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A harmonic function is considered in a three-dimensional bounded domain. Its normal derivative is given on nearly the entire boundary of the domain, while the value of the harmonic function is specified on the remaining small portion. The method of matched asymptotic expansions is used to construct a complete uniform asymptotic expansion of the function in powers of a small parameter characterizing...
The following two-dimension problem is considered: equation Δu = f(x) in some domain Ω ∈ ℝ2 with piecewise smooth boundary. The boundary condition is following: the derivative on a normal is equal zero everywhere, except a small segment γ, where function u(x) is given. The length of the segment equal to a small parameter ε. There is a problem to find the asymptotics of the solution u(x, ε) as ε →...
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