A Fast Algorithm to Solve the Bitsadze Equation in the Unit Disk

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 points on a polar coordinate grid by straightforward summation for the double integral would require floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of 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 [3].

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 points on a polar coordinate grid by straightforward summation for the double integral would require floating point operation per point. Evaluation of such integrals has been optimized in this paper giving an asymptotic operation count of per point on the average. In actual implementation, the algorithm has even better computational complexity, approximately of the order of 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 [3].

Cite this paper

nullD. Mashat and M. Alotibi, "A Fast Algorithm to Solve the Bitsadze Equation in the Unit Disk,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 118-122. doi: 10.4236/am.2011.21013.

nullD. Mashat and M. Alotibi, "A Fast Algorithm to Solve the Bitsadze Equation in the Unit Disk,"

References

[1] H. Begehr, “Boundary Value Problems in Complex Analysis I,” Boles’ de la Association’s Mathematical Venezuelan, Vol. 12, No. 1, 2005, pp. 65-85.

[2] H. Begehr, “Boundary Value Problems in Complex Analysis II,” Boles’ de la Association’ Mathematical Venezuelan, Vol. 12, No. 2, 2005, pp. 217-250.

[3] 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

[4] P. Daripa and D. Mashat, “An Efficient and Novel Numerical Method for Quasiconformal Mappings of Doubly Connected Domains,” Numerical Algorithms, Vol. 18, No. 2, 1998, pp. 159-175. doi:10.1023/A:1019169431757

[5] 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

[6] P. Daripa and L. Borges, “A Fast Parallel Algorithm for the Poisson Equation on a Disk,” Journal of Computational Physics, Vol. 169, No. 1, 2001, pp. 151-192. doi: 10.1006/jcph.2001.6720

[7] 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

[8] P. Daripa, “A Fast Algorithm to Solve the Beltrami Equation with Applications to Quasiconformal Mappings,” Journal of Computational Physics, Vol. 106, No. 2, 1993, pp. 355-365.

[9] 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

[10] 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

[11] A. V. Bitsadze, “Boundary Value Problems of Second Order Elliptic Equations,” North-Holland Publishing Company, Amsterdam, 1968.

[12] C. Miranda, “Partial Differential of Elliptic Type,” Springer, New York, 1970.

[13] M. Schechter, L. Bers and F. John, “Partial Differential Equations,” Interscience, New York, 1964.

[1] H. Begehr, “Boundary Value Problems in Complex Analysis I,” Boles’ de la Association’s Mathematical Venezuelan, Vol. 12, No. 1, 2005, pp. 65-85.

[2] H. Begehr, “Boundary Value Problems in Complex Analysis II,” Boles’ de la Association’ Mathematical Venezuelan, Vol. 12, No. 2, 2005, pp. 217-250.

[3] 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

[4] P. Daripa and D. Mashat, “An Efficient and Novel Numerical Method for Quasiconformal Mappings of Doubly Connected Domains,” Numerical Algorithms, Vol. 18, No. 2, 1998, pp. 159-175. doi:10.1023/A:1019169431757

[5] 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

[6] P. Daripa and L. Borges, “A Fast Parallel Algorithm for the Poisson Equation on a Disk,” Journal of Computational Physics, Vol. 169, No. 1, 2001, pp. 151-192. doi: 10.1006/jcph.2001.6720

[7] 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

[8] P. Daripa, “A Fast Algorithm to Solve the Beltrami Equation with Applications to Quasiconformal Mappings,” Journal of Computational Physics, Vol. 106, No. 2, 1993, pp. 355-365.

[9] 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

[10] 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

[11] A. V. Bitsadze, “Boundary Value Problems of Second Order Elliptic Equations,” North-Holland Publishing Company, Amsterdam, 1968.

[12] C. Miranda, “Partial Differential of Elliptic Type,” Springer, New York, 1970.

[13] M. Schechter, L. Bers and F. John, “Partial Differential Equations,” Interscience, New York, 1964.