Función de Green en una franja horizontal infinita de amplitud π con frontera mixta Dirichlet-Neumann usando el método parqueting-reflections
Resumen
En el presente artículo se aborda la construcción de una Función armónica de Green para un dominio consistente en una banda horizontal infinita en ℂ de amplitud π cuya frontera es mixta bajo las condiciones Dirichlet-Neumann, tal función se generará empleando procesos analíticos basados en el método de las reflexiones. Las ideas producidas en esta investigación se pueden utilizar en otros tipos de problemas mixtos y otras ecuaciones en derivadas parciales cuyos dominios sean delimitados o no. La función de Green generada permite la obtención del Kernel de Poisson, insumo importante en la producción de representaciones integrales de soluciones de problemas de valores de frontera.
Citas
Abdymanapov, H., S. Begehr, y Tungatarov, A. (2005). Some Schwarz problems in a quarter plane. Eurasian Math.J.3(3), 22–35. doi: 10.1142/97898127731590001
Akel, M., y Begehr, H. (2016). Neumann function for a hyperbolic strip and a class of related plane domains. Mathematische Nachrichten, 290(4), 490–506. doi:10.1002/mana.201500501
Anas, M., Abd Albasset, Y., y Hasan, B. (2018). The harmonic green function for a right isosceles triangle. International Journal of Novel Research in Physics Chemistry Mathematics Novelty Journals, 5, 9–18.
Descargado de https://www.noveltyjournals.com/upload/paper/ The%20Harmonic%20Green-1414.pdf
Axler, S. J., Bourdon, P., y Ramey, W. (2011). Harmonic function theory. Springer. doi: 10.1007/978-1-4757-8137-3
Begehr, H. (1994). Complex analytic methods for partial differential equations. Singapore: World Scientific Pub. Co. Inc. Descargado de https://www.worldscientific.com/doi/abs/10.1142/2162
Begehr, H. (2005). Boundary value problems in complex analysis ii. Boletín de la Asociación Matemática Venezolana, 65-85.
Begehr, H. (2014). Green function for a hyperbolic strip and a class of related plane domains. Applicable Analysis, 93(11), 2370–2385. doi: 10.1080/00036811.2014.926336
Begehr, H. (2015). The parqueting-reflection principle. Current Trends in Analysis and Its Applications, 77-84.
doi: 10.1007/978-3-319-12577-011
Begehr, H., Burgumbayeva, S., Dauletkulova, A., y Lin, H. (2020). Harmonic green functions for the almaty apple. Complex Variables and Elliptic Equations, 65(11), 1814-1825. Descargado de https://doi.org/10.1080/17476933.2019.1681413 doi: 10.1080/17476933.2019.1681413
Begehr, H., y Vaitekhovich, T. (2012). Harmonic dirichlet problem for some equilateral triangle. Complex Variables and Elliptic Equations, 57(2-4), 185–196. doi: 10.1080/17476933.2011.598932
Begehr, H., y Vaitekhovich, T. (2013). Schwarz problem in lens and lune. Complex Variables and Elliptic Equations, 59(1), 76–84. doi: 10.1080/17476933.2013.799152
Begehr, H., y Vaitsiakhovich, T. (2010a). Green functions, reflections, and plane parqueting. Eurasian Mathematical Journal, 17-31.
Begehr, H., y Vaitsiakhovich, T. (2010b). How to find harmonic green functions in the plane. Complex Variables and Elliptic Equations, 56(12), 1169–1181. doi: 10.1080/17476933.2010.534157
Begehr, H., y Vaitsiakhovich, T. (2013). The parqueting-reflection principle for constructing green functions.
Cedeño, R., y Vanegas, C. (2022). Función de green vía mapeo conforme para el semiplano superior agrietado. Revista MATEMÁTICA, 20(1). Descargado de http://www.revistas.espol.edu.ec/index.php/ matematica/article/view/1007/902
Driscoll, T. A., y Trefethen, L. N. (2002). Schwarz-christoffel mapping. Cambridge University Press. doi: 10.1017/CBO9780511546808
Gamelin, T. W. (2001). Complex analysis. Springer, New York, NY. doi: 10.1007/978-0-387-21607-2
Lin, H. (2020). Harmonic green and neumann functions for domains bounded by two intersecting circular arcs. Complex Variables and Elliptic Equations, 67(1), 79–95. Descargado de http://dx.doi.org/10.1080/
17476933.2020.1816984 doi: 10.1080/17476933.2020.1816984
Lin, H. (2021). Parqueting-reflection principle and boundary value problems in some circular polygons
(Tesis Doctoral). Descargado de http://dx.doi.org/10.17169/refubium-32948
Nagle, R. K. (2017). Fundamentals of differential equations and boundary value problems. Pearson.
Natanzon, S. (2019). Complex analysis, riemann surfaces and integrable systems. Springer International Publishing. doi: 10.1007/978-3-030-34640-9 2
Shupeyeva, B. (2013a). Harmonic boundary value problems in a quarter ring domain. Advances in Pure and Applied Mathematics, 3(4). doi: 10.1515/ apam-2012-0025
Shupeyeva, B. (2013b). Some basic boundary value problems for complex partial differential equations in quarter ring and half hexagon (Tesis Doctoral). Descargado de http://dx.doi.org/10.17169/refubium-12787
Shupeyeva, B. (2016). Dirichlet problem for complex poisson equation in a half hexagon domain. Journal of Complex Analysis, 2016, 1-8. doi: 10.1155/2016/8097095
Taghizadeh, N., y Mohammadi, V. S. (2017). Dirichlet and neumann problems for poisson equation in half lens. Journal of Contemporary Mathemati- cal Analysis (Armenian Academy of Sciences), 52(2), 61–69.
doi: 10.3103/ s1068362317020017
Trench, W. F. (1999). Conditional convergence of infinite products. The American Mathematical Monthly, 106(7), 646–651. Descargado de http://www.jstor.org/stable/2589494
Vaitekhovich, T. S. (2008a). Boundary value problems to first-order complex partial differential equations in a ring domain. Integral Transforms and Special Functions, 19(3), 211–233. doi: 10.1080/10652460701736569
Vaitekhovich, T. (2008b). Boundary value problems for complex partial diferential equations in a ring domain (Doctoral Thesis, Freie Universität Berlin). Descargado de www.diss.fu-berlin.de/diss/receive/FUDISS thesis 000000003859
Vergara, J., y Vanegas, C. (2021). A fundamental solution for the la place operator in a doubly connected domain. Bull. Comput. Appl. Math., 9, 85-96.
Villat, H. (1911). Sur le probléme de dirichlet relatif á une couronne circulaire. Comptes Rendus de l’ Académie des Sciences, 680-682.
Wang, Y. (2011). Boundary value problems for complex partial differential equations in fan-shaped domains (Doctoral Thesis, Freie Universitat Berlin). Descargado de www.diss.fu-berlin.de/diss/receive/FUDISS thesis 000000021359
Wang, Y., y Wang, Y. (2010). On schwarz-type boundary-value problems of polyanalytic equation on a triangle. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, 84.
Akel, M., y Begehr, H. (2016). Neumann function for a hyperbolic strip and a class of related plane domains. Mathematische Nachrichten, 290(4), 490–506. doi:10.1002/mana.201500501
Anas, M., Abd Albasset, Y., y Hasan, B. (2018). The harmonic green function for a right isosceles triangle. International Journal of Novel Research in Physics Chemistry Mathematics Novelty Journals, 5, 9–18.
Descargado de https://www.noveltyjournals.com/upload/paper/ The%20Harmonic%20Green-1414.pdf
Axler, S. J., Bourdon, P., y Ramey, W. (2011). Harmonic function theory. Springer. doi: 10.1007/978-1-4757-8137-3
Begehr, H. (1994). Complex analytic methods for partial differential equations. Singapore: World Scientific Pub. Co. Inc. Descargado de https://www.worldscientific.com/doi/abs/10.1142/2162
Begehr, H. (2005). Boundary value problems in complex analysis ii. Boletín de la Asociación Matemática Venezolana, 65-85.
Begehr, H. (2014). Green function for a hyperbolic strip and a class of related plane domains. Applicable Analysis, 93(11), 2370–2385. doi: 10.1080/00036811.2014.926336
Begehr, H. (2015). The parqueting-reflection principle. Current Trends in Analysis and Its Applications, 77-84.
doi: 10.1007/978-3-319-12577-011
Begehr, H., Burgumbayeva, S., Dauletkulova, A., y Lin, H. (2020). Harmonic green functions for the almaty apple. Complex Variables and Elliptic Equations, 65(11), 1814-1825. Descargado de https://doi.org/10.1080/17476933.2019.1681413 doi: 10.1080/17476933.2019.1681413
Begehr, H., y Vaitekhovich, T. (2012). Harmonic dirichlet problem for some equilateral triangle. Complex Variables and Elliptic Equations, 57(2-4), 185–196. doi: 10.1080/17476933.2011.598932
Begehr, H., y Vaitekhovich, T. (2013). Schwarz problem in lens and lune. Complex Variables and Elliptic Equations, 59(1), 76–84. doi: 10.1080/17476933.2013.799152
Begehr, H., y Vaitsiakhovich, T. (2010a). Green functions, reflections, and plane parqueting. Eurasian Mathematical Journal, 17-31.
Begehr, H., y Vaitsiakhovich, T. (2010b). How to find harmonic green functions in the plane. Complex Variables and Elliptic Equations, 56(12), 1169–1181. doi: 10.1080/17476933.2010.534157
Begehr, H., y Vaitsiakhovich, T. (2013). The parqueting-reflection principle for constructing green functions.
Cedeño, R., y Vanegas, C. (2022). Función de green vía mapeo conforme para el semiplano superior agrietado. Revista MATEMÁTICA, 20(1). Descargado de http://www.revistas.espol.edu.ec/index.php/ matematica/article/view/1007/902
Driscoll, T. A., y Trefethen, L. N. (2002). Schwarz-christoffel mapping. Cambridge University Press. doi: 10.1017/CBO9780511546808
Gamelin, T. W. (2001). Complex analysis. Springer, New York, NY. doi: 10.1007/978-0-387-21607-2
Lin, H. (2020). Harmonic green and neumann functions for domains bounded by two intersecting circular arcs. Complex Variables and Elliptic Equations, 67(1), 79–95. Descargado de http://dx.doi.org/10.1080/
17476933.2020.1816984 doi: 10.1080/17476933.2020.1816984
Lin, H. (2021). Parqueting-reflection principle and boundary value problems in some circular polygons
(Tesis Doctoral). Descargado de http://dx.doi.org/10.17169/refubium-32948
Nagle, R. K. (2017). Fundamentals of differential equations and boundary value problems. Pearson.
Natanzon, S. (2019). Complex analysis, riemann surfaces and integrable systems. Springer International Publishing. doi: 10.1007/978-3-030-34640-9 2
Shupeyeva, B. (2013a). Harmonic boundary value problems in a quarter ring domain. Advances in Pure and Applied Mathematics, 3(4). doi: 10.1515/ apam-2012-0025
Shupeyeva, B. (2013b). Some basic boundary value problems for complex partial differential equations in quarter ring and half hexagon (Tesis Doctoral). Descargado de http://dx.doi.org/10.17169/refubium-12787
Shupeyeva, B. (2016). Dirichlet problem for complex poisson equation in a half hexagon domain. Journal of Complex Analysis, 2016, 1-8. doi: 10.1155/2016/8097095
Taghizadeh, N., y Mohammadi, V. S. (2017). Dirichlet and neumann problems for poisson equation in half lens. Journal of Contemporary Mathemati- cal Analysis (Armenian Academy of Sciences), 52(2), 61–69.
doi: 10.3103/ s1068362317020017
Trench, W. F. (1999). Conditional convergence of infinite products. The American Mathematical Monthly, 106(7), 646–651. Descargado de http://www.jstor.org/stable/2589494
Vaitekhovich, T. S. (2008a). Boundary value problems to first-order complex partial differential equations in a ring domain. Integral Transforms and Special Functions, 19(3), 211–233. doi: 10.1080/10652460701736569
Vaitekhovich, T. (2008b). Boundary value problems for complex partial diferential equations in a ring domain (Doctoral Thesis, Freie Universität Berlin). Descargado de www.diss.fu-berlin.de/diss/receive/FUDISS thesis 000000003859
Vergara, J., y Vanegas, C. (2021). A fundamental solution for the la place operator in a doubly connected domain. Bull. Comput. Appl. Math., 9, 85-96.
Villat, H. (1911). Sur le probléme de dirichlet relatif á une couronne circulaire. Comptes Rendus de l’ Académie des Sciences, 680-682.
Wang, Y. (2011). Boundary value problems for complex partial differential equations in fan-shaped domains (Doctoral Thesis, Freie Universitat Berlin). Descargado de www.diss.fu-berlin.de/diss/receive/FUDISS thesis 000000021359
Wang, Y., y Wang, Y. (2010). On schwarz-type boundary-value problems of polyanalytic equation on a triangle. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, 84.
