An Approximate Solution

Let us suppose again that the right-hand side of a first order differential equation has the form of a polynomial of a degree k.

If the order of the integration method is less than k+1, we can get the numerical solution of the differential equation (11) only approximately. The accuracy, in such a case, is dependent on the integration step.

Example 3: Let us determine the value of the solution y of the following equation (19) at time t=10s

$y' = 6*t^5 \hspace{0.5cm} y(0)=0. \hfill (19)$

In the example we have k=5. If we use the 4th order numerical integration method, i.e. if only the Taylor series terms up to the fourth power of h are only taken into account, we obtain an approximate solution of equation (19). The approximate solution y(10) of equation (19), is shown in Tab.3.

h	y(10)	rel.error(%)	NRS
10	000000.000000000000	1.00E+0002	1
1.0	999720.000000000000	2.80E-0002	10
0.1	999999.970179020196	2.98E-0006	100
0.01	1000000.000020105120	3.34E-0008	1000
0.001	999999.994871600780	1.95E-0007	10000
0.0001	999999.977767592815	2.86E-0006	100000

Tab.3

It can be seen in Tab.3 that we can get close to the exact solution of y(10) by decreasing the integration step h (the exact solution is y =t⁶; for t =10s we have y(10) =1 000 000).

In the column "NRS" the number of computation steps is given after which, with the integration step h given, the point t_n = 10s has been reached.

Note: If we used the method of the corresponding order according to the exact numerical solution by the Taylor series method

y_n+1 = y_n + h*6*t_n⁵ + h²*15*t_n⁴ + h³*20*t_n³ + h⁴*15*t_n² + + h⁵*6*t_n + h⁶

we would obtain the exact solution y(10) for h=10s in one computation step.

Example 3 was chosen to stress

• the necessity of decreasing the integration step h (if the method order is insufficient),
• the inconvenient fact of the increasing number of computation steps NRS for y(10) ( in comparison to the exact numerical solution using the method of the corresponding order),
• the inconvenient influence of the accumulated error when decreasing the integration step h.

Similarly, if we use the 5th order integration method, again, we will obtain an approximate solution y(10) of equation (19) at t = 10s. Results with a higher accuracy ( in comparison to Tab.3) are shown in Tab.4.

h	y(10)	rel.error(%)	NRS
10.0	000000.000000000000	1.00E+0002	1
1.0	999990.000000000000	1.00E-0003	10
0.1	999999.999879020196	1.38E-0008	100
0.01	1000000.000020105120	3.37E-0008	1000
0.001	999999.994871600780	1.95E-0007	10000
0.0001	999999.977767592815	2.86E-0006	100000

Tab.4