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A polygon is said to be simple if its nonadjacent sides do not have common interior or endpoints. A polygon is said to be convex if it lies on one side of any straight line passing through two neighboring vertices (Fig. 3.13c). In what follows, we consider only plane simple convex polygons. | 14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables 633 Example. Consider a boundary value problem for the Laplace equation d2w d2w in a strip 0 x l -to y to with mixed boundary conditions r dw r w fi y at x 0 f2 y at x l. dx This equation is a special case of equation 14.9.1.1 with a x 1 and b x c x x t 0. The corresponding Sturm-Liouville problem 14.9.1.5 - 14.9.1.7 is written as u Xx Xy 0 u 0 at x 0 ux 0 at x l. The eigenfunctions and eigenvalues are found as u x Sin n 2n - 1 x 1. X T - 1 2 n 1.2. l l2 Using formulas 14.9.1.3 and 14.9.1.4 and taking into account the identities p Q 1 and yn 2 l 2 n 1 2 . and the expression for T from the first row in Table 14.8 we obtain the Green s function in the form 1 1 . . G x y Ç n sin a x sin a Ç e n 1 n 1 n n 2n - 1 n V n l 14.9.2. Representation of Solutions to Boundary Value Problems via the Green s Functions 14.9.2-1. First boundary value problem. The solution of the first boundary value problem for equation 14.9.1.1 with the boundary conditions w fi y at x xi w f2 y at x X2 w fs x at y 0 w f4 x at y h is expressed in terms of the Green s function as d fh d - G x y e n vde dy - a x2 Xi f2 n o - G x y e n ide j X2 xi dy C X2 d f4 e dn xy en r d fh w x y a xi Jo C X2 n G x y e n dyde. r r d i r J fs G x y y o de -J de 14.9.2-2. Second boundary value problem. The solution of the second boundary value problem for equation 14.9.1.1 with boundary conditions dxw fi y at x xi dy w fs x at y 0 dx w f2 y at x x2 dy w f4 x at y h 634 Linear Partial Differential Equations is expressed in terms of the Green s function as ch ch w x y - a xi f1 y G x y xi y dy a x2 f2 y G x y x2 n dy Jo Jo iX i-x2 - f3 e G x y e 0 de f4 G x y h de Jxi Jxi X h n G x y e n dyde . Jxi JO 14.9.2-3. Third boundary value problem. The solution of the third boundary value problem for equation 14.9.1.1 in terms of the Green s function is represented in the same way as the solution of the second boundary value problem the Green s .
