Figure 2-4 illustrate the region of absolute stability for first-order Huen's method while Figure 5-7 exhibit the regions of absolute stability for second-order Huen's method. The stability region is inside the boundary of the dotted lines. As noticeable that the stability region is larger when the step size is half (w=1/2) followed by when the step size being constant (w=1) and the smallest stability region is determined when the step size is double (w=2). As conclusion, the stability region is getting improve as the step size getting smaller.

**Figures**2-4 illustrate the region of absolute stability for first-order Huen's method while

**Figures**5-7 exhibit the regions of absolute stability for second-order Huen's method. The stability region is inside the boundary of the dotted lines.

**Notice**that the stability region is

**largest**when the step size is half (w=1/2)

**and smaller**when the step size

**is**constant (w=1)

**, with**the smallest stability

**region determined**

**by a double**step size (w=2)

**. In summary**, the stability region

**grows**as the step size

**becomes**smaller.

Comments

Thanks a a lot..

myviMister Micawber1- As mentioned earlier in chapter two, the analysation of the stability regions play a big role in determining a suitable method for a particular problem. Hence, this section will discuss the stability region of the GHI method for first and second order BVPs such use the steps applied in chapter two as well. Again the test equation (2.3) is employing into the first-order corrector formulae and then that formulae are representing in a matrix form.

2- Following the similar steps as w=2 for w=3 and w=4 and then solving the determinant of F=p+3w with p=2h, getting the stability polynomials as follows

3- The stability region obtained according to the stability polynomials (4.5, 4.6, 4.7) can be seen in Figure 23, 24 and 25.

4- In a similar manner as chapter two, the test equation (2.4) is employed to the corrector formulae of second-order GHI method with showing the obtained formulae as follows

5- Finally, solving the determinant of F=p+q+3w with p=2h and q=3h and the resulting stability polynomial is given as

6- The stability regions are defined by these stability polynomials (4.8, 4.9, 4.10) are shown as follows

In summary, the stability regions for first- and second-order GHI method are shown as Figure (23, 24, 25) and (26, 27, 28) respectively and it lies insides the dotted line contour. Looking at these stability regions obtained, as the step size becomes smaller, the size of stability regions increased. This pattern is happened as the earlier methods introduced which are ABC and DEF method.

myviYou overuse 'hence'.As mentioned earlier in

Chapter 2, theanalysisof the stability regions play a big role in determining a suitable method for a particular problem.Thissection will discuss the stability region of the GHI method forfirst- and second-orderBVPs, whichuse the steps applied inChapter 2as well. Again the test equation (2.3) isemployedinthe first-order corrector formulae and thenthoseformulae arerepresentedin a matrix form.Following similarsteps as w=2 for w=3 and w=4 and then solving the determinant of F=p+3w with p=2h,we obtainthe stability polynomials as follows.The stability region obtained according to the stability polynomials (4.5, 4.6, 4.7) can be seen in

Figures23, 24 and 25.In a similar manner as

in Chapter 2, the test equation (2.4) is employedforthe corrector formulae ofthesecond-order GHI method,producingthe obtained formulae as follows.Finally, solving the determinant of F=p+q+3w with p=2h and q=3h and the resulting stability polynomial is given as

The stability

regions definedby these stability polynomials (4.8, 4.9, 4.10) are shown as followsIn summary, the stability regions for

thefirst- and second-order GHI method are shownin Figures 23, 24, 25 and Figures 26, 27, 28respectively: they lie withinthe dotted line contour. Looking at these stability regions obtained, as the step size becomes smaller, the size ofthestability regionsincreases. This patternoccurs also inthe ABC and DEF methods introduced earlier.Mister Micawber