1. Poisson Equation Finite-Difference
EXAMPLE 2
Consider Poisson’s equation
The boundary conditions
Endpoint a=0, b=2, c=0, d=1
Integers m=5, n=6
Tolerance = 10-10

2. Poisson Equation Finite-Difference
Main function

3. Poisson Equation Finite-Difference

4. Poisson Equation Finite-Difference
Result

5. Poisson Equation Finite-Difference
Integers m=5, n=6
Integers m=10, n=12

6. Poisson Equation Finite-Difference
EXERCISE
Consider Poisson’s equation
The boundary conditions
Endpoint a=0, b=2, c=0, d=1
Integers m=10, n=10
Tolerance = 10-10

7. Poisson Equation Finite-Difference
Result

8. Heat Equation Backward-Difference
EXAMPLE 2
Backward-Difference method with h=0.1 and k=0.01
Subject to the constrains
Compare wi,50 to u(xi, 0.5)
Endpoint l=1
Maximum time T=0.5
Constant a=1
Integers m=10, n=50

9. Heat Equation Backward-Difference
Main function

10. Heat Equation Backward-Difference
Result
Integers m=10, n=50
Integers m=20, n=100

11. Crank-Nicolson Method
Main function

12. Crank-Nicolson Method
Result

13. Wave Equation Finite-Difference
EXAMPLE
Consider the hyperbolic problem
boundary conditions
Initial conditions
Easily verified that the solution to this problem
Endpoint l=1
Maximum time T=1
Constant a=2
Integers m=10, N=20

14. Wave Equation Finite-Difference
Main function

15. Wave Equation Finite-Difference
Result

16. Wave Equation Finite-Difference
EXERCISE
Consider the hyperbolic problem
boundary conditions
Initial conditions
Easily verified that the solution to this problem
Endpoint l=1
Maximum time T=1
Constant a=1
Integers m=4, N=4

17. Wave Equation Finite-Difference
Result

18. Finite-Element EXAMPLE Consider the problem boundary conditions
Actual solution to the boundary-value problem

19. Finite-Element
Main function

20. Finite-Element

21. Finite-Element

22. Finite-Element

23. Finite-Element
Result