example 1 Summary of the solution**

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Solution after 100 steps CISE301_Topic8L4&5 KFUPM

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**Example 2 4th-Order Runge-Kutta Method**

See RK4 Formula CISE301_Topic8L4&5 KFUPM

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**Example 2 Fourth Order Runge-Kutta Method**

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**Example 2 Fourth Order Runge-Kutta Method**

See RK4 Formula CISE301_Topic8L4&5 KFUPM

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Runge-Kutta Methods RK4 CISE301_Topic8L4&5 KFUPM

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**Example 2 Fourth Order Runge-Kutta Method**

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**Example 2 Summary of the solution**

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**Remaining Lessons in Topic 8**

Solving Systems of high order ODE Lesson 7: Multi-step methods Lessons 8-9: Methods to solve Boundary Value Problems CISE301_Topic8L4&5 KFUPM

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