Derivative Approximation

1
Derivative Approximation
Approximate u(1)
Forward difference
2
Derivative Approximation
Approximate u(1)
Forward difference
3
Derivative Approximation
Approximate u(1)
Forward difference
h error
0.1 2.1 0.1
0.01 2.01 0.01
0.001 2.001 0.001
4
Derivative Approximation
Approximate u(1)
Backward difference
h error
0.1 1.9 0.1
0.01 1.99 0.01
0.001 1.999 0.001
5
Derivative Approximation
Approximate u(1)
Central difference
h error
0.1 6.02 2e-2
0.01 6.0002 2e-4
0.001 1.999 2e-6
6
Derivative Approximation
Approximate u(1)
h Error Forward Error Backward Error central
0.1 1e-1 1e-1 2e-2
0.01 1e-2 1e-2 2e-4
0.001 1e-3 1e-3 2e-6
7
Numerical Solution for ODE
Solve the IVP
Exact Solution
x y(comp) y(exact)
0 1 1
0.1 1.1 1.1052
0.2 1.21 1.2214
0.3 1.331 1.3499
0.4 1.4641 1.4918
0.5 1.6105 1.6487
0.6 1.7716 1.8221
0.7 1.9487 2.0138
0.8 2.1436 2.2255
0.9 2.3579 2.4596
1 2.5937 2.7183
8
Exact sol
Solve the IVP
x y(comp) y(exact)
0 1 1
0.1 1.1 1.1052
0.2 1.21 1.2214
0.3 1.331 1.3499
0.4 1.4641 1.4918
0.5 1.6105 1.6487
0.6 1.7716 1.8221
0.7 1.9487 2.0138
0.8 2.1436 2.2255
0.9 2.3579 2.4596
1 2.5937 2.7183
comp sol
9
Exact
h0.1
h0.25