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  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 & Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228
References:
[1]

M. A. Al-JawaryJ. 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.  Google Scholar

[2]

A. BeshleyB. 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.  Google Scholar

[3]

A. BeshleyR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

O. ChkaduaS. 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.  Google Scholar

[7]

O. ChkaduaS. 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.  Google Scholar

[8]

O. ChkaduaS. 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.  Google Scholar

[9]

O. ChkaduaS. 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.  Google Scholar

[10]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

N. M. Gunter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967.  Google Scholar

[16]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, 1973.  Google Scholar

[18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[21]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems: With Application to Shells, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0094576.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. SladekV. 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.   Google Scholar

[29]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, 2007. doi: 10.1007/978-0-387-68805-3.  Google Scholar

show all references

References:
[1]

M. A. Al-JawaryJ. 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.  Google Scholar

[2]

A. BeshleyB. 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.  Google Scholar

[3]

A. BeshleyR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

O. ChkaduaS. 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.  Google Scholar

[7]

O. ChkaduaS. 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.  Google Scholar

[8]

O. ChkaduaS. 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.  Google Scholar

[9]

O. ChkaduaS. 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.  Google Scholar

[10]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

N. M. Gunter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967.  Google Scholar

[16]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, 1973.  Google Scholar

[18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[21]

S. E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients, Engineering Analysis with Boundary Elements, 26 (2002), 681-690.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems: With Application to Shells, Springer-Verlag, Berlin, 1998. doi: 10.1007/BFb0094576.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. SladekV. 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.   Google Scholar

[29]

O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, 2007. doi: 10.1007/978-0-387-68805-3.  Google Scholar

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