Internat. J. Math. & Math. Sci. VOL. ii NO. 4 (1988) 751-762 751 ON SOLUTION OF THE INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS OF TWO CIRCULAR-STRIPS C. SAMPATH and D.L. JAIN Department of Mathematics University of Delhi Delhi-f10007, India (Received May 28, 1986 and in revised form June 20, 1988) ABSTRACT. Solutions are given to some singular integral equations which arise in two- dimensional Dirichlet and Newmann boundary value problems of two equal infinite cocircular strips axial in various branches of potential theory. For illustration, these solutions are applied to solve some boundary value problems in electrostatics, hydrodynamics, and expressions for the physical quantities of interest are derived. KEY WORDS AND PHRASES. Inflnite co-axial circular strips, Chebychev polynomials, surface charge density, kinetic energy, elliptic integrals of the first and the second kinds. 1980 AMS SUBJECT CLASSIFICATION CODE. I. 31A. INTRODUCTION. Recently many authors [I-5] have presented solutions of various two dimensional boundary value problems of two infinite strips, by integral equation techniques [6Shall [9] has given the solution of the Fredholm singular integral equation of 8]. the first kind with logarithmic kernal{q + where a and q are known constants. logmla si(0 01)I},- < i, < , This type of singular integral equation governs the solutions of various two-dimensional Dirichlet boundary value problems involving an infinite circular acoustic scattering. strip in electrostatics, hydrodynamics, and low-frequency The solution to this singular integral equation is derived in a closed form from those of some well known integral equations of Carlemann type [I0, We present here solutions of Fredholm singular integral equations of the first kind of the type C. SAMPATH AND D.L. JAIN 752 the where function f(6)) is known inequality 0 B I01 equations of the type a 7, and of a class C q I, for the values known constants. are of 0 satisfying The singular the integral (I.I) govern solutions of various two-dimensional Dirichlet boundary value problems of two equal infinite co-axial circular strips in potential In the corresponding Newmann boundary value problems, the governing integral theory. equations are of the type + l(01)cosec 2 (6) < 16)I < f(0), 01)d01 (1.2) ’ where the unkown density function satisfies the edge condtlons ( ) ( =) o. (.B) Integrating by parts and using edge conditions (1.3), (1.2) becomes - / +/ 6)l)dO f(6)), IB < 16)( < (1.4) c (.5) l’(O). g(O) where >=or (o g(o We present here a simple technique of solving integral equations of the types 1. I) (1.2) and For illustration, this technique is applied solve to an electrostatic Dirichlet boundary value problem and a hydrodynamic Newmann boundary value problem of two equal infinite co-axial circular strips. We have also solved the two-dimensional problems of scattering of a low-frequency incident plane acoustic wa. by two equal infinite co-axial soft and rigid circular strips by the integral equation technique. The plan of this paper is as follows. This work will appear separately. In section 2, we first present a simple technique of solving integral equations of the type (I.I) without reducing it to some well known integral equations of the Carleman type [10, II, 12]. This is achieved by reducing the solution of (I.I) to that of two Fredholm singular integral equations of the first kind with (ii) (cosy- cosx) -I (i) kernels 0 < x,y < 7. (Const.) + logl2(cosx- cosy)l, 0 < x,y < 7, and The unknown and known functions are both even degree functions in each of these two integral equations. The first of these two equations readily yields the Fourier expansion of the unknown even degree funtion over the interval 0 < y < , when the well known expansion of the the kernel [1,7] and the Fourier expansion of the known even degree function over the interval 0 made in it. < x < n are Similarly, we obtain the series expansion of the unknown even degree function of the second integral equation in terms of Chebychev polynomials T (cosy) of n the first kind when we use the series expansion of its known even degree function in terms of Chebychev polynomials U (cosx)of n the second kind. The solution of this second integral equation contains an unknown constant which is evaluated by making its solution to satisfy an appropriate inner edge condition. Then we illustrate this INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS 753 (I.I), when the known function f(0) technique to solve the integral equation in it is Lastly, we explain how equation of a particular form, for our subsequent analysis. (1.2) can be transformed to the form of integral equation (i.i) and is therefore solvable by the above technique. In Section 2 we apply the integral equation technique given in Section the two-dimensional electrostatic Dirichlet of problem value boundary to solve two equal infinite co-axial perfectly conducting circular strips in a free space, when the total Section 3 is devoted to the study charge per unit height of the two strips is unity. of the two-dimensional hydrodynamic Newmann boundary value problem of uniform flow of an inviscid homogeneous liquid streaming past two equal infinite co-axial fixed rigid circular The strips. expression for the kinetic energy per unit of height the secondary fluid flow is derived. 2. SOLUTIONS OF INTEGRAL EQUATIONS. We present here a simple technique to derive solutions of equations (i.I) and (1.2). To solve equation (I.I) we first substitute in it [II] INTEGRAL EQUATION (1.1): fl () + f2 (0); f(0) g2(O), (2.1) and 2 represent the even degree and the odd degree parts of the where the subscripts These respectively. functions corresponding gl (0) + g(O) readily substitutions decouple the integral equation (I.I) into the following two integral equations f g2( for the loglma 2 (cosO sin( 0 gl (){2q + E )log determination substitutions cos 01 cos 0 ( + sl - = of gl(0) {(cosE {(nOsE i) and cos0 l)l}d01 E d01 f2 (0) We g2(0). =f (O) < first 0 < E 0 < a < (2.2) (2.3) solve equation cos=)cosy + (cosE + cos=)}, (2.2). The (2.4) cos=)cosx + (nose + cos=)}, (2.5) reduce the equation (2.2) to a simple form G 0 where Gl(Y) R(0) l(y){ A + log [(cosE [cosE 21cosx- cosy I} dy cos)g I(8 l)sinh]/sin81 cos0)(cos0 cosa)] 1/2 F l(x), 0 < x < , [gl(Ol)R(Ol)l/sin01’ (2.6) (2.7) (2.8) C. SAMPATH AND D.L. JAIN 754 2q + A fl(0), Fl(x log[a2(cosB cosa)/2], (2.9) and in (2.7) we have used the relations sin cos cos0 (cosB cosa I/2 I) cos = cos0 readily derivable from the substitution (2.4). Lastly, the well known formula [7] 0 n n;1 (2. lO) cosa cosny cosnx --2 r. log 2 cosy- cos x I/2 cosa cosB < x,y and the Fourier expansion of the known even degree function =a Fl(X)where a + E z o F n l(cosnx readily yield the solution G l(y) b O + Z dx, n < x (2.11) Fl(x) (2.12) 7, (2.13) 0 of the equation (2.6). Gl(Y) This is given by (2.14) b cosny, n n=l bo ao/(2A), bn -(nan)/ where < 0 a cosnx, n n < and the known coefficients a n n substituting the above value of I, n (2.15) O, are defined by the relations (2.13). Finally, Gl(Y) in the relation (2.7), we obtain the required solution of the equation (2.2) gl(01 [b 0 + Z n=l b cosny] [sin0 n I]/R(o I) B Now we take up the solution of the equation (2.3). of equation (2.3) with respect to 0 and obtain a g2(0 < O < a (2.16) We first differentiate both sides )sln0 B cos0- -cos0 d01 f2 (0)’ B < 0 < (2.17) a, where the integral in the left hand member is to be inerpreted as a Cauchy principal value. When we make the substitutions (2.4) and (2.5) in the above equations, we get C2(Y) cosy where and F2(x) cosx G2(Y) g2(01) F2(x) f2’(0), dy slny F2(x), 0 < x < [2g2(Ol)R(01)]/[cosB is a known even degree function of x. (2.18) cosa], (2.19) (2.20) Next, when we substitute INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS x in (2.18), cos -I (X), y cos (Y), (2.21) we obtain G (cos 2 -I Y) dY F2(cos-Ix) I/2(Y-X) -I (I _y2) The -I 755 above integral be can equation expansion of the known function F < -I readily X < I. (2.22) solved when we substitute it in the in terms of the Chebychev polynomials U (X) of the 2 n second kind F2(cos IX) Z c U (X), n n n:l < -I X < I, (2.23) and use the well known formula T (Y)dY (2.24) y2)I/2(y -I (I n--O, 0 n X) n n-I (X), I, U where T (Y) is the nth degree Chebychev polynomial n solution of the equation (2.22) is given by G2(Y + ATo(cOsy I Cn-ITn(csy)’ n=l 0 of < the y < first , f Cn Lastly, sinx sln(n + l)x 0 the (2.25) where A is an arbitrary constant and the constant coefficients c expansion (2.23) of the known function F Thus kind. n are defined by the These values are 2. F2(x)dx, n (2.26) 0. (2.19) and (2.25) lead to the required solution of the equation relations (2.3) (cos- g2(01) r. cos)[A + c n < n-I cosny]/R(0 where relation (2.4) gives the value of cosy in terms of cos 0 0 < (2 27) The value of the I. constant A in (2.27) is readily obtained by using the edge condition satisfied by the density function g2(01) g2(O1 The value of G2 y O([cosB -cosO 1] g2(01) O, as y as 0 This edge condition is + 0 (2.28) 0+, and hence l n g2(01) 1/2), i=. given by (2.27) satisfies this edge condition if in (2.25), A---- Therefore, at the inner edge c (2 29) n-I 2- cos cosa) l n=l Cn_ I(I cosny)]/R(0 I), < 01 < a. (2.30) 756 C. SAMPATH AND D.L. JAIN Since, cs01’) Yl (cosBCSB _-cosa sin2 infinite g2(01) for relations (2.1), where all B’s various (2.16) and the above equation, of right member satisfies the edge required (2.30) the yield condition solution g(0 required + loglma I) above Finally, of < equation [0[ < B, siq(0 01)[}d01 [Bl0 + BllCOS0] + [B200 + B21sin0],(2.31) Its solution is also very useful are known constants. problems boundary value the therefore, (2.28). We illustrate the above results by solving the integral equatlon, g(01){ q + the in series expression (I.I). is a common factor of all the terms occuring in the or presented in our subsequent analysis. in solving Comparing equation (2.31) with (I.I), we have, in this case, fl(0) Blo + BllCOS0, f2 (0) Fi(x) [Bio + Bil(COS8 G1(Y I_.{ B200 + B21sin0, + cosa)] + (cosB + cosa)] (sln01)[ gl(l BI0 + "( E + (cosB B cosy) + =-I cos cos0 I) I/2 + relation g(0 I) gl(01) + 01 < (2.34) cosa)](1 cos2y)}(2.35) gl,g 2 (2.36) a, cosa)] cosa)I/2 B21cos01}/(cosO 8 < e < (2.37) a from equations (2.36) and (2.37) in the we get the required solution of equation (2.31). g2(02), the above illustration, when cosa) cosy}, [B21 (cos < B {[B20 + B21 (cos and putting the values of the functions (2.33) 1,2, i + cosa)} l)Bll (cos -BIICOS01}/R(01) g2(01 cosa)cosx, (cos8 [B20 + 1/2B21 (cosS + cosa)](1 Cz(y (2.32) In 0, we readily obtain the solution of the equation, -<0<a, g(81)( q + loglZa siq(8 81) de (Blo + BllCOSO) + (B20 8 co s01] + B21 sinS), (2.38) 81 < (2.39) in the form ,-5. g(Ol) " sin ,’2tcos "’Bl{ (I___ ’1 B I0+ B 11 AI + l)cos 2 l{B20 + B21 [sin2 b + csOl ]}II(csol csa)ll2’ -a < a’ where A 2[q + log(a sin a)], (2.40) 757 INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS which agrees with results derived by Shail [9] by applying tedious inversion formulas. INTEGRAL EQUATION (1.2). equivalent form (1.4). ddo f + We present here the method of solving equation (1.2) or its Equation (1.4) can be rewritten in the form f g(O)loglXn (0 O1 )idOl Iol < f(O), l < , (2.41) which on integration of the both raembers of this equat[on yields -B a (2.42) where p’(0) f(0) and C in an unknown constant. also of the form (I.I), with f(0)= C- I) solution g(0 1/4P(0), Since the above inegral equation is q =-log2, and a can be derived as explained in Section 2.1. unknown constant C occuring in the even degree part g(0 I) - gl(01)+ g2(01), relation S can be obtained therefore, its Lastly, the value of the of this solution gl(01) by putting this value of -B a f-a + f g(O1)dO gl (O1)dO gl(01) i, in the a f + -a S I’(O1)dO B (2.43) O, where we have used relation (1.5) and the edge condition (1.3)). 3. ELECTROSTATIC POTENTIAL PROBLEM. We consider the electrostatic problem of two equal infinite co-axial perfectly conducting strips charged in a free space so that the total charge per unit height on the two strips is unity. In cylindrical polar co-ordinates (r,O,z), the two strips are defined by r -, < a, < 0 -B, B < 0 < a, B O, a < 7, z < . The electrostatic potential (r,O) of this boundary value problem is given by 6(r, where g(0 I) O) -a the - a + f g(O1)log[r2 + a 2 2arcos(O 01)}1/2 dO1, (3.1) is the unknown surface charge density per unit are defined by g(01) Since f of value the r=a+ (3.2) r=a- assumes potential strips, using this boundary condition in -B S a + S g(l)lOgl2a sin (0 (3.1), o )lao a value, constant say o’ on the two we obtain the integral equation o < Iol < , (3.3) C. SAMPATH AND D.L. JAIN 758 The solution of this equation is readily obtained from that of equation BI0 =-o/a, BII setting q--O, 0, in (2.36) and (2.37). 020 B21 (2.31), by Therefore, the solution of equation (3.1) is given by g,(o,)=- ;[,o/(= g(o,) A where log[a2(cos13 2 ,2)]l,nO, l/R(0,), Io, < < ,, (3.4) (3.5) cosa)/2]. in the above expression for the charge o of the strips, it is given that the total charge per unit height on each Finally, to evaluate the unknown constant density g(8 I) of the strip is unity and therefore g(8 f a + g(0 I) d01 I) must satisfy the condition (3.6) 2 We substitute the value of g(B I) from (3.4) in the above equation, and get A o (3.7) 2’ and, therefore, When 13 0, we < g(01) 4. the obtain a, -a strip r 0 < ,/2 cos 1 result [9] for the circular < z ,_ cosa)l/2 a(cosO limiting corresponding a, < -a < 01 < (3 9) (x. HYDRODYNAMICAL PROBLEM. We consider the problem of uniform flow of an inviscid homogeneous liquid in the - icos + jsinT of velocity U, streaming past two fixed rigid strips r direction n a, < 0 -, 4 (r,0) of ,s(r, O) where B I) 0 < a, 0 13 < a The secondary velocity potential function 7. this two-dimensional problem is given by a I(01 and I(0 < -B f a + f Cs(a 3 {-rllog[r I(0 + 2 + r 01 -Cs(a-, 01) 2 [9] 2rrlcos(O 13 < 1011 < , 01)] 1/2 }dO r =a (4.1) (4.2) satisfies the edge condition I(l) I(e=) 0. (4.3) 759 INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS Using the boundary condition r =a where 91(r,0) r =a -Ur cos(0 T) is -B the 2 llcosec g(0 0 To solve this integral equation, we f f + I(0 velocity potential of function the incident (4.1), the integral equation [9] uniform flow Un, we obtain from lld01 8,aUcos y), B (0 I01 < < (4.5) a, integrate the left member by first parts and obtain by using the edge conditions (4.3) f where g(0) can be + f g(8 l)cot I’(0). (8 8 -4,aUcos (8 I)d81 V), B < 181 < , (4.6) The above integral equation is of the type (1.4), and therefore solved by the method given in the Section 2.2. We first rewrite integral equation (4.6) in the form (2.41) which on integration yields f + f g( 81)l og s in (8 where C is an unknown constant. type (2.31), log2, a with q I, 8 l)ldS1 C ), B 2.aUsin (8 < 181 < a, (4.7) Fortunately, the aboe integral equation is of the BIO C, BII 2aUsinY, B20 0, B21 -2aUcosY. Therefore, substituting these values of the constants in equations (2.36) and (2.37), we obtain the required solution of the above integral equation in the form g(01) g,(01) + g2(01) (4.8) cosa)18}. (4.10) where and 6 log{(cosI3 The value of the unknown constant C occuring in the value of (4.9) is obtained by putting this value of gl(81) gl(81) given by equation in the relation (2.43). This yields the required value of the unknown constant C is given by C -aU sin "( (cosS + cosa). (4.11) We substitute this value of C in (4.9) to obtain the value of the function g1(81) aU sinvls+/-n 811[(cos + cosa)-2 cos 81]IR(01), <181< gl(81) as (4.12) C. SAMPATH AND D.L. JAIN 760 Using formulas (2.19) and A = A expression for g2(01) is given by [C- 2aU cos Y(B cos y + A cos 2y)], 8 g2(01)= R(I where (2.25), the (cos8 (cos8 + cos), cos cos=), B 81 <I01 < (4.13) A cos y + B, (4.14) and the unknown constant C is evaluated by using the condition (1.3) which yields A + J2 2aU cos Y [B J C /Jo’ (4.15) (cos ny) de where 0, I, 2, n R(0 (4.16) We substitute this value of C in (4.13) to obtain the following value of J1 2aUR(ol)COSY g2(01 2aUcos Y Sgn (0) Sgn 8 I[B(]--- cos y) + A o [_2 01 A + -j--(BJ +A J2 o J2 (--o cos2y)] + B cosO cos 2 01] 8 < [01[ The results (4.8), (4.12) and (4.17) yield the required solution g(0 equation (4.6). o where for we gl(01 Slnce g(0) f t(o have I’(0), 8 o +f used the [011 < relation ( Iol (4.8). f8 gl(01 I) < (4.17) a, of the integral a, therefore g(o) o I’ satisfies the relation g2(Ol). d01 ( Since ( e) the (4 12) expression O thereforef g2(01) d01 the expression (4.18) for I(0) satisfies all the required edge conditions (.3) if 0. We may remark here that there is no need of obtaining the solution I(0) of the (4.5) for finding the expressions for the physical quantities of interest. equation These expressions can be readily derived from the value of the function g(0 given by relations (4.8) to (4.12). For instance, the kinetic energy per unit height (K.E.) of the secondary fluid flow is given by -- K.E. =_1 aP Ua_P2 Ua__2_p 2 + I) -8 a + 8 I(01)[ ( )cos( o (rl, 01 8r dO rlfa y)d o -8 ./" g(O )sin(O y)dO Uap {d cosy d2sinY} where we have used the boundary condition (4.4), the relation g(0) (4.19) l’(O), the edge INTEGRAL EUATIONS FOR THE POTENTIAL PROBLEHS (4.3), and conditions and d constants d 2 p is the density of the ho,aogeneous liquid. d2 = + J2]/J o’ 2auUA cosT[BJ (4.20ab) gl (O1)esOldO gl(01) We substitute the values of the functions (4.17) in (4.20) to obtain d The values of the are given by the relations g2 (O1)sinO IdOl, dl 761 d cosa)2sinT. -aU(cosg 2 from equations (4.12) and g2(01) and (4 21ab) Finally, relations (4.19) and (4.21) give rise to the required expression for Kinetic energy and this is given by pa2U 2 A K.E. + A {[ o where the definite integrals J A + + ] J2 o (BJI +A J2 A -] cos2}, (4.22) n=O,l,2 are defined by relation (4.16) and A and B n are defined in the relation (4.14). We derive now some interesting lmltng results from the formula (4.2) for K.E. g When =, O, J n 0,I,2 and Jt J2 j j o - and therefore the formula (4.22) o yields the followlng corresponding limiting expression for the K.E.,o infinite circular rigid strip r ra2U2.sin2 o a, -a [1 + cos This seems to be a new result. < 0 < a a2(a 2 al) I and a8 < 02 in case of < y < al, < 0, a 2 pU 0 (4.23) 0, O, a =, such that the two equal parallel co-planar infinite rigid z < and it is given by 2 2 2 2 < in case of the =: we obtain the corresponding limiting expression for the Kinetic Equation strips x < z cos2Y], Similarly, when in formula (4.22) we let e aa a, al [(l+c 2 )-2E’/F’]cos 2 "f, (4.24) where c and F(, a2/a < c) and E(, I, F’ F(, (I _c 2 )I/2 ), g’ E(, (I-c2) I/2 ), (4.25) c) are elliptic integrals of the first and the second kind [14]. We have also solved the two-dlmenslonal problems of scattering of a low-frequency incident plane acoustic wave by the integral equation techniques presented here. This C. SAMPATH AND D.L. JAIN 762 work will appear separately. ACKNOWLEDGMENT. This research work was supported by a grant from the University Grants Commission, New Delhi (India). REFERENCES A note on electrostatic problem involving two I. GOEL, G.C. and JAIN, D.L. 2. GOEL, 3. strips, Indian J. Pure Appl. Math. 7(1976), 809-816. JAIN, D.L. and KANWAL, R.P. Acoustic diffraction of a plane wave by two co- strips, J. Pure Appl. Math. G.C. and JAIN, D.L. 7(1976), 751-756. Electrostatic problems of two co-planar parallel planar parallel perfectly soft or rigid strlps,Can. J. Phys. 50(1972), 929- 939. Diffraction of elastic waves by two co-planar and 4. JAIN, D.L. and KANWAL, R.P. 5. parallel rigid strips, Int. J. Engg. Sci. 10(1972), 925-937. 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