# American Institute of Mathematical Sciences

November  2020, 19(11): 5097-5114. doi: 10.3934/cpaa.2020228

## A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

 School of Engineering, Computing and Mathematics, Wheatley Campus, Oxford Brookes University, OX33 1HX, Wheatley, UK

Received  March 2020 Revised  May 2020 Published  November 2020 Early access  July 2020

Fund Project: This research was supported by the grants 1636273 from the EPSRC

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [25] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

Citation: Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228
##### References:
 [1] M. A. Al-Jawary, J. Ravnik and L. C. Wrobel, Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients, Comput. Math. Appl., 64 (2012), 2695-2711.  doi: 10.1016/j.camwa.2012.08.002. [2] A. Beshley, B. Chapko and T. Johansson, On the alternating method and boundary-domain integrals for elliptic Cauchy problems, Comput. Math. Appl., 78 (2019), 3514-3526.  doi: 10.1016/j.camwa.2019.05.025. [3] A. Beshley, R. Chapko and B. T. Johansson, An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients, J. Eng. Math., 112 (2018), 63-73.  doi: 10.1007/s10665-018-9965-7. [4] R. Chapko and B. T. Johansson, A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems, Appl. Numer. Math., 129 (2018), 104-119.  doi: 10.1016/j.apnum.2018.03.004. [5] J. Choi and D. Kim, Estimates for Green functions of Stokes systems in two dimensional domains., J. Math. Anal. Appl., 471 (2019), 102-125.  doi: 10.1016/j.jmaa.2018.10.067. [6] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Singular localised boundary-domain integral equations of acoustic scattering by inhomogeneous anisotropic obstacle, Math. Method. Appl. Sci., 41 (2018), 8033-8058.  doi: 10.1002/mma.5268. [7] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅰ: Equivalence and invertibility, J. Integral Equ. Appl., 21 (2009), 499-543.  doi: 10.1216/JIE-2009-21-4-499. [8] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅱ: Solution regularity and asymptotics, J. Integral Equ. Appl., 22 (2010), 19-37.  doi: 10.1216/JIE-2010-22-1-19. [9] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Anal. Appl., 11 (2013), 1-33.  doi: 10.1142/S0219530513500061. [10] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043. [11] M. Costabel and E. P. Stephan, An improved boundary element Galerkin method for three dimensional crack problems, Integr. Equat. Oper. Th., 10 (1987), 467-507.  doi: 10.1007/BF01201149. [12] T. T. Dufera and S. E. Mikhailov, Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D In: Integral Methods in Science and Engineering: Theoretical and Computational Advances., C. Constanda and A. Kirsh, eds. Springer (Birkhäuser): Boston, (2015), 163-175. [13] O. Gonzalez, A theorem on the surface traction field in potential representations of Stokes flow, SIAM J. Appl. Math., 75 (2015), 1578-1598.  doi: 10.1137/140978119. [14] R. Grzhibovskis, Mikhailov S.E. and Rjasanow S.: Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Comput. Mech., 51 (2013), 495-503.  doi: 10.1007/s00466-012-0777-8. [15] N. M. Gunter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967. [16] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, 1973. [18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [19] S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027. [20] S. E. Mikhailov, Analysis of Segregated Boundary-Domain Integral Equations for BVPs with Non-smooth Coefficient on Lipschitz Domains, Bound Value Probl, 87 (2018), 1-52.  doi: 10.1186/s13661-018-0992-0. [21] S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690. [22] S. E. Mikhailov and C. F. Portillo, Analysis of Boundary-Domain Integral Equations to the Mixed BVP for a compressible Stokes system with variable viscosity, Communications on Pure and Applied Analysis, 18 (2019): 3059–3088. doi: 10.3934/cpaa.2019137. [23] S. E. Mikhailov and C. F. Portillo, Analysis of boundary-domain integral equations based on a new parametrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain, J. Integral Equ. Appl., 87 (2018). doi: 10.1186/s13661-018-0992-0. [24] A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems: With Application to Shells, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0094576. [25] C. F. Portillo, Boundary-Domain Integral Equations for the diffusion equation in inhomogeneous media based on a new family of parametrices, Complex Var. Elliptic, 65 (2020), 558-572.  doi: 10.1080/17476933.2019.1591382. [26] C. F. Portillo and Z. W. Woldemicheal, On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix, Complex Var. Elliptic, (2019), 1–15. doi: 10.1080/17476933.2019.1687457. [27] J. Ravnik and J. Tibaut, Fast boundary-domain integral method for heat transfer simulations, Eng. Anal. Bound. Elem., 99 (2019), 222-232.  doi: 10.1016/j.enganabound.2018.12.003. [28] J. Sladek, V. Sladek and Ch .Zhang, Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients, J. Eng. Math., 51 (2005), 261-282. [29] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, 2007. doi: 10.1007/978-0-387-68805-3.

show all references

##### References:
 [1] M. A. Al-Jawary, J. Ravnik and L. C. Wrobel, Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients, Comput. Math. Appl., 64 (2012), 2695-2711.  doi: 10.1016/j.camwa.2012.08.002. [2] A. Beshley, B. Chapko and T. Johansson, On the alternating method and boundary-domain integrals for elliptic Cauchy problems, Comput. Math. Appl., 78 (2019), 3514-3526.  doi: 10.1016/j.camwa.2019.05.025. [3] A. Beshley, R. Chapko and B. T. Johansson, An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients, J. Eng. Math., 112 (2018), 63-73.  doi: 10.1007/s10665-018-9965-7. [4] R. Chapko and B. T. Johansson, A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems, Appl. Numer. Math., 129 (2018), 104-119.  doi: 10.1016/j.apnum.2018.03.004. [5] J. Choi and D. Kim, Estimates for Green functions of Stokes systems in two dimensional domains., J. Math. Anal. Appl., 471 (2019), 102-125.  doi: 10.1016/j.jmaa.2018.10.067. [6] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Singular localised boundary-domain integral equations of acoustic scattering by inhomogeneous anisotropic obstacle, Math. Method. Appl. Sci., 41 (2018), 8033-8058.  doi: 10.1002/mma.5268. [7] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅰ: Equivalence and invertibility, J. Integral Equ. Appl., 21 (2009), 499-543.  doi: 10.1216/JIE-2009-21-4-499. [8] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, Ⅱ: Solution regularity and asymptotics, J. Integral Equ. Appl., 22 (2010), 19-37.  doi: 10.1216/JIE-2010-22-1-19. [9] O. Chkadua, S. E. Mikhailov and D. Natroshvili, Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains, Anal. Appl., 11 (2013), 1-33.  doi: 10.1142/S0219530513500061. [10] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043. [11] M. Costabel and E. P. Stephan, An improved boundary element Galerkin method for three dimensional crack problems, Integr. Equat. Oper. Th., 10 (1987), 467-507.  doi: 10.1007/BF01201149. [12] T. T. Dufera and S. E. Mikhailov, Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet BVP in 2D In: Integral Methods in Science and Engineering: Theoretical and Computational Advances., C. Constanda and A. Kirsh, eds. Springer (Birkhäuser): Boston, (2015), 163-175. [13] O. Gonzalez, A theorem on the surface traction field in potential representations of Stokes flow, SIAM J. Appl. Math., 75 (2015), 1578-1598.  doi: 10.1137/140978119. [14] R. Grzhibovskis, Mikhailov S.E. and Rjasanow S.: Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D, Comput. Mech., 51 (2013), 495-503.  doi: 10.1007/s00466-012-0777-8. [15] N. M. Gunter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967. [16] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, 1973. [18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. [19] S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027. [20] S. E. Mikhailov, Analysis of Segregated Boundary-Domain Integral Equations for BVPs with Non-smooth Coefficient on Lipschitz Domains, Bound Value Probl, 87 (2018), 1-52.  doi: 10.1186/s13661-018-0992-0. [21] S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690. [22] S. E. Mikhailov and C. F. Portillo, Analysis of Boundary-Domain Integral Equations to the Mixed BVP for a compressible Stokes system with variable viscosity, Communications on Pure and Applied Analysis, 18 (2019): 3059–3088. doi: 10.3934/cpaa.2019137. [23] S. E. Mikhailov and C. F. Portillo, Analysis of boundary-domain integral equations based on a new parametrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain, J. Integral Equ. Appl., 87 (2018). doi: 10.1186/s13661-018-0992-0. [24] A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems: With Application to Shells, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0094576. [25] C. F. Portillo, Boundary-Domain Integral Equations for the diffusion equation in inhomogeneous media based on a new family of parametrices, Complex Var. Elliptic, 65 (2020), 558-572.  doi: 10.1080/17476933.2019.1591382. [26] C. F. Portillo and Z. W. Woldemicheal, On the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix, Complex Var. Elliptic, (2019), 1–15. doi: 10.1080/17476933.2019.1687457. [27] J. Ravnik and J. Tibaut, Fast boundary-domain integral method for heat transfer simulations, Eng. Anal. Bound. Elem., 99 (2019), 222-232.  doi: 10.1016/j.enganabound.2018.12.003. [28] J. Sladek, V. Sladek and Ch .Zhang, Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients, J. Eng. Math., 51 (2005), 261-282. [29] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, 2007. doi: 10.1007/978-0-387-68805-3.
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