ON SOLUTION OF THE INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS OF TWO CIRCULAR-STRIPS

Solutions are given to some singular integral equations which arise in two- dimensional Dirichlet and Newmann boundary value problems of two equal infinite co- axial circular strips 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.

where the known function f( 6)) is of class C I, for the values of 0 satisfying the inequality 0 B I01 a 7, and a q are known constants.The singular integral equations of the type (I.I) govern solutions of various two-dimensional Dirichlet boundary value problems of two equal infinite co-axial circular strips in potential theory.
In the corresponding Newmann boundary value problems, the governing integral equations are of the type 2 + l(01)cosec (6) 01)d01 f(0), < 16)I < ' (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 / +/ g(o >=or -(o 6)l)dO f( 6)), IB < 16)( < c where g(O) l'(O). (1.4) We present here a simple technique of solving integral equations of the types 1. I) and (1.2) For illustration, this technique is applied to solve 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.This work will appear separately.
The plan of this paper is as follows.
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 first kind with kernels (i) (Const.)+ logl2(cosx-cosy)l, 0 < x,y < 7, the 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 < x < n are made in it.
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 the second kind.
The solution of this n 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 technique to solve the integral equation (I.I), when the known function f(0) in it is of a particular form, for our subsequent analysis.
Lastly, we explain how equation (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 to solve the two-dimensional electrostatic Dirichlet boundary value problem of two equal infinite co-axial perfectly conducting circular strips in a free space, when the total charge per unit height of the two strips is unity.Section 3 is devoted to the study 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 strips.
The expression for the kinetic energy per unit height of 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).
and the Fourier expansion of the known even degree function Fl(x) where (2.12) (2.13) readily yield the solution Gl(Y) of the equation (2.6).This is given by G l(y) b + Z b cosny, Now we take up the solution of the equation (2.3).We first differentiate both sides of equation (2.3) with respect to 0 and obtain a g2(0 )sln0 B cos0--cos0 d01 f2 (0)' B < 0 < a, (2.17) 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 (2.20) and F2(x) is a known even degree function of x.Next, when we substitute -I -I x cos (X), y cos (Y), in (2.18), we obtain -I G 2 (cos Y) dY -I (I _y2) I/2(Y-X) F2(cos-Ix) -I < X < I. (2.28) The value of g2(01) given by (2.27) satisfies this edge condition if in (2.25), G2 y O, as y 0+, and hence Therefore, (2 29) n-I n g2(01) 2-cos cosa) l Cn_I(I cosny)]/R(0 I), < 01 < a. (2.30) n=l Since, sin2 Yl or (cosBCSB _-cosa cs01') is a common factor of all the terms occuring in the infinite series in the right member of the above equation, therefore, the above expression for g2(01) satisfies the required edge condition (2.28).
Its solution is also very useful in solving various boundary value problems presented in our subsequent analysis.
INTEGRAL EQUATION (1.2).We present here the method of solving equation (1.2) or its equivalent form (1.4).Equation (1.4) can be rewritten in the form ddo f + f g(O)loglXn (0 O1 )idOl f(O), l < Iol < , (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.Since the above inegral equation is also of the form (I.I), with f(0)= C-1/4P(0), q =-log2, and a i, therefore, its solution g(0 I) can be derived as explained in Section 2.1.Lastly, the value of the unknown constant C occuring in the even degree part gl(01) of this solution g(0 I) gl(01)+ g2(01), can be obtained by putting this value of gl(01 where we have used relation (1.5) and the edge condition (1.3)).
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.
The electrostatic potential (r,O) of this boundary value problem is given by a 2 6(r, O) -a f + f g(O1)log[r2 + a 2arcos(O 01)}1/2dO1, (3.1) where g(0 I) is the unknown surface charge density per unit are defined by r=a+ g(01) -r=a- (3.2) Since the value of the potential assumes a constant value, say o' on the two strips, using this boundary condition in (3.1), we obtain the integral equation The solution of this equation is readily obtained from that of equation (2.31), by setting q--O, BI0 =-o/a, BII 020 B21 0, in (2.36) and (2.37).Therefore, the solution of equation (3.1) is given by g(o,) g,(o,)=-;[,o/(= ,2)]l,nO, l/R(0,), < Io, < ,, Finally, to evaluate the unknown constant in the above expression for the charge o density g(8 I) of the strips, it is given that the total charge per unit height on each of the strip is unity and therefore g(8 I) must satisfy the condition a f + g(0 I) d01 We substitute the value of g(B I) from (3.4) in the above equation, and and, therefore, When 13 0, we obtain the corresponding limiting result [9] for the circular strip r a, -a < 0 < a, < z < ,/2 cos 1 g(01) ( cosa)l/2 ,_ -a < 01 < (x.
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)  The value of the unknown constant C occuring in the value of gl(81) given by equation (4.9) is obtained by putting this value of gl(81) 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 gl(81) as g1(81) aU sinvls+/-n 811[(cos + cosa)-2 cos 81]IR(01), <181< (4.12)Using formulas (2.19) and (2.25), the expression for g2(01) is given by where A A g2(01)= R(I [C-2aU cos Y(B cos y + cos 2y)], 8 <I01 < A = (cos8 cos=), B (cos8 + cos), cos 81 A cos y + B, We substitute this value of C in (4.13) to obtain the following value of g2(Ol).
We may remark here that there is no need of obtaining the solution I(0) of the equation (4.5) for finding the expressions for the physical quantities of interest.
These expressions can be readily derived from the value of the function g(0 I) given by relations (4.8) to (4.12).For instance, the kinetic energy per unit height (K.E.where we have used the boundary condition (4.4), the relation g(0) l'(O), the edge conditions (4.3), and p is the density of the ho,aogeneous liquid.The values of the constants d and d 2 are given by the relations dl g2 (O1)sinO IdOl, d2 = gl (O1)esOldO (4.20ab) 21ab) o' 2 Finally, relations (4.19) and (4.21) give rise to the required expression for Kinetic energy and this is given by K.E.where the definite integrals J 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.
Similarly, when in formula (4.22) we let e 0, O, a =, such that aa a I and a8 a2(a 2 < al) we obtain the corresponding limiting expression for the Kinetic Equation 02 in case of the two equal parallel co-planar infinite rigid < z < and it is given by strips x 0, a 2 < y < al,   , c) and E(, 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 (2A), bn -(nan)/ n I, (2.15) and the known coefficients a n O, are defined by the relations (2.13).Finally, n substituting the above value of Gl(Y) in the relation (2.7), we obtain the required solution of the equation (2.2) gl(01 [b + Z b cosny] [sin0 I]/R(oI) B < O above integral equation can be readily solved when we substitute it in the expansion of the known function F 2 in terms of the Chebychev polynomials U (X) of the n second kind F2(cos IX) Z c U (X), -I < X < I, 24)where T (Y) is the nth degree Chebychev polynomial of the first kind.Thus the n solution of the equation(2.22) is given by G2(Y ATo(cOsy + I Cn-ITn(csy)' 0 < y < , (2.25) n=l where A is an arbitrary constant and the constant coefficients c are defined by the n expansion (2.23) of the known function F 2. These values are Cn f sinx sln(n + l)x F2(x)dx, n 0. 0 (2.26)Lastly, relations(2.19)and (2.25) lead to the required solution of the equation (2.3) g2(01) (coscos)[A + 2.4) gives the value of cosy in terms of cos 0 I.The value of the constant A in (2.27) is readily obtained by using the edge condition satisfied by the density function g2(01) at the inner edge i=.This edge condition is 1 constant C is evaluated by using the condition (1.3) formula (4.22)    yields the followlng corresponding limiting expression for the K.E.,o infinite circular rigid strip r