## 6 Numerical Examples
### Batch of 6 Numerical Examples  
In this part, all the six problems originally solved in Ref. [13] by using the [PERSON] method with cubic (\(m=3\)) splines as basis functions and subsequently solved in Ref. [20] by using the quadratic condensed element (\(m=2\)) on given uniform meshes will be resolved by the proposed adaptive FE algorithm using the reduced element.  
The 6 problems, presented in Ref. [13], are rearranged in sequence order in Ref. [20], and this paper adopts the presentation in Ref. [20] as follows.  
* Problem 1: \(u^{\prime}-u=0,\ u(0)=1,\ u(t)=e^{t},\ 0\leq t\leq 10\), Problem 4 in Ref. [13]
* Problem 2: \(u^{\prime}+u=0,\ u(0)=1,\ u(t)=e^{-t},\ 0\leq t\leq 100\), Problem 5 in Ref. [13]
* Problem 3: \(u^{\prime}=-2 tu^{2},\ u(0)=1,\ u(t)=1/(1+t^{2}),\ 0\leq t\leq 1\), Problem 1 in Ref. [13]
* Problem 4: \(u^{\prime}=1/(1+\tan^{2}u),\ u(0)=0,\ u(t)=\text{arcant},\ 0\leq t\leq 1\), Problem 2 in Ref. [13]
* Problem 5: \(u^{\prime}-u=-(2t/u),\ u(0)=1,\ u(t)=(2t+1)^{1/2},\ 0\leq t\leq 1\), Problem 3 in Ref. [13]
* Problem 6: \(u_{1}^{\prime}=u_{1}^{2}u_{2},\ u_{2}^{\prime}=-1/u_{1},\ u_{1}(0)=1,\ u_{2}(0)=1,\ u_{1}=e^{t},\ u_{2}=e^{-t},\ 0\leq t\leq 1\), Problem 6 in Ref. [13]  
Among the above problems, Problem 1 and 2 are examples of single linear ODEs with constant coefficients; Problems 3 to 5 are examples of single nonlinear ODEs; and Problem 6 is a system of nonlinear ODEs. The computed results are shown in Tables 2-7 and Figures 3-14. In the following, some brief remarks for each individual problem are given.  
Figure 4: The error ratio variation of Problem 1 (\(T_{l}=10^{-5}\)).  
Figure 5: The step-size distribution of Problem 2 (\(T_{l}=10^{-5}\)).  
Figure 6: The error ratio variation of Problem 2 (\(T_{l}=10^{-5}\)).  
Figure 7: The step-size distribution of Problem 3 (\(T_{l}=10^{-5}\)).  
Figure 11: The step-size distributions of Problem 5 (\(T_{l}=10^{-5}\)).  
Figure 8: The error ratio variation of Problem 3 (\(T_{l}=10^{-5}\)).  
Figure 10: The error ratio variations of Problem 4 (\(T_{l}=10^{-5}\)).  
Figure 9: The step-size distributions of Problem 4 (\(T_{l}=10^{-5}\)).  
Problem 1 itself is an intrinsically unstable problem. The exact solution of Problem 1 is the exponential function, which soars up rapidly over time and reaches the magnitude of 22,026.47 at \(t=10\). This presents a severe challenge to any algorithm with the stringent tolerance \(T_{l}=10^{-3}\sim 10^{-6}\) in maximum norm, which means the algorithm should be able to gain at least 8 to 11 exact figures towards the terminal. Table 2 shows the computed results for the problem, and Figures 3 and 4 show the adapted step sizes and error distributions respectively, from which it is seen that proposed algorithm performs very well and all the FE solutions strictly meet the error tolerances.  
Problem 2 characterizes the opposite behavior to Problem 1, as its exact solution decays rapidly over time, resulting in a very large final step size as shown in Figure 5. This reflects the adaptivity capability of the proposed algorithm.  
Problems 3, 4, and 5 are nonlinear problems and are solved with no difficulties. Problem 6 is a system of nonlinear equations and, although the two solution components  
Figure 14: The error ratio variations of Problem 6 (\(T_{l}=10^{-5}\)).  
Figure 12: The error ratio variations of Problem 5 (\(T_{l}=10^{-5}\)).  
Figure 13: The step-size distributions of Problem 6 (\(T_{l}=10^{-5}\)).  
comprise a rapidly increased and a rapidly decayed functions, the proposed algorithm solves the problem without difficulties either.  
For all the above problems, the proposed algorithm performs satisfactorily and all the computed FE solutions from the reduced elements are able to strictly satisfy the tolerances and hence have made full achievements of the adaptivity objective established in Equation (19).