CLASSICAL RK4 METHOD Consider the initial-value problem in Examples 7.4 and 7.5 : y′ - x²y = 2x², y(0) = 1, 0 ≤ x ≤ 1, h = 0.1 1. Using hand calculations, compute the estimated value of [latex]y_1 = y(0.1)[/latex] by the classical RK4 method. 2. Solve the initial-value problem by executing the user-defined function RK4.

1. Noting that f(x, y) = x²(2 + y), the calculations are carried out as follows: [latex]k_1=f(x_0,y_0)=f(0,1)=0[/latex] [latex]k_2=f(x_0+\frac{1}{2} h,y_0+\frac{1}{2}k_1h) =f(0.05,1)=0.0075[/latex] [latex]k_3=f(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_2h )=f(0.05,1.0004)=0.0075[/latex] [latex]k_4=f(x_0+ h,y_0+k_3h) =f(0.1,1.0008)=0.0300[/latex] [latex]y_1=y_0+\frac{1}{6}h(k_1+2k_2+2k_3+k_4)=\boxed{1.0010}[/latex]

Question 7.6: CLASSICAL RK4 METHOD Consider the initial-value problem in E...

Numerical Methods for Engineers and Scientists Using MATLAB®

CLASSICAL RK4 METHOD

Consider the initial-value problem in Examples 7.4 and 7.5:

y′ – x²y = 2x², y(0) = 1, 0 ≤ x ≤ 1, h = 0.1

1. Using hand calculations, compute the estimated value of $y_1 = y(0.1)$ by the classical RK4 method.

2. Solve the initial-value problem by executing the user-defined function RK4.

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>> f = @(x,y)((x^2)*(2+y));
>> x = 0:0.1:1;
>> y0 = 1;
>> y = RK4(f,x,y0); y = y'
y =
1.0000
1.0010
1.0080
1.0271
1.0647
1.1276
1.2240
1.3634
1.5583
1.8252
2.1868

A summary of all calculations is provided in Table 7.3 where it is easily seen that the global % relative error for the classical RK4 method is significantly lower than all previous methods used up to this point. As expected, starting with Euler’s method, which is indeed a first-order Runge–Kutta method, the accuracy improves with the order of the RK method.

TABLE 7.3
Summary of Calculations in Example 7.6
x	$y_{\text{RK4}}$	RK4 $e_{\text{RK4}}$	RK3 $e_{\text{RK3}}$	RK2 $e_{\text{Heun}}$	RK1 $e_{\text{Euler}}$
0.0	1.000000	0.000000	0.0000	0.00	0.00
0.1	1.001000	0.000001	-0.0000	-0.05	0.10
0.2	1.008011	0.000002	-0.0001	-0.10	0.50
0.3	1.027122	0.000003	-0.0004	-0.15	1.18
0.4	1.064688	0.000004	-0.0010	-0.19	2.12
0.5	1.127641	0.000005	-0.0018	-0.23	3.27
0.6	1.223966	0.000006	-0.0030	-0.25	4.56
0.7	1.363377	0.000007	-0.0044	-0.27	5.96
0.8	1.558286	0.000010	-0.0059	-0.28	7.40
0.9	1.825206	0.000016	-0.0074	-0.27	8.87
1.0	2.186837	0.000028	-0.0087	-0.25	10.37