The computation of zeros of Ahlfors map for doubly connected regions

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1 The computation of zeros of Ahlfors map for doubly connected regions Kashif Nazar, Ali W. K. Sangawi, Ali H. M. Murid, and Yeak Su Hoe Citation: AIP Conference Proceedings 1750, (2016); doi: / View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Solving Robin problems in bounded doubly connected regions via an integral equation with the generalized Neumann kernel AIP Conf. Proc. 1750, (2016); / Some integral equations related to the Ahlfors map for multiply connected regions AIP Conf. Proc. 1691, (2015); / Computing Green's function on unbounded doubly connected regions via integral equation with the generalized Neumann kernel AIP Conf. Proc. 1605, 215 (2014); / Radial slits maps of unbounded multiply connected regions AIP Conf. Proc. 1522, 132 (2013); / New exact solutions for Hele-Shaw flow in doubly connected regions Phys. Fluids 24, (2012); /

2 The Computation of Zeros of Ahlfors Map for Doubly Connected Regions Kashif Nazar 1, 4, b), Ali W.K. Sangawi 2, c), Ali H.M. Murid 1, 3, a) 1, d) and Yeak Su Hoe 1 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM Johor Bahru, Johor, Malaysia. 2 Computer Department, College of Basic Education, Charmo University, Chamchamal, Sulaimani, Kurdistan, Iraq. 3 UTM Centre for Industrial and Applied Mathematics(UTM-CIAM), Ibnu Sina Institute of Science and Industrial Research, Universiti Teknologi Malaysia, UTM Johor Bahru, Johor, Malaysia. 4 Department of Mathematics, COMSATS Institute of Information Technology, P.O.Box Defence Road Off Raiwind Road Lahore Pakistan. a) Corresponding author: alihassan@utm.my b) knazar@ciitlahore.edu.pk c) ali.kareem@charmouniversity.org d) s.h.yeak@utm.my Abstract. The Ahlfors map and Szeg kernel are both classically related to each other. Ahlfors map can be computed using Szegö kernel without relying on the zeros of Ahlfors map. The Szegö kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The exact zeros of the Ahlfors map are unknown except for the annulus region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded doubly connected region. The method depends on the values of Szegö kernel, its derivative and the derivative of boundary correspondence function of Ahlfors map. Using combination of Nyström method, GMRES method, fast multiple method and Newton s method, the numerical examples presented here prove the effectiveness of the proposed method. INTRODUCTION Ahlfors map is the conformal map from a multiply connected region onto the unit disk. If the region is simply connected then the Ahlfors map reduces to the Riemann map. Many of the geometrical features of a Riemann mapping function are shared with Ahlfors map. For a multiply connected region of connectivity, the Ahlfors map is analytic map that is onto, and, where is fixed in. It maps onto a unit disk D in -to-one manner because it has branch points in the interior and is no longer one-to-one there. Using the integral equation method, most of the conformal mapping of multiply connected regions can be computed efficiently. The integral equation method has been used by many authors to compute the one-to-one conformal mapping from multiply connected regions onto some standard canonical regions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In [9], Nasser et al. presented a fast multipole method, which is fast and accurate method for numerical conformal mapping of bounded and unbounded multiply connected regions with high connectivity and highly complex geometry. For computing the Ahlfors map, some integral equations have been given in [4, 18, 19, 20, 21, 22]. In [1], Kerzman and Stein have derived a uniquely solvable boundary integral equation for computing the Szegö kernel of a bounded region and this method has been generalized in [18] to compute Ahlfors map of bounded multiply Advances in Industrial and Applied Mathematics AIP Conf. Proc. 1750, ; doi: / Published by AIP Publishing /$

3 connected regions without relying on the zeros of Ahlfors map. In [22], Tegtmeyer and Thomas computed the exact zeros of Ahlfors map using Szegö and Garabedian kernels for the annulus region, where the authors have used series representations of both Szegö kernel and Garabedian kernel. In [20], Nasser and Murid computed Ahlfors map using a boundary integral equation related to a Riemann-Hilbert problem. In [4, 19, 20] the integral equations for computing Ahlfors map of doubly connected regions requires knowledge of zeros of Ahlfors map, which are unknown in general. In [23], Ahlfors described a method for finding the zeros of Ahlfors map by the minimization problem involving the Green s function of the domain and harmonic measures. In [21], Tegtmeyer extended Ahlfors method with the intersection of level curves of harmonic measure and found the exact values of zeros of Ahlfors map for a particular triply connected region. In [4], Nazar et al. extended the approach of Sangawi [11, 15, 16] to construct an integral equation for solving where is the boundary correspondence function of Ahlfors map of multiply connected region onto a unit disk. However the integral equation is solved numerically by assuming the zeros of the Ahlfors map are known. In this paper, we extend the approach of [4] to compute the zeros of Ahlfors map for general doubly connected regions. AUXILIARY MATERIALS Let be a bounded multiply connected region of connectivity. The boundary consists of smooth Jordan curves such that lie in the interior of, where the outer curve has counterclockwise orientation and inner curves have clockwise orientation. The positive direction of the contour is usually that for which is on the left as one traces the boundary as shown in Figure 1. FIGURE 1. A bounded multiply connected region of connectivity. The curves are parameterized by -periodic triple continuously differentiable complex-valued functions with non-vanishing first derivatives The total parameter domain is defined as the disjoint union of intervals. The notation must be interpreted as follows [20]: For a given, to evaluate the value of at, we should know in advance the interval to which belongs, i.e., we should know the boundary Γ contains, then we compute. FORMULA FOR COMPUTING Since the Ahlfors map can be written as, where is analytic in and in, Nazar et al. [4] have formulated an integral equation for computing as

4 where Im The kernel is the classical Neumann kernel. We want to find an alternate formula for, which will enable us to apply (1) to find the zeros of Ahlfors map. We next show how to compute from the knowledge of and. Note that the Ahlfors map is related to the Szegö kernel and the Garabedian kernel by [18] The Szegö kernel and Garabedian kernel are related on as so that (4) becomes The Szegö kernel is known to satisfy the integral euation [18] where The kernel is also known as the Kerzman-Stein kernel. Thus the boundary values of the Ahlfors map in (5) are completely determined from the boundary values of the Szegö kernel. With and and by differentiating (6) with respect to, we get Similarly, we can get Since

5 we get and Using the solution of (6) and (9), we can find the derivative of Szegö kernel from (8). We next show how to find. Differentiating both sides of (5) with respect to, the result is Using the fact for Ahlfors map, we get Substituting (5) and (12) into (13), we get From (10) and (11), we have Using the fact that, we have which is the alternative formula for computing

6 COMPUTING THE SECOND ZERO OF THE AHLFORS MAP FOR DOUBLY CONNECTED REGION In particular, if is a doubly connected region, i.e., then (2) becomes or As is known from (1), and the zero can be freely prescribed, the only unknown in (17) is the second zero of Ahlfors map. If we take, then can be viewed as containing two unknowns namely and. To determine them, we need two equations. For this we choose any two values of the parameter in the given interval. At these two random values of, we have two equations and, equivalently To solve the above equations, we use Newton s method [24] which is described below. NEWTON S METHOD The idea of Newton s method in two variables is the same as in one variable. Just like the one variable case, we choose a starting point, locally linearize, solve the system of linear equations and repeat. First, we formalize the notion of a system of equations as a single, vector-valued function Like the one variable case, we find the local linearization of at a vector is given by. Here, represents the Jacobian matrix of at, which is a multivariable generalization of the derivative: where is the partial derivative of with respect to, and likewise for the other entries of the matrix. Assuming is invertible, we can solve the linear system for. We calculate: As in the one variable case, this motivates definition of the Newton function of, given by:

Thus, Newton s method in two variables can be seen as iteration of. Essentially, start at a vector, then recursively define until within the desired accuracy of a root.

7 Thus, Newton s method in two variables can be seen as iteration of. Essentially, start at a vector, then recursively define until within the desired accuracy of a root. NUMERICAL EXAMPLES For solving the integral equation (6) numerically, the reliable procedure is by using the Nyström method with the trapezoidal rule with n equidistant nodes in each interval, [5, 6, 7, 8, 9]. The trapezoidal rule is the most accurate method for integrating periodic functions numerically [25, pp ]. In [9] an iterative method GMRES have been applied for solving the linear systems powered by the fast multipole method (FMM). We use the similar approach as [9] to solve our linear system. For evaluating the Cauchy integral formula numerically, we use the equivalent form which also works very well for Ω near the boundary Γ. When the trapezoidal rule is applied to the integrals in (24), the term in the denominator compensates the error in the numerator (see [14]). Example Consider a doubly connected region bounded by the two non-concentric circles with and radius. The test region is shown in Figure 2. Given a first zero of the Ahlfors map, the exact second zero is unknown for this region. We compute from (17) which is the approximate value of. The acuracy is measured by computing from (24). The theoretical value of is zero. See Table 1 for the numerical values of and. FIGURE 2. The region for Example 1. Figure 3 shows the 3D plot of for the region in Figure 2. The plot provides good initial guess for the second zeros of the Ahlfors map. This initial guess is then used in Newton s method in Matlab to solve a 1 g the nonlinear equations and for the second zero of Ahlfors map

FIGURE 3. plot of for Example 1 with initial guess a g 0.8 1. 0i. 1 TABLE 1. Numerical values of and for Example 1.

Given a first zero of the Ahlfors map, the exact second zero is also unknown for this region. We compute from (17) which is the approximate value of.

8 FIGURE 3. plot of for Example 1 with initial guess a g i. 1 TABLE 1. Numerical values of and for Example 1. Example Consider a doubly connected region bounded by and The test region is shown in Figure 4. Given a first zero of the Ahlfors map, the exact second zero is also unknown for this region. We compute from (17) which is the approximate value of. The acuracy is measured by computing from (24). The theoretical value of is zero. See Table 2 for the numerical values of and. The convergence of equations in Newton s method is strongly depend on the condition number of Jacobian,. Figure (6) shows the stability graph of Jacobian. FIGURE 4. The region for Example

9 Figure 5 shows the 3D plot of for the region in figure 4. The plot provides good initial guess for the second zeros of the Ahlfors map. This initial guess a 1 is then used in Newton s method in Matlab to solve the nonlinear equations eq and eq for the second zero of Ahlfors map. g FIGURE 5. plot of for Example 2 with initial guess a g i. 1 TABLE 2. Numerical values of and for Example 2. FIGURE 6. Jacobian Stability

10 CONCLUSION In this paper, we have constructed a numerical method for finding the zeros of the Ahlfors map of doubly connected regions. We used two formulas for the derivative of the boundary correspondence function of the Ahlfors map, and the derivative of the Szego kernel function. These formulas were then used to find the zeros of the Ahlfors map for any smooth doubly connected regions. We used fast multipole method (FMM) for each iteration of the iterative method GMRES to solve the linear system obtained by discretization of integral equation. By Newton s method which is a quick and efficient method, we solved the obtained system of equations to find the second zero of Ahlfors map. Analytical method for computing the exact zeros of Ahlfors map for annulus region and a particular triply connected regions are presented in [21] and [22] but the problem of finding zeros for arbitrary doubly connected regions is the first time presented in this paper. The numerical examples presented have illustrated that our method is reliable and has high accuracy. ACKNOWLEDGMENT This work was supported in part by the Malaysian Ministry of Education through the Research Management Centre (RMC), Universiti Teknologi Malaysia (FRGS Ref. No. R.J F637). This support is gratefully acknowledged. REFERENCES 1. N. Kerzman and E. M. Stein, Mathematische Annalen 236, (1978). 2. N. Kerzman and M.R. Trummer, Journal of Computational and Applied Mathematics 14, (1986). 3. B. Lee and M. R. Trummer, Electronic Transactions on Numerical Analysis 2, (1994) 4. K. Nazar, A. H. M. Murid and A. W. K. Sangawi, Jounal Teknologi, 73, 1 9 (2015). 5. M. M. S. Nasser, Computational Methods and Function Theory 9, (2009). 6. M. M. S. Nasser, SIAM Journal on Scientific Computing 31, (2009). 7. M. M. S. Nasser, Journal of Mathematical Analysis and Applications 382, (2011). 8. M. M. S. Nasser, Journal of Mathematical Analysis and Applications 398 (2), (2012). 9. M. M. S. Nasser and F. A. A. Al-Shihri, SIAM Journal on Scientific Computing 35, A1736 A1760 (2013). 10. S. T. O Donnell and V. Rokhlin, SIAM journal on scientific and statistical computing 10, (1989). 11. A. W. K.Sangawi, Boundary Integral Equations for Conformal Mappings of Bounded Multiply Connected Regions, PhD thesis, Universiti Teknologi Malaysia, A. W. K. Sangawi, A. H. M. Murid and M. M. S. Nasser, Applied Mathematics and Computation 218, (2011). 13. A. W. K. Sangawi, A. H. M. Murid and M. M. S. Nasser, Abstract and Applied Analysis. Hindawi Publishing Corporation, A. W. K. Sangawi, A. H. M. Murid, International Journal of Science & Engineering Research 4(10), (2013). 15. A. W. K. Sangawi, Applied Mathematics and Computation 228, (2014). 16. A. W. K. Sangawi, Advances in Computational Mathematics, 1 17 (2014). 17. A. A. M. Yunus, A. H. M. Murid and M. M. S. Nasser, Abstract and Applied Analysis, Hindawi Publishing Corporation, S. Bell, Journal of mathematical analysis and applications 120, (1986). 19. A. H. M. Murid and M. R. M. Razali, Matematika 15, (1999). 20. M. M. S. Nasser and A. H. M. Murid, Clifford Anal. Clifford Algebr. Appl, (2013). 21. T. J. Tegtmeyer, The Ahlfors Map and Szeg o Kernel in Multiply Connected Domains, PhD thesis, Purdue University, T. J. Tegtmeyer and A. D. Thomas, Rocky Mountain J. Math, (1999). 23. L. V. Ahlfors, Duke Math. Journal 14, 1 11(1947). 24. N. Eagan, G. Hauser and T. Flaherty, Newton s Method on a System of Nonlinear Equations, ( revised paper.pdf). 25. P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, Orlando,

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