# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021114
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## An indirect BIE free of degenerate scales

 a. Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan b. Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan c. Department of Civil Engineering, National Cheng-Kung University, Tainan, 70101, Taiwan d. Bachelor Degree Program in Ocean Engineering and Technology, National Taiwan Ocean University, Keelung, 20224, Taiwan e. Bachelor Degree Program in Ocean Engineering and Technology, National Taiwan Ocean University, Keelung, 20224, Taiwan

* Corresponding author

Received  November 2020 Revised  May 2021 Early access August 2021

Fund Project: The financial support from the Ministry of Science and Technology, Taiwan under Grant No. MOST 106-2221-E-019-009-MY3 and No. MOST 107-2221-E-019 -003, and we also appreciate Mr. Kuen-Ting Lien to provide numerical results

Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.

Citation: Jeng-Tzong Chen, Shing-Kai Kao, Jeng-Hong Kao, Wei-Chen Tai. An indirect BIE free of degenerate scales. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021114
##### References:
 [1] A. J. Burton and G. F. Miller, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 323 (2016), 4367-4382. doi: 10.2307/2152750.  Google Scholar [2] R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging., 6 (2012), 25-38. doi: 10.2307/2152750.  Google Scholar [3] G. Chen and J. X. Zhou, Boundary Elements Methods with Applications to Nonlinear Problems, Atlantis press, Paris, 2010.  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [4] I. L. Chen, J. T. Chen, S. R. Kuo and M. T. Liang, A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation method, J. Acoust. Soc. Am., 109 (2001), 982-998. doi: 10.2307/2152750.  Google Scholar [5] J. T. Chen, J. W. Lee, Y. L. Huang, C. H. Shao and C. H. Lu, On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale, J. Mech., 37 (2021), 339-345. doi: 10.2307/2152750.  Google Scholar [6] J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale for a torsion bar problem using BEM, Third International Conference on BeTeQ, (2002). doi: 10.2307/2152750.  Google Scholar [7] J. T. Chen, L. W. Liu and H. K. Hong, Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply-connected problem, R. Soc. London Ser. A-Math. Phys. Eng. Sci., 459 (2003), 1891-1924. doi: 10.2307/2152750.  Google Scholar [8] J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chiu, Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants, Eng. Anal. Bound. Elem., 25 (2001), 819-828. doi: 10.2307/2152750.  Google Scholar [9] J. T. Chen and W. C. Shen, Degenerate scale for multiply connected Laplace problems, Mech. Res. Commun., 34 (2007), 69-77. doi: 10.2307/2152750.  Google Scholar [10] J. T. Chen, Y. T. Lee, S. R. Kuo and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM, Eng. Anal. Bound. Elem., 36 (2012), 1397-1405. doi: 10.2307/2152750.  Google Scholar [11] J. T. Chen. S. R. Kuo, Y. L. Huang and S. K. Kao, Linkage of logarithmic capacity in potential theory and degenerate scale in the BEM for the two tangent discs, Appl. Math. Lett., 102 (2020), 106-135. doi: 10.2307/2152750.  Google Scholar [12] J. T. Chen. S. R. Kuo. K. T. Lien and Y. L. Huang, On the degenerate scale of an infinite plane containing two unequal circles, Adv. Appl. Math. Mech., 12 (2020), 1280-1300. doi: 10.2307/2152750.  Google Scholar [13] J. T. Chen. S. R. Kuo and Y. L. Huang, Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM, Eng. Anal. Bound. Elem., 120 (2020), 238-245. doi: 10.2307/2152750.  Google Scholar [14] J. T. Chen, C. F. Lee, I. L. Chen and J. H. Lin, An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation, Eng. Anal. Bound. Elem., 26 (2002), 559-569. doi: 10.2307/2152750.  Google Scholar [15] J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale problem when solving Laplace equation by BEM and its treatment, Int. J. Numer. Meth. Eng., 62 (2002), 559-569. doi: 10.2307/2152750.  Google Scholar [16] J. T. Chen, H. D. Han, S. R. Kuo and S. K. Kao, Regularized methods for ill-conditioned system of the integral equations of the first kind, Inverse Probl. Sci. Eng., 22 (2014), 1176-1195. doi: 10.2307/2152750.  Google Scholar [17] J. T. Chen, S. R. Kuo, S. K. Kao and J. Jian, Revisit of a degenerate scale: A semi-circular disc, Comp. Appl. Math., 283 (2015), 182-200. doi: 10.2307/2152750.  Google Scholar [18] J. T. Chen, Y. T. Lee, Y. L. Chang and J. Jian, A self-regularized approach for rank-deficiency systems in the BEM of 2D Laplace problems, Inverse Probl. Sci. Eng., 25 (2017), 89-113. doi: 10.2307/2152750.  Google Scholar [19] J. T. Chen, S. K. Kao and J. W. Lee, Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates, Engng. Anal. Bound. Elem., 85 (2017), 70-86. doi: 10.2307/2152750.  Google Scholar [20] C. Constanda, On the solution of the Dirichlet problem for the two dimensional Laplace equation, Proc. Am. Math. Soc., 119 (1993), 877-884. doi: 10.2307/2152750.  Google Scholar [21] C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656. doi: 10.2307/2152750.  Google Scholar [22] C. L. Epstein, L. Greengard and T. Hagstrom, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 36 (2016), 4367-4382. doi: 10.2307/2152750.  Google Scholar [23] G. Fichera, Boundary Problems in Differential Equations, The University of Wisconsin Press, Madison (WI), (1959). doi: 10.1007/978-1-4612-0873-0.  Google Scholar [24] M. A. Golberg, Solution methods for integral equations, Plenum press, New York and London, (1978).  Google Scholar [25] H. C. Hu, Necessary and sufficient boundary integral equations for plane harmonic functions, China Ser. A-Math. Phys. Astron., 35 (1992), 861-869. doi: 10.2307/2152750.  Google Scholar [26] M. A. Jaswon and G. T. Symm, Integral equation method in potential theory and elastostatics, in Computational mathematics and applications, Academic Press, (1977).  Google Scholar [27] S. R. Kuo, J. T. Chen, J. W. Lee and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for regular N-gon domains in BEM, Appl. Math. Comput., 219 (2012), 1397-1405. doi: 10.2307/2152750.  Google Scholar [28] S. R. Kuo and J. T. Chen, Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs, Appl. Math. Lett., 29 (2013), 929-938. doi: 10.2307/2152750.  Google Scholar [29] S. R. Kuo, S. K. Kao, Y. L. Huang and J. T. Chen, Revisit of the degenerate scale for an infinite plane problem containing two circular holes using conformal mapping, Appl. Math. Lett., 92 (2019), 99-107. doi: 10.2307/2152750.  Google Scholar [30] G. Li, F. Gu and F. Jiang, Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation, Commun. Pure Appl. Anal., 19 (2020), 1449-1462. doi: 10.2307/2152750.  Google Scholar [31] C. K. Lin and K. C. Wu, Fundamental solution of the Laplace equation (dimensional analysis viewpoint), Proceedings of the 7th Taiwan-Philippine Symposium in Mathematics, (2007), 94-102. doi: 10.2307/2152750.  Google Scholar [32] H. A. Schenck, Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Am., 44 (1976), 41-58. doi: 10.2307/2152750.  Google Scholar [33] G. Strang, Introduction to Linear Algebra, 5$^{nd}$ edition, Wellesley Cambridge Press, Wellesley, 2016. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

show all references

##### References:
 [1] A. J. Burton and G. F. Miller, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 323 (2016), 4367-4382. doi: 10.2307/2152750.  Google Scholar [2] R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging., 6 (2012), 25-38. doi: 10.2307/2152750.  Google Scholar [3] G. Chen and J. X. Zhou, Boundary Elements Methods with Applications to Nonlinear Problems, Atlantis press, Paris, 2010.  doi: 10.1007/978-1-4612-0873-0.  Google Scholar [4] I. L. Chen, J. T. Chen, S. R. Kuo and M. T. Liang, A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation method, J. Acoust. Soc. Am., 109 (2001), 982-998. doi: 10.2307/2152750.  Google Scholar [5] J. T. Chen, J. W. Lee, Y. L. Huang, C. H. Shao and C. H. Lu, On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale, J. Mech., 37 (2021), 339-345. doi: 10.2307/2152750.  Google Scholar [6] J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale for a torsion bar problem using BEM, Third International Conference on BeTeQ, (2002). doi: 10.2307/2152750.  Google Scholar [7] J. T. Chen, L. W. Liu and H. K. Hong, Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply-connected problem, R. Soc. London Ser. A-Math. Phys. Eng. Sci., 459 (2003), 1891-1924. doi: 10.2307/2152750.  Google Scholar [8] J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chiu, Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants, Eng. Anal. Bound. Elem., 25 (2001), 819-828. doi: 10.2307/2152750.  Google Scholar [9] J. T. Chen and W. C. Shen, Degenerate scale for multiply connected Laplace problems, Mech. Res. Commun., 34 (2007), 69-77. doi: 10.2307/2152750.  Google Scholar [10] J. T. Chen, Y. T. Lee, S. R. Kuo and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM, Eng. Anal. Bound. Elem., 36 (2012), 1397-1405. doi: 10.2307/2152750.  Google Scholar [11] J. T. Chen. S. R. Kuo, Y. L. Huang and S. K. Kao, Linkage of logarithmic capacity in potential theory and degenerate scale in the BEM for the two tangent discs, Appl. Math. Lett., 102 (2020), 106-135. doi: 10.2307/2152750.  Google Scholar [12] J. T. Chen. S. R. Kuo. K. T. Lien and Y. L. Huang, On the degenerate scale of an infinite plane containing two unequal circles, Adv. Appl. Math. Mech., 12 (2020), 1280-1300. doi: 10.2307/2152750.  Google Scholar [13] J. T. Chen. S. R. Kuo and Y. L. Huang, Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM, Eng. Anal. Bound. Elem., 120 (2020), 238-245. doi: 10.2307/2152750.  Google Scholar [14] J. T. Chen, C. F. Lee, I. L. Chen and J. H. Lin, An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation, Eng. Anal. Bound. Elem., 26 (2002), 559-569. doi: 10.2307/2152750.  Google Scholar [15] J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale problem when solving Laplace equation by BEM and its treatment, Int. J. Numer. Meth. Eng., 62 (2002), 559-569. doi: 10.2307/2152750.  Google Scholar [16] J. T. Chen, H. D. Han, S. R. Kuo and S. K. Kao, Regularized methods for ill-conditioned system of the integral equations of the first kind, Inverse Probl. Sci. Eng., 22 (2014), 1176-1195. doi: 10.2307/2152750.  Google Scholar [17] J. T. Chen, S. R. Kuo, S. K. Kao and J. Jian, Revisit of a degenerate scale: A semi-circular disc, Comp. Appl. Math., 283 (2015), 182-200. doi: 10.2307/2152750.  Google Scholar [18] J. T. Chen, Y. T. Lee, Y. L. Chang and J. Jian, A self-regularized approach for rank-deficiency systems in the BEM of 2D Laplace problems, Inverse Probl. Sci. Eng., 25 (2017), 89-113. doi: 10.2307/2152750.  Google Scholar [19] J. T. Chen, S. K. Kao and J. W. Lee, Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates, Engng. Anal. Bound. Elem., 85 (2017), 70-86. doi: 10.2307/2152750.  Google Scholar [20] C. Constanda, On the solution of the Dirichlet problem for the two dimensional Laplace equation, Proc. Am. Math. Soc., 119 (1993), 877-884. doi: 10.2307/2152750.  Google Scholar [21] C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656. doi: 10.2307/2152750.  Google Scholar [22] C. L. Epstein, L. Greengard and T. Hagstrom, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 36 (2016), 4367-4382. doi: 10.2307/2152750.  Google Scholar [23] G. Fichera, Boundary Problems in Differential Equations, The University of Wisconsin Press, Madison (WI), (1959). doi: 10.1007/978-1-4612-0873-0.  Google Scholar [24] M. A. Golberg, Solution methods for integral equations, Plenum press, New York and London, (1978).  Google Scholar [25] H. C. Hu, Necessary and sufficient boundary integral equations for plane harmonic functions, China Ser. A-Math. Phys. Astron., 35 (1992), 861-869. doi: 10.2307/2152750.  Google Scholar [26] M. A. Jaswon and G. T. Symm, Integral equation method in potential theory and elastostatics, in Computational mathematics and applications, Academic Press, (1977).  Google Scholar [27] S. R. Kuo, J. T. Chen, J. W. Lee and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for regular N-gon domains in BEM, Appl. Math. Comput., 219 (2012), 1397-1405. doi: 10.2307/2152750.  Google Scholar [28] S. R. Kuo and J. T. Chen, Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs, Appl. Math. Lett., 29 (2013), 929-938. doi: 10.2307/2152750.  Google Scholar [29] S. R. Kuo, S. K. Kao, Y. L. Huang and J. T. Chen, Revisit of the degenerate scale for an infinite plane problem containing two circular holes using conformal mapping, Appl. Math. Lett., 92 (2019), 99-107. doi: 10.2307/2152750.  Google Scholar [30] G. Li, F. Gu and F. Jiang, Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation, Commun. Pure Appl. Anal., 19 (2020), 1449-1462. doi: 10.2307/2152750.  Google Scholar [31] C. K. Lin and K. C. Wu, Fundamental solution of the Laplace equation (dimensional analysis viewpoint), Proceedings of the 7th Taiwan-Philippine Symposium in Mathematics, (2007), 94-102. doi: 10.2307/2152750.  Google Scholar [32] H. A. Schenck, Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Am., 44 (1976), 41-58. doi: 10.2307/2152750.  Google Scholar [33] G. Strang, Introduction to Linear Algebra, 5$^{nd}$ edition, Wellesley Cambridge Press, Wellesley, 2016. doi: 10.1007/978-1-4612-0873-0.  Google Scholar
Row space, column space, null space and range of $[U]$ and $[T]$ matrices in the indirect and direct BEMs when the degenerate scale occurs
Numerical evidences for the degenerate scale of four cases in the BEM
Comparison of four BIEs
Study on the degenerate scale by using different approaches
Degenerate scales appearing in the conventional BEM, the approach of adding rigid body mode, the Burton and Miller approach and the method of introducing the fictitious source point
Comparison of the four kernel functions
 [1] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [2] Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 [3] Georg Ostrovski, Sebastian van Strien. Payoff performance of fictitious play. Journal of Dynamics & Games, 2014, 1 (4) : 621-638. doi: 10.3934/jdg.2014.1.621 [4] István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089 [5] Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 511-529. doi: 10.3934/dcds.2014.34.511 [6] P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807 [7] Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357 [8] Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767 [9] Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control & Related Fields, 2011, 1 (1) : 1-20. doi: 10.3934/mcrf.2011.1.1 [10] Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091 [11] Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055 [12] H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127 [13] Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 [14] Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487 [15] Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075 [16] João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 [17] Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 [18] Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707 [19] Saeed Assani, Jianlin Jiang, Ahmad Assani, Feng Yang. Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1357-1382. doi: 10.3934/jimo.2020025 [20] Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200

2020 Impact Factor: 1.916