Adaptive Fourth-Order RK Method

Use the adaptive fourth-order RK method to integrate $y^{\prime}=4 e^{0.8 x}-0.5y$ from x = 0 to 2 using h = 2 and an initial condition of y(0) = 2. This is the same differential equation that was solved previously in Example 25.5. Recall that the true solutions is y(2) = 14.84392.

Question

Adaptive Fourth-Order RK Method

Use the adaptive fourth-order RK method to integrate [latex]y^{\prime}=4 e^{0.8 x}-0.5y[/latex] from x = 0 to 2 using h = 2 and an initial condition of y(0) = 2. This is the same differential equation that was solved previously in Example 25.5. Recall that the true solutions is y(2) = 14.84392.

Accepted Answer

The single prediction with a step of h is computed as

[latex]y(2)=2+\frac{1}{6}[3+2(6.40216+4.70108)+14.11105] 2=15.10584[/latex]

The two half-step predictions are

[latex]y(1) = 2 + \frac{1}{6}[3 + 2(4.21730 + 3.91297) + 5.945681]1 = 6.20104[/latex]
and

[latex]y(2)=6.20104+\frac{1}{6}[5.80164+2(8.72954+7.99756)+12.71283] 1=14.86249[/latex]

Therefore, the approximate error is

[latex]E_{a}=\frac{14.86249-15.10584}{15}=-0.01622[/latex]

which compares favorably with the true error of

[latex]E_{t}=14.84392-14.86249=-0.01857[/latex]

The error estimate can also be used to correct the prediction

y(2) = 14.86249 - 0.01622 = 14.84627

which has an [latex]E_{t}=-0.00235[/latex].

Question 25.12: Adaptive Fourth-Order RK Method Use the adaptive fourth-orde...

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