Tham khảo tài liệu 'mixed boundary value problems episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 228 Mixed Boundary Value Problems Substituting for A k from Equation into Equation we find after interchanging the order of integration that Ớ r f . J f rf r d g C Q t de rf dr dt _Ja J Jt Vr2 -12 where mg p Í J k cos kt Yi k sin kt Q e t Vo J2 k Y2 k X Y1 k cos ke - Ji k sin k k or A ư ỉ . . dk f . . dk Q e t - ựo sin k e - t k - Ịữ sin k e t k 3 jr 1 d L Ự0 p b k J k J Our final task is to evaluate the contour integral H 2 z H 1 z ei Ị t z ị z where the contour r consists of the real axis from the origin to R an arc in the first quadrant z R 0 6 n 2 and imaginary axis from iR to the origin. As R TO we find that H 2 k H 1 k i i t k edk k p Ii k Jo Ki k Therefore Equation becomes -k t Ị dk k . t t Ii k k t p dk Ki k k rf r Vĩ2-2 dr 1 or a t Ỉ a kJ r e-k t V dk de 2 r rf r 9 Ja ưo Ki k e dk e nJt Vr -t2dr 2008 by Taylor Francis Group LLC Transform Methods 229 Figure The solution u r z to the mixed boundary value problem governed by Equation through Equation with a 2. for a t . Figure illustrates the solution when a 2. Example Let us solve44 2 1 du. TT G 1 0 0 r 0 z dr2 r dr z ơz dz subject to the boundary conditions lim 2 r z x lim u r z 0 r 0 r 0 z x lim u r z 0 0 r x and z u r 0 1 0 znUz r z z 0 0 1 r 1 r x where K 0. 44 Taken from Brutsaert W. 1967 Evaporation from a very small water surface at ground level Three-dimensional turbulent diffusion without convection. J. Geophys. Res. 72 5631 5639. 1967 American Geophysical Union. Reproduced modified by permission of American Geophysical Union. 2008 by Taylor Francis Group LLC 230 Mixed Boundary Value Problems Using transform methods or separation of variables the general solution to Equation Equation and Equation is u r z z 1-n 2 A k J-2LUz 1-n 2v Jo L1 - n Jo kr dk where V 1 n p n 2 . Substituting Equation into Equation we have that
