1. Numerical Solutions of ODE
Dr. Asaf Varol

2. What is ODE and PDE
A differential equation is an equation which involves derivatives of one or more dependent variables. If there is only one independent variable involved in the equation(s), then the derivatives are referred to as ordinary derivatives. If, however, there is more than one independent variable in the equation, then partial derivatives (PDE) with respect to each of the independent variables are used.

3. Linear first-order ODEs
dy/dx = x + y
y’ = x + y
du/dx + u = 2
u’ + u = 2

4. Non-Linear first-order ODEs
dy/dx = x + cos(y)
y’ = x + cos(y)
du/dt + u2 = 2
u’ + u2 = 2

5. Linear, second-order ODEs
d2y/dx2 – dy/dx = xy
y’’= -2y + 0.1y’

6. Non-Linear, second-order ODEs
d2y/dx2 – dy/dx = xy-y
y’’= -2y + 0.1(y’)2

7. Homogeneous ODEs d2y/dx2 – dy/dx = xy-y y’’= -2y + 0.1(y’)2
Homogeneous ODE is an equation which contains the dependent variable or its derivatives in every term.
d2y/dx2 – dy/dx = xy-y
y’’= -2y + 0.1(y’)2

8. Partial Differential Equation
First order, linear PDE
where for a given function u = u( x, t)
x and t are the independent variables
Ф is the independent variable.
Second-order linear PDE
Here, x and y are the independent variables.

9. Euler’s Method

15. MATLAB (Euler)

16. Plot (Euler)

18. Example EULER METHOD’S Solving a simple ODE with Euler’s Method
Consider the differential equation y’ = f( x, y ) on a≤ x≥ b. Let
y’ = x + y; ≤ x ≥ a = 0, b = 1, y(0) = 2.
First, we find the approximate solution for h=0.5 (n = 2), a very large step size.
The approximation at x1 = 0.5 is
y1=y0 + h (x0 + y0)= ( ) = 3.0
Next, we find the approximate solution, we use n = 20 intervals, so that h = 0.05.

19. Solution with MATLAB (Euler)

20. Plot (Euler)

21. Modified Euler Method

22. Higher Order Taylor Methods
One way to obtain a better solution technique is to use more terms in the Taylor series for y in order to obtain higher order truncation error. For example, a second-order Taylor method uses
y(x+h)=y(x)+hy’(x)+(h2/2)y’’(x)+O(h3)
O(h3) is the local truncation error

23. Solving a Simple ODE with Taylor’s Method
Consider the differential equation
y’=x + y; 0≤ x ≤1 with a initial condition y(0) = 2.
To apply the second order Taylor method to the equation, we find
y’’=d/dx( x+ y) = 1 + y’ = 1 + x + y
This gives the approximation formula
y(x + h)=y(x)+hy’(x)+(h2/2)y’’(x)

24. Cont’d yi+1=yi+h(xi+yi)+(h2/2)(1+xi+yi) For n=2 (h=0.5), we find
y1=y0+h(x0+y0)+(h2/2)(1+x0+y0)=
=2+0.5(0+2)+((.5)2/2)(1+0+2)=3.375
y2=y1+h(x1+y1)+(h2/2)(1+x1+y1)= = ( )+((0.5)2/2)( )=5.9219

25. MATLAB Program f. Taylor

26. Plot (Taylor)

27. RUNGE-KUTTA METHODS
Runge-Kutta methods are the most popular methods used in engineering applications because of their simplicity and accuracy. One of the simplest Runge-Kutta methods is based on approximating the value of y at xi + h/2 by taking one-half of the change in y that is given by Euler’s method and adding that on to current value yi. This method is known as the midpoint method.

28. Midpoint Method k1=hf(xi,yi) Change in y given by Euler’s method.
k2=hf(xi+0.5h,yi+0.5k1) Change in y using slope estimate at midpoint

29. Solving a Simple ODE with Midpoint Method
Consider the differential equation
y’=x + y; 0≤ x ≤1 with a initial condition (a=0.0, b=0.0), y(0) = 2.
First, we find the approximate solution for h=0.5 (n=2), a very large step size.
k1=hf(x0,y0)=0.5( )=1.0
k2=hf(x0+0.5h,y0+0.5k1)=0.5( * *1.0)=1.375
Y1=y0+k2= =3.375
Next, we find the approximate solution y2 at point x2=0.0+2h=1.0

30. Cont’d k1=hf(x1,y1)=0.5(x1,y1)=0.5(0.5+3.375)=1.9375
k2=hf(x1+0.5h,y1+0.5k1)=0.5( * *1.9375)=2.547
y2=y1+k2= =5.922

31. MATLAB Prog. f. Midpoint

32. Plot (Midpoint)

33. References
Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”, Ararat Books & Publishing, LCC., Morgantown, 2001
Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice Hall, 2003 by Pearson Education, Inc., Upper Saddle River, NJ 07458
Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002 Prentice Hall, Upper Saddle River, NJ 07458
Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth Edition, 2004 Prentice Hall, Upper Saddle River, NJ 07458
Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat University, 2001