Fabrikant mixed BC

Applications of Mathematics Valery I. Fabrikant Exact solutions to some external mixed problems in potential theory Applications of Mathematics , Vol. 31 (1986), No. 3, 224–246 Persistent URL: http://dml.cz/dmlcz/104200 Terms of use: © Institute of Mathematics AS CR, 1986 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use . This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 31(1986) APLIKACE MATEMATIKY No. 3,224—246 EXACT SOLUTIONS TO SOME EXTERNAL MIXED PROBLEMS IN POTENTIAL THEORY VALERY I. FABRIKANT (Received December 7, 1984) Summary. A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved: i) An arbitrary potential is specified at the boundary outside a circle, and its normal de­ rivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and the potential is zero inside. Several illustrative examples are considered. Certain methods of appli­ cation of the proposed technique to the solution of a few complex problems are also discussed. INTRODUCTION AND PRELIMINARIES Various applications of the potential theory in electrostatics, fluid flow, heat transfes, linear elasticity, etc. are well known [1]. A majority of these solutions dealing with classical mixed problems is "constructed" rather than derived. The derivation, if it exists, is very complicated while the final result is simple and is often expressed in terms of elementary functions. It seems logical that since the solution is simple, there should exist an elementary and straightforward procedure for obtaining it. On the basis of this logic, this investigation presents a new method for obtaining such results. Some preliminary considerations are necessary to understand the approach proposed. Introduce the following function 1 T(2 co (1) X{K, *) = — \ * = £ K"' e** • I + K Z — 2K COS 1/7 n = - oo Define the integral L-operator (2) L(K)f(<t>) = 1 f 2 V- 4 ~ *)IW # = 27t n 00 1 £ £MeiB* — = - oo 2TE >2л e-^/^d^ £ К |n|Le xnф 224 Here fn is the Fourier coefficient of the function f. The following properties of the L-operators are obvious from (2): (3) L{K)L(K l) = L{KK 1), L{l)f = f. The properties (3) allow a construction of the operator inverse to L as (4) L-\K) = L(K'i). The following two integrals will be widely used in this paper: (5) (6) where x2 Ч-*ß)àx \pq i i łnя -iw) tan" pq V(P 2 - * 2) V(« 2 - * 2) R„ R *•&>»)** , n V* / 1 _i y2(^) _ _ ____ = — tan l Í ^-J- V(x2 - p 2) V(* 2 - q 2) * Pq R Pq (7) R 2pq = p 2 + q 2 - 2pq cos p . Both integrals (5) and (6) are easily verifiable by the substitution (8) yi(x) = J(p 2-x2)J(q2-x2)lx, y2(x) = J(x 2-p2)J(x2-q2)lx, respectively. Two non-axisymmetric external mixed problems of the potential theory for a homogeneous half-space with a circular line of division of boundary conditions are solved in the next section. The problems are called external because nonzero boundary conditions are prescribed outside a circle while zero conditions are given inside. Various illustrative examples are considered in the third section. All the solutions obtained are exact and expressed in terms of elementary functions. The last section is devoted to the dicsussion of the results and possibilities of further applica* tions of the technique introduced. SOLUTION OF EXTERNAL MIXED PROBLEMS Problem 1. Consider a homogeneous half-space z ^ 0. It is necessary to find a harmonic function W subject to the boundary conditions at z = 0, namely dW (9) — = 0 for Q < a , 0 = (j> < 2n , dz W = W(Q, (j>) for Q ^ a , 0 ^ 0 < 2K . 225 Here a set of cylindrical coordinates Q, <j>, z is used. It is well known that the potential W can be represented as w(в, Ф, Z) = Г f °Ш r ér # + Г Г fí-üö r dr # Jo Jo -R Jo Ja R (10) where 1 dW (11) (7 = - , z = 0 and R 2 = O 2 + r 2 - 2Or cos (0 - ij/) + z 2 . 2TI dz Introducing the quantities (12) /,.( r) = i[V(( r + e)2 + z2) + (-l)y(( r-^ + z2)] f 0r 1 = 1,2, one can verify that (13) h(r) l 2(r) = Or , / 2(r) + / 2(r) = O 2 + r 2 + z 2 and (14) lim lj(r) = min (r, O) , lim l 2(r) = max (O, r) . z-,0 z->0 Making use of (5) and (6), and substituting p by /_, q by l 2 and ft by 0 - i/>, the following integral representations become possible: (15) (16) 1 _ 2 R Я 'ПW A ( — , (/> — t^ } đx ^ßГ 0 vww-^vта-* 2)' ___Ґ" U2 / * *Jь WV(*2-l2(t-))V(* 2-^ M) Substituting (15) in the first term of (10), and (16) in the second, one obtains after changing the order of integration and making necessary transformations due to (13) rh(a) (17) W( Гhì Q, ф, z) = 4 Jo dx + 4 Г00 dx_ J,2V(*2-Vte2 - *2) rg(x) r ár Q2). «X) s/(r 2 - d 2(x)) \gr r ár a(r, ф) + „ VИ*)-' 2) b(^a(r,ф). Hereafter, for the sake of simplicity l x is understood as l t(a) and l 2 as /2(a); the operator L is understood according to its definition (2) as L(k) a(r, i» = — f \k, 4>-ý) a(r, _ ) 2nJ0 dф and 226 (18) «x)шiJ(l + 7T*)-One can notice that the function g(x) is inverse to l t for x < Q, and is inverse to l2 for x 2 > Q 2 + z 2. Now the substitution of the boundary conditions (9) in (17) leads to the integral equation Equations of the type (19) were treated in [2]. Here, a different type of solution is derived. Let the operator -Г Q dO dtJ.V^ 2-'2) \Q. be applied to both sides of (19). The use of properties of the Abel-type operators and properties of the L-operators (3) results in 2тc r ár L(-) <r(r, 4>) = - - r2) \t 2 { V) dř LV0 The next operator to apply is _d dyj <*> ф)Mfi.Ф). tăt . ЛУ 2 - ň , V(e 2 - t2) Ve. and the result is -n2yL(y)a(y, ф) = Г tát dv L(ň- aҖy2-t2) " >dt Q áQ , V(e 2 - t2) Ve V(Q, ф) . Finally, using (4), the solution takes the form (»)+,«=44'): d[' '*' n2y \y/dyj a ЭДІ.Г Q ÚQ W(Q, ф) . ,V(/-t 2) 'dt} t V(e 2-t2) \QJ Differentiation under the integral sign gives another form of the solution, namely (21) where (22) <Hľ, Ф) 1 f x(a, У, Ф) f X(a, У, Ф) Г ÌЛЃ-a 1) ] a ár n2 (V(/ - a 2) J a V(ľ 2 - r 2) дr X(r, У, Ф) x(r, y, ф) = г dO V(e 2 - r 2) dg [ H-The following transformation can now be performed: (23) дr l(r, У, Ф) = »-i{ dв , V(e2 - ň (Lw)'\ = 227 Г dg )r vV - ň QàQ [(Lw)' + Q(LW)" - 2(Ľ.w)'] = -I^K-H-H') Here the primes (') indicate partial derivatives with respect to O, L stands for I\**2/yO), w = W(Q, <£), and the following identity is used: д I JL_ дr 2*- r Q Õ УQ As 1 õ2w 'Є~2"ôф2 1 ð2L w , _2#2 an addition to and subtraction from (23) transforms it into QdQ r vV - r2) [LAw - (AL)w], where A is the Laplace operator in the polar coordinates. Since X is harmonic AL = 0, and (23) finally simplifies to (24) — x(r, y, Ф) = cr Q dQ Substitution of (24) in (21) gives (25) 1 f xfa, y, $ a(y, ф) = ЛУ2 ~ a 2 r^Q2 dr r2) LЛw Q dO J(y2-r2))rJ(Q 2-r2) \ ye_ Лw(O, ф). Expression (25) presents a new type of solution for the integral equation (19). It may be noticed that the first term in (25) becomes singular, and the second term tends to zero when y -> a. In the case of w being a harmonic function, the second term in (25) vanishes, and the solution is represented by the first term only. Further, integration with respect to r becomes possible in (25) after changing the order of integration and using (5). The result is (26) ~f» ^ - _ 1 \j^iM) <*«--;?{; + 2яJ0 . W(y2 - a 2) Aw(Q,\l/)gdQd\l/ t__! V(g2 tan a2)V(j; 2-a2) s/(Q 2 + Г - ЬQ CОS (ф - ф)) 7(e 2 + У 2 ~ 2УQ cos (Ф -*))! Solutions in the forms (20) or (25) are appropriate to use when exact evaluation of integrals is possible, while the solution in the form (26) has certain advantages when numerical integration is to be employed. 228 It is of interest to express the potential W in the half-space directly through its boundary value w. As a inside the circle is zero, expression (17) takes the form •_(*) ydy f00 dx C 9 (27) W(Q, d>, z) - 4J^ V(x2 _ Q2) 1 j{g2{x) _ y2) -^2 Substitution of (20) in (27) and integration with respect to y yields (28) Ae4\o(y,4>). W(g,ф,z) ăx l2Љ2 - ñ Q g 2(x) õg(x) -тÂ-Hr,Ф). 9{x) V(r2-ö2(x)) V Here the properties (3) of the L-operators were used along with the following identity valid for the Abel-type operators: ày I mtét _« Лŕ)! (29) ) aJ(92-y2)dyLJ(y 2-t*) 2" A change of the order of integration in (28) and integration with respect to x give (30) W(Q, <f>, z) = — f * [X — ["- + tan" x -1 w(r, ty rdrdip . n2Jo L #3U Rj Here R is defined by (ll), and S, can be presented in several equivalent forms, namely, (31) { _ V(''2 - °2)M ~ « 2) ... V(r2 ~ « 2) V(g 2 ~ If) _ a l t = Vffl'-)-/QV(/ 22(r)-/D = zj(r 2-a2) h V(« 2 - ft ' each form being useful in different specific transformations. It must be noted that throughout this paper, l^ and l2 are understood as l t(a) and l 2(a) according to the general definition (12). Details of the derivation of (30) are given in Appendix A. In the particular case of z = 0, expression (30) simplifies to (32) = W^ 'Jo lV('- 2-«2)[ [W(Q, ф) W(Q, ф, 0) = w(r,ф) r dr dф for Q ^ a , [r2 + Q2 - 2rQ cos (0 - x//)] for Q ^ a . Expression (32) corresponds to the result previously reported in [2]. The solution can now be interpreted. The charge density o is given by the two equivalent expressions (20) and (26); the potential is given by (28) and (30), the former being more convenient for exact evaluation of the integrals while the latter is better suited for numerical integration. 229 Problem 2. Consider the following external mixed problem: to find a harmonic function W, satisfying the boundary conditions at z = 0 W = 0 for Qйa, 0 йф <2к, дW (33) = ~2KG(Q, cj)) for Q > a , 0 = <j> < 2K . Oz Substitution of the boundary conditions (33) in (17) leads to the integral equation (34) JoV(^ 2)J,V(^ 2)Lt)ff(^) = JflV(*2-e2)J«V(* 2-'-2) vW i ; One should notice that cr on the right side of (34) is known from (33), while the value of a on the left side of (34) is yet unknown. Using (15) and changing the order of integration, the right side of (34) can be transformed so that Eq. (34) takes the form r d* r rdr -L^VM)= JoVte 2-*2^^2-*2) w K } Vte2-*2^^2-*2 dx f °° r dг JoV(e 2-*2)J« V(f 2-*2) with the immediate result r dr b(~j<r,ф). (35) Лr2 i(')*«=-fA l(')*" The application of the operator ml x áx es/(> 2) to both sides of (35) gives, after necessary transformations, 2 7i ^(a 2 - Q 2)J (36) <J(Q,Ф) = -~^L(^)<т(r,ф)rdr for g<a, Г«Ч> (37) or, interpreting the L-operator, one obtains m2n r°° V(r2 - a 2) a(r, ifi r dr di/> r2 + Q 2 — 2rQ cos ((/> — i/f) Now the value of a is known all over the plane z = 0, and (17) can be used for expressing the potential W directly through the given value of a. Substitution of (36) in the first term of (17) gives, after integration with respect to r, •^--iW-rtľ. 230 (38) W(Q, ф, Z + ) = 4 JoV(e 2-*2)j 4 r - dx [g(x) ii>s/(x 2-Q2)L : ydy aj(y2-92(x)) \QУJ r dr . ( QГ -)o(y,ф) + o(r, ф). V(« 2(x)-r 2) V^ 2 The second term in (38) is equivalent to the second term in (10) which, in turn, can be represented, using (15), as rh(r) J. rd'í dx o(r, ф) . V(e2-*2)VO2-02M) W Now the following scheme of change of the order of integration can be used: /•GO fh(r) fh(a) /*oo fh(<x>) (*<x> dr dx = \ dx\ dr + \ dx\ dr, J a Jo JO Jrt J h(a) J g(x) and the second term in (38) can be rewritten as (39) + 4 h dx "»GC oV(e 2-*2)J a Ii(oo) dx r dr V('2 - 92(x)) TL(~V(r' x)) \Qr) p r )«>W-dr Ф) + o(r, ф) . I„ V^-^J.wV^-^x)) \ Qr Substitution of (39) in (38) gives, by virtue of /t(oo) = Q, (40) W(Q, <f>, z) = 4 r r — L(—\ a(r, </>). 1 ' ' J „ V(^2 - x2) J *-> V(^2 - 92(x)) W A change of the order of integration in (40) and integration with respect to x, using (5), results in (41) W( Q,cj),z) = 2[ " ^Jo J, where R is defined by (11) and £ is any one of the expressions in (31). The solution can now be explained. Expression (37) defines the charge density a inside a circle directly through its values outside, the potential W is given by the two equivalent expressions (40) and (41), the first one being recommended for an exact evaluation of the integrals involved while the second has certain advantages for numerical integration. ^tan-^rdrd* R R ILLUSTRATIVE EXAMPLES Several particular cases of general solutions, obtained in the previous section, are considered here in order to illustrate the effectiveness of the proposed technique. 231 Єз = Z(V(Є 2 + 2 2) + Z) , e.= z(V(e 2 + , - z) e For the case of an odd n = 2k + 1 the integration can be performed by using the substitution t2 — x 2 — O 2 — z 2, and the final result is ,47) w{e,*,2),^níi±m\í cш + k+ \ / i \m p)m - 1 + y D -5——-- ,„ . (m - 1)! .n m-1LV'/ V-x Г(n/2) [,„ . (2m - l)(a 2 - /2)m"1/2 for n = e 2 + z 2 1 -IV. - cot ^ ^-2-'2) where ď1-"' Г (t + z 2)* (k - m)! dř fe~m L(ř + O 2 + Z 2) г__±___l for ř_0, Цt + Є 2 + 2 2)*+1J (48) Ð m = __^Г_____П i+l-mL ř* J (k + 1 - m)! d. Substitution of (42) in (21) yields, after integration [3], (49) a( Q, *) = F^n + M -o { ! v ; { ' i. r(n/2) V'V(e 2-«2) _nW- fl 2) /n + 1 3 a 2 e"+2 V 2 22 and the Gauss hypergeometric function can be expressed through elementary functions [4], namely, for even n = 2fc, fc = 1, 2, 3,... , , 1 3 \ 1 ď Fl + k, - ; - ; ř ) = - 2 2/ 2k! dtfc ,*-l/2 l„ 1 + Vt ІП i-Vü and for odd n = 2k + 1, k = 0, 1, 2, ... , (50) V* 3 1 3 \ F k + -,-;-; n =- 2 2 2/ 2F(3/2 + k) V d* г . 1/2 ř d<* LVO - ř The solution can now be presented for several specific values of n: n = 1 (51) (52) W(Q, ф, z) = - я V(Є 2 + z 2) <г(e, </>) = -w°ri„-iV(в__ /, *V V (Í?2 - . ' 233 n = 2 (53) w( e, <p, z) = At the plane z = O, W(Q, <j>, 0) = ?['-1 - 00 + vV + *2) Ч i - v^2 ~ Q2У (54) и = 3 (j(O, ø) for O < a for _• __ O , i, e + vv - fl2. - -ln (55) Wfo ф, z) = 2тт_> 2 (V(Í2 2 ~ ö2) Q 4w0 Чe2 + ^2)2 U/V - /ï) /;- vva2 -1\) 2a2 2z2 . _lч/ _2 + At the plane z = 0, W(Q, tf>, 0) = W(0, 0,0) = 4w0/3rrO 3, 2 V(^ 2 + ^2) / 2 í2w0f . _! 0 _? // 2 —I sin ~V(O 2 -\TZQ Ó L O O for Q __ Я , for g __ O , 2w0 2O 2 — (56) *(<?>.) = -+; 71 Q a ч/(_? — я ) п = 4 (57) -„„.)-__^j£z^,l- 0o)-i<l- ЄЛ + + E 2(g2 + z2) _a 2 V At the plane z = 0, 2../П 2 - /v 2 (/2 - a 2 z(2z 2 - Зg 2) 2'Є2 + z2)3'2 lnQU W\_. </>, 0) = _ГІ___ІZІ_;_>!_J_П for e_й, >4 L - 2a 3 J for Q _ a , řҒ(0, 0, 0) = Зw0/8a4, 234 (58) G{Q, ф) = ЗWn ЗO2 - O 2 3 O + V(^ 2 ~ a 2) 8тrO 4 [a2 X /(O2 - O 2) O The total charge at the plane z = 0 is equal to \v 0 for n = 1, and it is zero for n = 2. Figure 1 shows the dimensionless charge density GO ,, + 1/w0 versus Qja for n = = V 2, 3 4 (formulae 52, 54, 56, 58). The charge density is nonnegative for n = 1, and changes sign when n = 2, its negative maximum increases with n while the total charge stays at zero. DENSITY 0.25 0.00 0.20 0. 15 0. 10 - 0.05 0.00 ц\\n=i \ \ >v? -0.05 1 .00 2.00 З.øø 4.øø 5.øø 6 . 00 7 . 00 RADIUS Fig. 1. Charge density for n = 1, 2, 3, 4 Equipotential lines for the case n = 2 (formula 53) are presented in Fig. 2. The range of values of the dimensionless potential WQanjw is taken as 005 to 0-6. Because of symmetry of the problem, only a quarter of each equipotential line is presented. Example 2. Consider the boundary conditions (59) W -5 Qinф for o = a , 235 ÕW õz = 0 for Q < a where w„ is a constant. The solution is given by (28) and (20). Substitution of (59) in (28) gives after first integration [3] W(Q, ф, Z) = 2Г(И + l/2) „ dx Лг' V(*2 - ñ Z-C00RDINATE 7.00 ø.øø 1 .00 2.øø З.øø 4.00 5 .08 RADIUS Fig. 2. Equipotential lines for n = 2 The second integration yields (60) W(Q 9 4>9Z) = — ^ txn* Y i" 1)"1"1^ + V 2) / 1 _ Q2«-1) V ; V ^ V 71 e" »-i r(m) r(n + l- m) (2m - 1) V ^° ' and Q 0 is defined by (46). Substituting (59) in (20), one gets, after integration [3], (бi) O(Q, ф) inф Г(n + 1 /2) w„e *3,2Г(n) eV(/-« 2) Evidently, it can be noticed that (61) can be obtained by differentiation of (60) with respect to z for z = 0, instead of integration of (20). 236 Here are some explicit expressions for several particular values of n. W'Q,Ф,Z) = -i Q iф Q M ~ ťï n = 2 3 w? (62) W {Q,ф,z) = ~^є 2 2O2 n = 3 PV(>, <_ , z) -15 w 4 O 3 з _з Ж - g2)] _ 1 [. _ /V(/2 - g2)^ IH^-ин^ľh Equipotential lines at the plane 0 = 0 for n = 2 due to (62) are presented in Fig. 3. The curves correspond to a set of values of the dimensionless potential (Wa 2jw2) in the range of 0-05 to 0-4. Z-COORDINATE I .50 І .25 1.00 0.75 0.50 0.25 ø.øø / \w Wa 2/w? = .05 \J07 \ .1 ^чX \.15 \ \ \ / л\ \\ \ \ 0.00 1 .00 2.øø З.øø 4.øø 5.øø RADIUS Fig. 3. Equipotential lines for n = 2 237 Example 3. Consider several cases related to problem 2, with the boundary con­ ditions (63) w= 0 for O < a , дW _ cz 2к°° Q" for O > a . The solution is given by (40) and (36). Substitution of (63) in (40) yields, after first integration, using (44), r[(n - l)/2] ( Q dx (64) W O, ф, z) = 2 yjҡ a 0 Г(n/2) J ,,Vu? 2 - x 2) ű-Ҷx) where #(x) is defined by (18). The technique, used in the previous examples, can be employed here for further integration. The final result depends on the value of n being even or odd. For n = 2k, k = V 2, 3, ... , the potential is (65) Wg, ф, z) = 2 > *c Г[í ": 1)/2] í 2ß, ln ô гv"/2) l íc - 1 +1 m = l ^2w — 1) Z 2 A- 1 •-+-FfT1 + "m + 1 + I -~t [Є7 - eт - (Є7 - e:)] •• i mz Here 0, 0„ £> 2, Q3, QA, are defined by (46), and dm-l p („ _z2)*"l (66) (m - ť)!dn m"1 + z2 - цf "k-m+X ~ (ř2 - z 2) (m - 1)! dt m_1 For /r = 2k + 1, the result is ЃЛJ '• - z 2)"- 1 - V + z2)-t)fc. for n = 0 , for ř = -V(ő 2 + z 2 (67) W(O, 0, z) __ 2 Vn Y^{n " 1)/2] - -Í2- V J—^ ^ " fl r(n/2) z"" 1 m = i (2m - 1 \ O Г(m) õif 1'1 Vч tan" -a2) 7 for ГҪ = e2 + z 2 Here, (68Y fc — m + i — Д«) dŕ 1 Г_+ ___"] 1L (ч + t)* J for ř = 0 , n = Є2 + z 2 я^+1 = J_i___Г____ГП f0 г(m)dí m-1 L t k r ř = - 238 The solutions for several specific values of n can now be given. n = 2 (69) WÍQ,4>,z)=-r*p—\TiQ vV + * ) 2JI õz I z = 0 and Q is defined by (46). The charge density can be found by differentiation (70) ' dW> and as (71) we have (72) a — ln 0 = Reí- - + ^Z ( Q O X/(O2 — Q 2 for z = 0 . <<O, ф) = Ц Re í 1 - . đ } , O v0, 0) = - -pL O2 ( V(я ~ e )i 2a and Re means that only the real part of the expression inside the brackets is taken. n = 3 (73) W'Q,Ф,Z) = 4<гn Q2 + Z 2 V<I2 - « 2) У(e2 + ; 2) Differentiating (73) and using (70) (74) 2f7n OQ, <f>) = J - sin — O O yj(a 2 — Q 2) <__o . -i Vig!+i! ) for Q < a , for O > ű and c(0, 0) = — 2G0/37U/ 3. The following formula was used in the derivation of (74): dz n = 4 (75) tľ (76) ?z ( vy2 - Є 2)J foг z = 0 W,Q, ф, z) = KGr, Q2 - 2z 2 2(в2 + z 2)2 У Q 2 + z 2' lnЄ + + 2 l^a2-l2)-Зz+ Ѓ\^l22-a2)i. At the plane z = 0 we have ЪOQ flti g + Vtø 2 ~ Д 2) __ O yV - fl2> 2O3 239 and W O, (j>9 o) = 0 for O ___\ a. Further, Җa2 + W(0, 0, z) = —° K ' z 3 12a 2 a The charge density a can be found from (76) using (70), (71) and (75): (77) O(Q, *) = a-l Re (l - -^-f^l, <0, 0) = - ^ . Q [ 2a \'\ a - Q )) % a Example 4. Consider the boundary conditions at the plane z = 0 (78) W = 0 for Q S a , — = - In 5-5 e1"* for <> > O . dz O" Substitution of (78) in (40) gives after in.egration (79) Wye, <j>,z) = 2 > L^M ^ e-* L/(/ 2 - a 2) - z + E(n) g" I + z"y „_!____) [1 _ (i _ ,2/ a2y.-i/2-il » = i r(m + l)r(n - m)(2m - 1) L V " ; J J and at the plane z = 0, W(Q, (j>, o) = 2 VTT r( " ~ ^2) £2 e in* Re V(<? 2 - n 2) It") <?" with the charge density ff(g^) ________ _2e'"* Re II • (-i) mrfn (80) -I F(m 4- 1) F(/t — m) (2m — 1) The results for several particular values of n are n = 1 V(«2 - e2) [1 _(1 _^ a2)m-l/2JI (81) n = 2 (82) 1T( e,^z)=2 Ir^e i*[V(/ 2-fl2)-z], <Л_S íìe'*ReГl « -1; Q L V(« 2 - e2)J w\e, </>, z) Ä2І*Г V'(/ 2-a2)-2z+lV(« 2-'2) a 240 1 ^ C? 2 L 2aV(a 2-Є2)J и = 3 (83) W(Q, Ф,*) = ђ Ц e зi* L. - a 2) - 8 z + 2 Z V(« 2 - П) ~ 4 Q Л l Ъ a z(a2 - l]fl 2} _ Зa3 J ^ 3 °Ä eЗІ* Re l 8 - 2 >2 ~ ^) + (*' ~ g 2)372 (84) O-(O,.) 8 O3 (3 ^/(/r - O 2) a 3a 3 A more general case of boundary conditions, namely, (85) W = 0 for O _ O , T1IF rT _!_ _ _27C^e im^ for O _ a dz O n can also be considered using the same technique as in the previous examples, and the final result can always be expressed in elementary functions. The form of the result will be different for (m + n) even and for (m + n) odd. As an example, the following expression can be obtained by substituting (85) in (36), namely for m + n = = 2k, (86) a( Q, _ = _- e- Re (l _— fl - V ^-^ («f "ll, 1 } [ } Q n I V(« 2 - e 2) L >=2 2 > r(j) W Jj and for m + n = 2k + 1 and Q < a, (87) ,,«.) = 2_» ei».•;_-!£—g—fi___-_-iYgy j"2ii. *.<?" 1 « V(« 2-e2)L ,=2 4T(j + 1/2)W Jj Expressions (86) and (87) represent general formulae which cover all the particular cases considered in Example 3 and Example 4. The form of the solution may some­ times be different, for example, for m = n — 3 formula (S6) gives (88) . , _ - 2S e 3i* Re (l - - / • + ___ + <?3 I V(A 2 - e 2) 2a V(a 2 - e 2) 8a 3 V(« 2 - Q2). which looks different from (84), but as one can easily verify expressions (84) and (88) are equivalent. Using (36) one can obtain the following expression for the total charge I directly through the given charge density a: 2 C2n f °° n (89) I = - a(r, <f>) cos" 1 - r dr d(j> . *Jo J. r 241 In the particular case of a = a 0r n, (89) results in (90) s _ 2<T0 VIT r(n - l)/2] (n - 2) E(n/2) a""" 2 The total charge for n = 3 is 4a 0ja, and for n = 4, .27 = na 0\2a2. Figure 4 presents the charge density versus radius defined by formulae (72), (74) and (77). The equipotential lines, defined by expression (73), are presented in Fig. 5. DENSITY 1 .00 - ø.5ø- 0 00 n = 2 ^^ 4 •0.50 -—-_^n = 2 1.00 - V 1.50 _ 4 2.00 _ 0.00 1 .00 2.00 З.øø 4.00 5.00 6.00 7.00 RÁDIUS Fig. 4. Charge density for n = 2, 3, 4, The dimensionless potential w* = Wa 2\a0 was varied from 0-5 to 1-3. One can notice that the equipotential lines for w* <. 92 have two branches. The corresponding equipotential surfaces can be seen in Fig. 6. DISCUSSION No new attention has been paid in literature to the basics of the potential theory after classical contributions in this field by Green, Lord Kelvin, Hobson and others. This paper presents a novel approach to the problem and the advantages of this new approach become evident when compared with the previous classical results. 242 Z-COORDINATE 7.00 6.00 5.00 J 4.00 J З.øø J w*=.5 0.00 RADIUS Fig. 5. Equipotential lines for n Fig. 6. Equipotential surfaces for n === 3 243 A formula similar to (30) was first published without proof by Lord Kelvin in 1847, and 22 years later he proved it in his "Papers on Electrostatics and Magnetism" using the method of images which requires a lot of ingenuity in geometric considera­ tions. Later the same formula was derived by Hobson [5] who used Sommerfeld's method of a potential in a Riemannian space. Later several solutions were also published using various integral transforms. Unfortunately, practical applications of these results are rather difficult due to the complexity of the integrals involved. The main advantage of the new approach is its simplicity where the generation of the solution is reduced to a straightforward procedure, and the integrals involved tre elementary as demonstrated in the previous section. | The technique of this paper can be used for the solution of many different problems, ind, among various immediate applications, one can indicate the punch and crack problems in linear elasticity. For example, expression (25) is proportional to the normal stress distribution for the external punch problem when the normal displace­ ments w are given outside a circle, and the normal stresses are zero inside; expression (37) can give the normal stress distribution in the crack neck through its values on the crack faces; the total force at infinity in external crack problems can be defined by (89), etc. :• This new approach can be generalized for more complicated boundary conditions. It can be used not only for a disk, but also for a spherical bowl, and possibly for an arbitrary surface of revolution. The investigation reported in this paper was supported by grants A 7104 and A 0791 from the Natural Science and Engineering Research Council of Canada. References [1] I. N. Sneddon: Mixed boundary value problems in potential theory. North-Holland Publishing Company, Amsterdam, 1966. [2] T. S. Sankar, V. /. Fabrikant: Investigations of a two-dimensional integral equation in the theory of elasticity and electrostatics. Journal de Mecanique Theorique et Appliquee, Vol. 2, No. 2, 1983, pp. 285-299. [3] I. S. Gradshtein, I. M. Ryzhik: Table of Integrals, Series and Products. AP, New York, 1965. [4] H. Bateman, A. Erdelyi: Higher transcendental functions, Vol. 1, McGraw-Hill, 1953. [5] E. W. Hobson: On Green's function for a circular disk, with application to electrostatic problems. Transactions of Cambridge Philosophical Society, Vol. 18, 1900, pp. 277—291. APPENDIX A Here some details of the derivation are presented which allow transformation of (28) into (30). Introduction of a new variable u = g(x) changes (28) into /A. IT// , ^ 2 f 00 dl 2(u) T(l](u)\ d f 00 rdr /1\ , . Al) W(Q,^Z)=--\ - lhl( T 2\ L[ )T\ ITI K L~W r'*)' * J« VO2OO - Q) \ Q J duL V( r " u) VJ 244 Change of the order of integration in (Al) yields (A2) W(Q,ф,z) LÍ-) w(r, 0) dr — \rl ár u ál 2(u) Here the following general formula was used: rf(r) dr (AЗ) F(u) du — du .v(r2--a)va.(-)-в a) d Г u F(u) du m I. V(^ 2 - «2) Differentiation under the integral sign in (A2) gives r°° d C r /(r)drf drja V(r2 - и 2) (A4) lҒ(в -ţ /*2я ŕa >Ф>*)-M l Я Jo Ja ľ(a) X QГ , ф - ф Ш-a 2)Ш")-Q 2) + + áu ^J(r2 — u 2) áu l\(u) Г2{a)Xp-2, *-* V('K«) - Q 2) w(r, ф) r àr åф Formula (2) was used here along with the following rule of differentiation under the integral sign: (A5) _d_ dr «j(«)dи f(a) r + r r d/(и) LV(- 2-"2) V(^ a-«a) ' J.VO" 2-"2) Introducing the notation (A6) F(u) = — [— + t 7,3 U(«) z R3 R ,(tt) _V(^2-»2)V(l 2И-« 2). expression (A4) can be rewritten as (A7) W(Q, ф, Z) = 0 J r2 - a 2 áF(a) a áa + + í áu d Г(r 2 - u 2)3/2 dF(u) л/(r2 — u 2) du L w du \iw(r, i/t) r ár áij/ . Now integration with respect to u in (A7) can be performed elementarily by parts, and the result is (A8) ґ»2я /»oo W(Q, ф,z)ш — F(a) w(r, ф) r dr dф Taking into consideration (A6), one can see that expression (A8) is equivalent to (30). 245 Souhrn EXAKTNÍ ŘEŠENÍ NĚKTERÝCH VNĚJŠÍCH SMÍŠENÝCH PROBLÉMŮ V TEORII POTENCIÁLU VALERY I. FABRIKANT Je navržena nová metoda řešení smíšeného problému potenciálu v poloprostoru s okrajovými podmínkami na částech hranice rozdělených kruhovým obloukem. Postup je založen na novém typu integrálních operátorů se speciálními vlastnostmi. Řeší se dva obecné vnější problémy: 1) libovolný potenciál je specifikován na hranici vně kružnice, a jeho normálová derivace uvnitř je nulová; 2) vně kružnice je dána libovolná normálová derivace, uvnitř je potenciál roven nule. Je uvedeno několik ilustrativních příkladů a diskutují se některé metody aplikace navržených postupů na řešení několika složitých problémů. Резюме ТОЧНОЕ РЕШЕНИЕ НЕКОТОРЫХ ВНЕШНИХ СМЕШАННЫХ ЗАДАЧ ТЕОРИИ ПОТЕНЦИАЛА УАГЕйУ I. РАВЯ1КАЭТ Предлагается новый метод решения смешанных задач теории потенциала в полупро­ странстве с круговой линией раздела граничных условий. Метод основан на новом типе интегральных операторов со специальными свойствами. Рассмотрены несколько иллюстра­ тивных примеров и обсуждены возможности применения нового метода к решению сложных задач теории упругости. АиХког'з айАгезз; Ог. Уа1егу РаЪпкап1, Оерт.. оГ Меспатса1 Еп§теепп§, Сопсогша ^п^- уешгу, 1455 с!е Ма1$оппеш!е В1уа\ ХУезг, Мопггеа!, ОиеЪес НЗС 1М8, Сапаёа. 246