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The subject of the paper is the construction of a solution of the differential equation delta u (x) - c(x)u(x) = f(x, u(x)), in the spherical domain D = {x = (xt, x2, x,): | x | < R}, satisfying the Dirichlet boundary-value condition u(x) = h(x) for x e B(D). Let (x, y) -ť G(x, y) denote the Green function to the Laplace equation deltaG(x, y) = 0 in the sphere D and to the homogeneous boundary-value condition G(x, y) = 0 for x belong B(D), y belong D. Applying the change of the unknown function x -ť u(x) = u (x) + w(x), where u is a solution to the equation delta u (x, y) = 0 0, x belong D, with the boundary-value condition u(x) = h(x), x belong B(D), and w is a a new unknown function, we obtain the equation deltaw(x) = u + w(x)) + f(x,u(x) + w(x)) or the equation deltaw(x) = F(1)(x,w(x) = c(x)(u(x) + F(x,w(x)) with F(x, w(x)) = c(x)~w(x) + f(x, u(x) + w(x)), where x e D. Inverting the last problem by the Green function G, we obtain the integral equation w(x) = f, (x) + +fffF(y(w(y))G(x,y)dy, xbelongD, with f, (x) = fff c(y)u (y)G(x, y)dy, x belong D, and the homogeneous boundary-value condition w(x) = 0 for x e B(D). Solving by the Banach fixed point method the last equation we obtain w and u = u + w.
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Tom
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21-26
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Bibliogr. 1 poz.
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- Institute of Mathematics, Cracow University of Technology, Warszawska 24, Kraków
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Bibliografia
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bwmeta1.element.baztech-article-BPP1-0028-0077