Abstract
We investigate existence and uniqueness conditions for solution to one Robin type problem for inhomogeneous biharmonic equation in the unit ball. We construct polynomial solution to the problem when the boundary functions of the problems are polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lai, M-C., and Liu, H.-C. “Fast Direct Solver for the Biharmonic Equation on a Disk and its Application to Incompressible Flows”, Appl.Math. Comput. 164, No. 3, 679–695 (2005).
Dang, Q. A. “IterativeMethod for Solving the Neumann Boundary Problemfor Biharmonic Type Equation”, J. Comput. Appl.Math. 196, No. 2, 634–643 (2006).
Dang, Q. A. and ThaoMai Xuan, “Iterative Method for Solving a Problem withMixed Boundary Conditions for Biharmonic Equation Arising in Fracture Mechanics”, Boletim da Sociedade Paranaense de Matem. 31, No. 1, 65–78 (2013).
Karachik, V. V. and Antropova, N. A. “Polynomial Solutions of the Dirichlet Problem for the Biharmonic Equation in the Ball”, Differ. Equ. 49, No. 2, 251–256 (2013).
Karachik, V. V. “On Solvability Conditions for the Neumann Problem for a Polyharmonic Equation in the Unit Ball”, J. Appl. Ind. Math. 8, No. 1, 63–75 (2014).
Gomez-Polanco, A., Guevara-Jordan, J. M. and Molina, B. “A Mimetic Iterative Scheme for Solving Biharmonic Equations”, Math. and Comput.Model. 57, No. 9–10, 2132–2139 (2013).
Gazzola, F. and Sweers, G. “On Positivity for the Biharmonic Operator under Steklov Boundary Conditions”, Arch. Rat. Mech. Anal. 188, 399–427 (2008).
Karachik, V. V., Sadybekov, M. A. and Torebek, B. T. “Uniqueness of Solutions to Boundary-Value Problems for the Biharmonic Equation in a Ball”, Electronic J. Diff. Equat., No. 244 (2015).
Karachik, V. V. “Solvability Conditions for the Neumann Problem for the Homogeneous Polyharmonic Equation”, Differ. Equ. 50, No. 11, 1449–1456 (2014).
Gulyashchikh, I. A. “Homogeneous Dirichlet–Riquier Problem for Inhomogeneous Biharmonic Equation in a Ball”, Vestnik YuUrGU. Ser.Matem.Mekhan. Fizika 8 (1), 57–60 (2016) [in Russian].
Karachik, V. V. and Torebek, B. T. “On One Mathematical Model Described by Boundary Value Problem for the Biharmonic Equation”, Bulletin of South Ural Univ. Ser. “Matem. Model. and Programm.” 9, No. 4, 40–52 (2016).
Karachik, V. V. “On theMean Value Property for Polyharmonic Functions in a Ball”, Siberian Adv.Math. 24, No. 3, 169–182 (2014).
Bitsadze, A. V. Equations of Mathematical Physics (Nauka, Moscow, 1982) [in Russian].
Bateman, H. and Erdelyi, A. Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1953; Nauka, Moscow, 1966).
Karachik, V. V. “Construction of Polynomial Solutions of Some Boundary Value Problems for the Poisson Equation”, Comput. Math.Math. Phys. 51, No. 9, 1567–1587 (2011).
Karachik, V.V. “Polynomial Solutions of Partial Differential Equations withConstant Coefficients. I”, Vestnik YuUrGU. Ser.Matem.Mekhan. Fizika, No. 10 (227), 4–17 (2011).
Karachik, V. V. Method of Normalized Systems of Functions (South Ural State University, Chelyabinsk, 2014) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.V. Karachik, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 39–53.
About this article
Cite this article
Karachik, V.V. Solving a Problem of Robin Type for Biharmonic Equation. Russ Math. 62, 34–48 (2018). https://doi.org/10.3103/S1066369X18020056
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X18020056