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

Poisson Equation Finite-Difference Main function

Poisson Equation Finite-Difference

Poisson Equation Finite-Difference Result

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

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

Poisson Equation Finite-Difference Result

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

Heat Equation Backward-Difference Main function

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

Crank-Nicolson Method Main function

Crank-Nicolson Method Result

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

Wave Equation Finite-Difference Main function

Wave Equation Finite-Difference Result

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

Wave Equation Finite-Difference Result

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

Finite-Element Main function

Finite-Element

Finite-Element

Finite-Element

Finite-Element Result