A Scalar Homotopy Method for Solving an Over ... - TechScience

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62 Copyright © 2009 Tech Science Press CMES, vol.53, no.1, pp.47-71, 2009Table 3: The numerical solutions of Example 6.x1 x2 x3 x4 x5-0.280404179177 -0.117172528010 -0.069880205750 -0.058442152525 -0.061261838916x6 x7 x8 x9 x10-0.072054214387 -0.090429926667 -0.120061711893 -0.170914641178 -0.269370642228
A Scalar Homotopy Method for Solving an Over/Under-Determined System 63may have multiple solutions, but Krzyworzcka (1996) did not give solution for thisexample. Obviously, our method converges faster than that by Mo, Liu and Wang(2009).3.7 Example 7In this example we apply the scalar homotopy method to solve the following boundaryvalue problem, which is solved by Liu (2006) using the Lie group shootingmethod:u ′′ = 3 2 u2 ,u(0) = 4, u(1) = 1. (42)The exact solution is4u(x) =(1 + x) 2 . (43)By introducing a finite difference discretization of u at the grid points we can obtainF i = 1(∆x) 2 (u i+1 − 2u i + u i−1 ) − 3 2 u2 i ,u 0 = 4, u n+1 = 1, (44)where ∆x = 1n+1 is the grid length. We select n= 25 and ε = 10−10 . The initialguess is generated randomly as shown in Fig. 5(a). After 32262 steps, the solutionis obtained. It can be seen from Fig. 5(a) that the numerical solution perfectlycoincides with the exact solution, and the maximum absolute error is about 7×10 −4as shown in Fig. 5(b).3.8 Example 8In this example, we consider the following equations:F 1 (x,y,z) = x 2 + y 2 + z 2 − 1 = 0,F 2 (x,y,z) = x24 + y24 + z2 − 1 = 0. (45)It can be easily seen that this system has two equations in three variables and thesolutions are (x, y, z) = (0, 0, 1) and (0, 0, -1). Although this system is very simple,the conventional Newton method and a vector homotopy method fail, since they all
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