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Historical developments of computing of the inhomogeneous Dirichlet problem in bidimensional or multidimensional domains. (English) Zbl 1391.35004
Summary: We present the question of solving and computing nonhomogeneous Dirichlet problems in domains in \(\mathbb R^2\) or \(\mathbb R^n\) (\(n \geq 2\)). Using complex analysis we present the Kutta-Joukowski method of computing a bidimensional flow around a profile. In the case [G. K. Batchelor, An introduction to fluid dynamics. 2nd pbk-ed. Cambridge: Cambridge University Press (1999; Zbl 0958.76001); L. D. Landau and E. M. Lifshits, Fluid mechanics. 2nd ed. Transl. from the Russian by J. B. Sykes and W. H. Reid. Oxford etc.: Pergamon Press (1987; Zbl 0655.76001)] of a three-dimensional flow around a cylindrical profile, we determine Sobolev spaces concerned and calculate by optimization methods an approximation of the solution by the use of Galerkin approximations [M. Chipot, Variational inequalities and flow in porous media. New York, NY: Springer (1984; Zbl 0544.76095); Elements of nonlinear analysis. Basel: Birkhäuser (2000; Zbl 0964.35002); W. Rudin, Real and complex analysis. 2nd ed. McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Comp. XII, 452 p. (1974; Zbl 0278.26001)]. This problem arises in engineering science, thermal physics or dynamics of flows in porous media [Zbl 0544.76095; R. A. Silverman, Complex analysis with applications. Englewood Cliffs, N. J.: Prentice-Hall, Inc. X, 274 p. (1974: Zbl 0348.30001)].
MSC:
35-03 History of partial differential equations
01A60 History of mathematics in the 20th century
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
30C35 General theory of conformal mappings
35A35 Theoretical approximation in context of PDEs
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