 Applied Mathematics, 2011, 2, 118-122 doi:10.4236/am.2011.21013 Published Online January 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM A Fast Algorithm to Solve the Bitsadze Equation in the Unit Disk Daoud Mashat, Manal Alotibi Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia E-mail: dmashat@kau.edu.sa, dmashat63@gmail.com Received October 19, 2010; revised November 18, 2010; accepted November 24, 201 0 Abstract An algorithm is provided for the fast and accurate computation of the solution of the Bitsadze equation in the complex plane in the interior of the unit disk. The algorithm is based on the representation of the solution in terms of a double integral as it shown by Begehr [1,2], some recursive relations in Fourier space, and Fast Fourier Transforms. The numerical evaluation of integrals at 2N points on a polar coordinate grid by straightforward summation for the double integral would require 2ON floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of In ON per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of 1O per point. The algorithm has the added advantage of working in place, meaning that no additional memory storage is required beyond that of the initial data. This paper is a result of application of many of the original ideas described in Daripa . Keywords: Singular Integrals, Fast Algorithm, Bitsadze Equation 1. Introduction The solutions of many elliptic partial differential equa-tions represent in terms of singular integrals in the com-plex plane in the interior of the unit disk, as the nonho-mogeneous Cauchy-Riemann equations, the Beltrami eq- uation, the Poisson equation, etc. Then solving these eq-uations requires computing the values of the singular integrals. There are two main difficulties in the straight- forward computation of these integrals using quadrature rules. Firstly, straightforward computation of each of these integrals requires an operation count of the order2ON per point. This gives a net operation count of 4ON for 2N grid points which is computationally very in-tensive for large N. Secondly, this method also gives poor accuracy due to the presence of the singularities in the integrand. Daripa and Co-workers ([3-9]) presented fast algorithms to solve the singular integrals that arise in such solutions. By these algorithms evaluation of singu-lar integrals has been optimized, giving an asymptotic operation count of 2ln lnON N for2N points. More- over, these algorithms have the added advantages of working in place, meaning that no additional memory storage is required beyond that of th e initial data. In this paper we follow his method to present an algo-rithm to solve an elliptic partial differential equation called the Bitsadze equation which define in the unit disk 0;1: 1Bzz in complex plane  for any com-plex valued function w of complex variables ,zz B by 2221;,,4wiwfzxiyxyxyz (1) Where f is a complex valued function on B.  The Bitsadze equation arises in many areas including structural mechanics, electrostatics, magneto statics, po- wer electromagnetic, conductive media, heat transfer and diffusion ([11-13]). In [1,2], Begehr introduced some boundary value problems for Bitsadze equation under some solvability conditions and presented their solutions as results for applying the Dirichlet and Neumann boun-dary value problems all those solutions have singular in- tegrals in their context. For that, we will consider one of these problems to apply our numerical method for eva-luating the singular integrals. Problem (1.1): zzwfin B, 0,w1 zzzwon B, 0zwc, Where 101;, ,; and fLBCB c . D. MASHAT ET AL. Copyright © 2011 SciRes. AM 119Its unique solution is given by Equation (1.2) (Below) Let us rewrite it in the form   ,wzczuzvzGf z  (1.3) Where  011,2duz iz  21111log 1,2zdvz ziz  and  2211 .()zGf zfddz (1.4) Our method is basically a recursive routine in Fourier space that divides the interior of the unit disk into a col-lection of annular regions and expands the integral in Fourier series in angular direction with radius dependent Fourier coefficients. A set of exact recursive relations are derived which are used to produce the Fourier coeffi-cients of singular. These recursive relations involve ap-propriate scaling of one-dimensional integrals in annular regions. The integrals in (1.4) at all grid points are then easily obtained from the Fourier coefficients by the FFT. The rest of the paper is structured as follows. Section 2 presents the solution of the linear integrals u and v. Sec-tion 3 develops the mathematical foundation of the effi-cient algorithm to evaluate (1.4) within the unit disk. The fast algorithms for solving problem (1.1) is discussed in Section 4. We carry out numerical results with this me-thod in section 5. Finally, we summarize and conclude in Section 6. 2. Evaluating the Integrals u and v 1)  0112duz iz Let ie and izre, where ,0r then  2001,2iiiiiiedure eiere Since the integral is on the unit disk then 1, 20020001,211 2iiininneur dreere d Let 0iinnneae, 20001,2 ,nin ininnnnninnnurareeedare   Then we can write ,innnur ue (2.1) Where if 0, 0 if 0.nnnar nun 2)  21111log 1,2zdvz ziz   221022102211011,log1,21 log1, 21 .2iiiiiiiiiniininriedvrere eire ereredrerreednre  Let 1iinnnebe, 221011 1111,,2 .nin ininninnnninnnbrrvreee dnrebrren    put n = m+1 2101., mmimmmbvrrr em Then we can write ,,immmvr ve (2.3) Where 21 if 0,10 if 0. mmmmbrr mvmm   2220111 1111 1log 1.22 ()zzddwz czzfddiziz z     (1.2) D. MASHAT ET AL. Copyright © 2011 SciRes. AM 120 3. Mathematical Foundation of the Algorithms In this section, we develop the theory n eed e d to construct an efficient algorithm for evaluation the singular inte-grals (1.4). The following theorem is crucial for later development of the algorithm. Theorem 3.1. The value of the integral (1.4) with izre, 0r can be written as  innnGfzgr e (3.1) Where 2220;10;2220; ; 01 ; 1 nnnBBrnnnBrrrfddngr rrfddn (3.2) Proof. If we denote 22201,.2()inzQre dz (3.3) Then it follows from (1.4) and (3.1) that   (0;1)1, .nBgrfQrdd (3.4) The integral in (3.3) can be evaluated by first expand- ing 22()zzin power of z and z to get 2222 202221; ,; .ninnnninnnrrerzzrr er  By comparing this result with the Fourier series coef- ficients of 22()zz, we get 2222220; 0, ,; 0, ,,; 1, ,0; 1, .nnnnnnnrrr nrQr rrnrnr   (3.5) Substitution of (3.5) into (3.4) yields the desired result. Corollary 3.1. Suppose that ie and f has Fourier coefficient nf; then the coefficients ngr in Equation (3.2) can be written as: 1222122210–; 0,=2; 1.nnnnrnrnnnnrr fdngr rrfdn (3.6) Corollary 3.2. It follows directly from Equation (3.6) that 10ng for 0n, 00ng for 1n. Corollary 3.3. Let ijrr. Define 21,21; 0,; 1.jijirnnrnij rnnrfdnCfdn (3.7) and 21,21; 0, ; 1.jijirnnrnij rnnrfdnBfdn (3.8) After some algebraic manipulation it follows from Equations (3 .7) and (3.8) that .nini nigrCrBr (3.9) Where ,2; 0,nnn inii ijn jjrCrrCCrnr (3.10) 22,2; 1,nnninjnij ijjrCrCrrCnr (3.11) 22,2; 0,nnn iniiijn jjrBrr BBrnr (3.12) and ,2; 1.nnninjni jijjrBrBrrB nr (3.13) Corollary 3.4. Let 01 201Mrrr r . It follows from recursive applications of (3.10)-(3.13) and from using Corollary 3.2 that 12,1 ,121, 1,12; 0,2; 1.Mnnnnliil iiilnl lnn nnliiliiirCrBngrrC rBn (3.14) D. MASHAT ET AL. Copyright © 2011 SciRes. AM 121for 1, 2,3,,1lM. Corollary 3.5. It follows directly from Equations (3.6) and (3.9) that 110; 0,nnCB n (3.15)  000; 1.nnCB n (3.16) 4. The Fast Algorithm We construct the fast algorithm based on the theory of section 3. The unit disk is discredited using MN lat- tice points with M equidistant points in the radial direc-tion and N equidistant points in the circular direction. The following is a formal description of the fast algo-rithm useful for pr ogramming pur p oses. Algorithm 4.1. (The fast algorithm for evaluating the singular integral (1.4)) Input: M, N and 2,1,1, 1,ik Nlfrel MkN Output: 2,1,1,1, ik NlGf relMkN . Step 1. Set 8KN, 00r and 1Mr. Step 2. Compute the Fourier coefficients nlfr, 0, 1lM and ,nKK from known values of 2,1,2,,ik Nlfre kN using the FFT. Step 3. Compute ,1 1, 1niiCiM and 2,1[0,2] nKK  using Equation (3.7). Step 4. Compute,1 1, 1niiBiM and 2, 1[0,2]nK K  using Equation (3.8). Step 5. Compute nlgr, 2, 2nKK , 1, 1lM using relations (3.1 4). set  0,0,0, 2nMnMCrBrn K do 0,1, ,2nK do 1, ,1lM  ,1 112,nnn lnll llnllrCr rCCrr  22,1 112,nnn lnll llnllrBrr BBrr .nlnl nlgrCrBr enddo enddo set  000,0,2, 1nnCrBrn K 2110,12,nnnCrr C 110,12,nnnBr rB 111.nnngrCrBr do 2,,1nK  do 2, ,1lM  22111,2,nnnlnlnll lllrCrCrr Cr  111,2,nnnlnlnll lllrBrBr rBr . nlnl nlgrCrBr enddo enddo Step 6. Finally, compute 2222;1,1 1,, .Kik Nik NlnlnKGfzregr elM kN  Algorithm 4.2 (The fast algorithm for solving the problem (1.1)): Input: M, N , 20ik Ne, 21ikNe and 2ik Nlfre, 1,1 ,1,lM kN . Output: 2,1, 1,1,.ik Nlwrel Mk N Step 1. Set 8KN, 00r and 1Mr. Step 2. Compute 2ik NlGf zre, 1, 1lM, 1,kN using algorithm 4.1. Step 3. Compute the Fourier coefficients ,nan KK from known values of 20ik Ne, 1, 2,,kN using the FFT. Step 4. Compute the Fourier coefficients 1, 1nbnK K  from known values of 21ikNe, 1, 2,,kN using the FFT. Step 5. Compute 2(),1,1,1,ik Nlurel MkN using the relation (2.1). Step 6. Compute 2(),1,1,1,ik Nlvrel MkN using the relation (2.3). Step 7. Compute 2(),1,1,1,ik NlwrelMk N . 22222()ik Nik Nik Nll lik Nik Nllwrecre urevre Gfre 5. Numerical Results In this section we solve a boundary value problem for the inhomogeneous Bitsadze equation in the unit disk by using the algorithms that presented in section 4. Example Consider the problem 2zzwzin B,,wz2,zzzw ,zB 00zw, The exact solution for this problem is given by 2,wzzz zB By using algorithm 4.2 we have the max error as illu-strate in the following Table 1. D. MASHAT ET AL. Copyright © 2011 SciRes. AM 122 Table 1. Maximum relative error. MAX ERROR M = 10 M = 50 M = 100 M = 200 N=64 3.371768E-07 2.980596E-07 2.588641E-07 2.245232E-07 N=128 3.371753E-07 2.980553E-07 2.588640E-07 2.245231E-07 N= 256 3.371734E -07 2.980548E-07 2.588639E-07 2.245196E-07 N= 512 3.371718E -07 2.980526E-07 2.588628E-07 2.245153E-07 N= 1024 3.371717E-07 2.980501E-07 2.588627E-07 2.245142E-07 6. Conclusions We presented a fast algorithm to solve the Bitsadze equ-ation in the unit disk under special boundary conditions in the complex plane, by constructed the fast algorithm to evaluate the singular integral transform (1.4). The method divides the interior of the unit diskB :1zz into a collection of annular regions. The in- tegrals and the function f (z) are expanded in terms of Fourier series with radius dependent Fourier coefficients. The good performance of the algorithm is due to the use of scaling one-dimensional integral in the radial direction to produce the value of the singular integral over the en-tire domain. Specifically, scaling factors are used to de-fine exact recursive relations which evaluate the radius dependent Fourier coefficients of the singular integral (1.4). The inverse Fourier transform are applied on each circle to obtain the value of the singular integrals on all circles. 7. References  H. Begehr, “Boundary Value Problems in Complex Analysis I,” Boles’ de la Association’s Mathematical Ve-nezuelan, Vol. 12, No. 1, 2005, pp. 65-85.  H. Begehr, “Boundary Value Problems in Complex Analysis II,” Boles’ de la Association’ Mathematical Ve-nezuelan, Vol. 12, No. 2, 2005, pp. 217-250.  P. Daripa, “A Fast Algorithm to Solve Nonhomogeneous Cauchy–Riemann Equations in the Complex Plane,” SIAM Journal on Scientific and Statistical Computing, Vol. 13, No. 6, 1992, pp. 1418-1432. doi:10.1137/0913080  P. Daripa and D. Mashat, “An Efficient and Novel Nu-merical Method for Quasiconformal Mappings of Doubly Connected Domains,” Numerical Algorithms, Vol. 18, No. 2, 1998, pp. 159-175. doi:10.1023/A:1019169431757  P. Daripa and D. Mashat, “Singular Integral Transforms and Fast Numerical Algorithms,” Numerical Algorithms, Vol. 18, No. 2, 1998, pp. 133-157. doi:10.1023/A: 1019117414918  P. Daripa and L. Borges, “A Fast Parallel Algorithm for the Poisson Equation on a Disk,” Journal of Computa-tional Physics, Vol. 169, No. 1, 2001, pp. 151-192. doi: 10.1006/jcph.2001.6720  P. Daripa and L. Borges, “A Parallel Version of a Fast Algorithm for Singular Integral Transforms,” Numerical Algorithms, Vol. 23, No. 1, 2000, pp. 71-96. doi:10.1023/ A:1019143832124  P. Daripa, “A Fast Algorithm to Solve the Beltrami Equ-ation with Applications to Quasiconformal Mappings,” Journal of Computational Physics, Vol. 106, No. 2, 1993, pp. 355-365.  P. Daripa, “On a Numerical Method for Quasiconformal Grid Generation,” Journal of Computational Physics, Vol. 96, No. 2, 1991, pp. 229-236. doi:10.1016/0021-9991(91) 90274-O  A. H. Babayan, “A Boundary Value Problem for Bitsadze Equation in the Unit Disc,” Journal of Contemporary Mathematica Analysis, Vol. 42, No. 4, 2007, pp. 177-183. doi:10.3103/S1068362307040012  A. V. Bitsadze, “Boundary Value Problems of Second Order Elliptic Equations,” North-Holland Publishing Company, Amsterdam, 1968.  C. Miranda, “Partial Differential of Elliptic Type,” Springer, New York, 1970.  M. Schechter, L. Bers and F. John, “Partial Differential Equations,” Interscience, New York, 1964.