Numerical Methods by Dr. Laila Fouad

Course Outline References Introduction to Matlab Numerical Solution of ODE Numerical Solution of PDE References W. Young Yang, W.Cao, T.-Sang Chung, J.Morris, Applied Numerical Methods Using MATLAB, John Wiley & Sons, Inc., Hoboken, New Jersey, 2005. (Ch5, Ch6 and Ch9) J.Kiusalaas, Numerical Methods in Engineering with MATLAB, Cambridge University Press, New York, 2005. (Ch6, Ch7 and Ch8)

Lecture 3 Numerical Solution of Ordinary Differential Equations ((continued

Recall our Lecture’s Goals SOLUTION TO SINGLE 1ST ORDER INITIAL VALUE PROBLEMS (IVP’s) The methods suggested comes from integrating the function.

The principle behind the numerical methods is to approximate the integral For example in Modified Euler

Trapezoidal rule is applied for approximating integrals f(x)

One Step Methods We have dealt with: Euler Method Modified Euler These methods are called single step methods, because they use only the information from the previous step. Euler Method Modified Euler Runge-Kutta Method

Multi-Step Methods The principle behind a multi-step method is to use past values, y and/or dy/dx to construct a polynomial that approximate the derivative function.

The integral can be represented. Two Point Three Point Adam Bashforth Four Point Adam Bashforth

These methods are known as explicit schemes because the use of current and past values are used to obtain the future step. The method is initiated by using either a set of know results or from the results of a Runge-Kutta to start the initial value problem.

Adams Bashforth Method (4th Point) Example Consider Exact Solution The initial condition is: The step size is:

From the 4th order Runge Kutta x y 0 1 0.1 1.1048 0.2 1.2186 0.3 1.3401 From the 4th order Runge Kutta The 4 Point Adam Bashforth is:

Upgrade the values

The explicit Adam Bashforth method gave solution gives good results without having to go through large number of calculations.

Multi-Step Methods There are second set of multi-step methods, which are known as implicit methods. The implicit methods use the future steps to modify the future steps. What is used to do iterative method, which will make an initial guess and use it until stability is reached. The method is initiated by using either a set of known results or from the results of a Runge-Kutta to start the initial value problem.

Implicit Multi-Step Methods The main method is Adam -Moulton Method Three Point Adam-Moulton Method Four Point Adam-Moulton Method

The method uses what is known as a Predictor-Corrector technique The method uses what is known as a Predictor-Corrector technique. It uses the explicit scheme to estimate the initial guess and uses the value to guess the future y* and dy/dx= f*(x,y*) values. Using these results, the Adams- Moulton method can be applied.

3 Point Adams-Moulton Predictor-Corrector Method Adams third order Predictor-Corrector scheme. Use the Adam Bashforth three point explicit scheme for the initial guess. Use the Adam Moulton three point implicit scheme to take a second step.

Example Consider Exact Solution The initial condition is: The step size is:

From the 4th order Runge Kutta The 3 Point Adams Bashforth is:

The results of explicit scheme is: The results of implicit scheme is: The functional values are:

The values for the Adam Moulton

Summary Euler, Modified Euler 4th Order Runge-Kutta Method Explicit Multi-Step Methods(Adams-Bashforth) Implicit Multi-Step Methods(Adam Moulton )

Matlab Built-In Functions Matlab ODE Suite The Matlab ODE suite contains three explicit methods for nonstiff problems: • The explicit Runge–Kutta pair ode23 of orders 3 and 2, • The explicit Runge–Kutta pair ode45 of orders 5 and 4, of Dormand– Prince, • The Adams–Bashforth–Moulton predictor-corrector pairs ode113 of orders 1 to 13, and four implicit methods for stiff systems: • The implicit Runge–Kutta pair ode23s of orders 2 and 3, • ode23t is an implementation of the trapezoidal rule, • ode23tb is a two-stage implicit Runge-Kutta method, • The implicit numerical differentiation formulae ode15s of orders 1 to 5.