Singular Integral Equations arising in Water Wave Problems

Singular Integral Equations arising inWater Wave Problems Aloknath Chakrabarti Department of Mathematics Indian Institute of Science Bangalore-560012, India Email:alok@math.iisc.ernet.in

Abstract • Mixed Boundary Value Problems occur, in a natural way, in varieties of branches of Physics and Engineering and several mathematical methods have been developed to solve this class of problems of Applied Mathematics. • While understanding applications of such boundary value problems are of immense value to Physicists and Engineers, analyzing these problems mathematically and determining their solutions by utilizing the most appropriate analytical or numerical methods are the concerns of Applied Mathematicians. • Of the various analytical methods, which are useful to solve certain mixed boundary value problems arising in the theory of Scattering of Surface Water Waves, the methods involving complex function theory and singular integral equations will be examined in detail along with some recent developments of such methods.

Literature Review • Sneddon [1972]: Varieties of mixed boundary value problems of mathematical physics can be solved by reducing them to integral equations of one type or the other. • Muskhelishvili [1953],Gakhov [1966], Mikhlin [1964] : Certain singular integral equations and their methods of solution in detail. • Chakrabarti [2006] : Development of above recently. • Chakrabarti [1997] and Mandal and Chakrabarti [2000: Book]: Occurrences of such singular integral equations in studies on problems of scattering of surface water waves by barriers, present in the fluid medium.

1.Solution of Abel’s Integral equation and its generalization Consider here the general form of Abel’s Integral equation (1.1) where h(t) is a strictly monotonically increasing and differentiable function in (a, b) and Solution of (1.1) by using a very simple method : Consider (1.2) By using (1.1), we can express (1.2), after interchanging the orders of integration, as

(1.3) Using the transformations (1.4) we obtain, from (1.3) and (1.2), the following results (1.5) giving, on differentiation : (1.6) which solves the Abel’s integral equation (1.1) completely.

Examples of Abel’s integral equation equation (1.1) Example-1.1 (1.7) Solution: (1.8) (1.9) Example-1.2: (1.10) Solution:

Example 1.3: (1.11) Solution: (1.12) Here we have chosen (1.13) Example 1.4 : Solution: (1.14)

A direct method of solution of Abel’s integral equation (1.15) Writing (1.16) Then the integral equation (1.15) can be expressed as (1.17) where (1.18) If we look at the given equation (1.15) as (1.19) with k(x) = x-, and if we recall the convolution theorem in the following form

(1.20) Now, we easily find that (1.21) Then, if we set (1.22) so that (1.23) Then we obtain, (1.24) By utilizing the convolution theorem (1.20) once more, in a clever manner, we find that (1.24) gives: (1.25)

which, on using the identity (1.26) gives (1.27) By using integration by parts we can also rewrite this as (since f(0) = 0) (1.28) We shall next consider the general form of Abel’s integral equation which is given by the relation (1.29) where (1.30) The method of solution of the general Abel’s integral equation (1.29) involves the theory of functions of a complex variable leading to Rieman Hilbert type boundary value problems.

Some Important Theorems and Results in Complex Function Theory Theorem -1. If the function () satisfies the Hölder condition: (A1) where A is a positive constant, for all pairs of points on a simple closed, positively oriented contour  of the complex z – plane (z = x+iy, i2 = -1), then the Cauchy-type as given by the relation: (A2) represents a “sectionally analytic” (analytic except for points z lying on ) function of the complex variable z. The function (), in the relation (A2) is called the “density function” of the Cauchy-type integral (z).

Theorem -2:(The Basic Lemma) If the density function () satisfies a Hölder condition, then the formula on passing through the point z = t, of the simple closed contour , behaves as a continuous function of z, i.e., exists and is equal to (t). [Note : Theorem also holds even if  is an arc in the z- plane, provided that the point g does not coincide with any end point of ]. (A3) (A4)

Theorem-3(Plemelj-Sokhotski Formulae) If represents a sectionally analytic function, as in Theorem (*), then and exist, then the following formulae hold good: (A5) (A6) where means that the points z approach the point t on  from the left of the positively oriented contour , and means that z approaches t from the right of  . The formulae (A5) are known as the Plemelj-Sokhotskiformulae (also referred to as just the Plemelj formulae) involving the Cauchy-type integrals (z). The formulae (A5) can also be expressed as :

(A7) (Plemelj formulae)

Generalized Abel Integral Equation and its Solution The generalized Abel integral equation (G1) whereas the forcing term f(x) and the unknown function (x) belong to those classes of functions which admit representations of the form (G2) where possesses a Hölder continuous derivative in and satisfies Hölder’s condition in

(G3) With (G4) so that (G5) and the associated Riemann – Hilbert problem is finally solved by utilizing the Plemelj – Sokhotski formulae involving Cauchy-type singular integrals.

Particular Example Integral Equation: (G6) Solution (by Gakhov [1996]): (G7) where (G8)

The Detailed Method (G9) and below As z tends to a point x  , from above Then the sectionally analytic function  (z) ) tends to the following limiting values: (G10) Where (G11) and The relation (D2) can also be expressed as and (G12)

By using the relations (G12) in the given integral equation (G1), we obtain (G13) Relation (13) represents the special Riemann-Hilbert type problem (G14) with (G15) and (G16) Method of solution of the new Riemann-Hilbert type problem (14): (z), given by equation (G9), satisfies the following condition at infinity: (G17)

First solve the homogeneous problem (14), satisfying the relation (G18) Giving (G19) where and (G20) Now we can express the function satisfying (19), as : (G21) where (G22) with (G23) Next, by utilizing (19) in (14), we obtain (G24) where

(G25) with being obtainable by suing the relations (21)-(23). Then, by utilizing the first of the formulae (G12), we can determine the solution of the Riemann-Hilbert type problem (G24), as given by : (G26) where (G27) The relation (27) takes the equivalent form: (G28) with (G29)

Next we obtain the following limiting values of the function (z), as z approaches the point [see (G10)]: (G30) giving (G31) where (G32) Finally, by utilizing the first formula in (G12), once again, we obtain the required solution of the given generalized Abel integral equation (G1) in the form (G33) The result (G33) can also be expressed in the equivalent form: (G34)

2. Solution of Singular Integral Equations of the Cauchy type The general theory of a single linear singular integral equation of the type (2.1) Defining a sectionally analytic function (2.2) Utilizing the Plemelj-Sokhotski formulae we can rewrite (2.1) as i.e. (2.3) provided

The relation (2.3) is a particular case of the most general such relation, as given by (2.4) Consider the case c = p in equation (2.1) and is the open interval 0 < x < 1: We first solve the homogeneous Riemann-Hilbert problem (2.3) in this particular case. Here (2.5) of the homogeneous Then, by the aid of any suitable solution problem (2.5), we can cast the original Riemann-Hilbert problem as: (2.6) The general solution of the Riemann-Hilbert problem (2.6) can be written down by the aid of the Plemelj-Sokhotski formulae. The general solution is given by (2.7) where E(z) is an arbitrary entire function of z.

Then we find that the general solution of our integral equation (2.1) can be determined by means of the relation (2.8) We thus find that the general solution of the integral equation (2.1) depends on an arbitrary choice of an entire function E(z) appearing in the relation (2.7). A special choice of E(z) can be made depending on the class of the forcing functions f(x) and the selection of the function representing the solution of the homogeneous problem (2.5). To illustrate the above procedure we take up the special case such that and f(x) are bounded at x=0 but unbounded at x=1 , with an integrable singularity there. We select (2.9) (2.10) so that we have (2.11)

Then, observing that [by fixing the idea that (i) (2.12) (ii) (2.13) as well as the fact that (2.14) We find that we must select (2.15) giving (2.16) Using the Plemeji-Sokhotski formulae on the relation (2.16), together with the results (2.12), we find that the relation (2.8) produces the unique solution of our integral equation (2.1) in this special circumstance. It is given by:

(2.17) NOTE: The limiting case of the integral equation (2.1) with is the integral equation of the first kind as given by (2.18) This limiting case gives and the limit of the solution (2.17) is obtained as (2.19)

3.Hyper-singular Integral Equation (singular integral equation having a higher order singularly in the integral) (3A) is considered for its solution for The hypersingular integral Hf appearing in the equation (3A) is understood to be equal to the Hadamard finite part (see Martin [1992]) of this divergent integral, as given by the relation: (3B) The equation (3A) has been solved by Martin [1992] and Chakrabarti and Mandal [1998], under the circumstances when, in the following closed form: (3C)

A Direct Function Theoretic Method and The detailed analysis Consider the sectionally analytic function (3.1) Then if we utilize the following standard limiting values (3.2) and (3.3) we obtain the following Plemelj – type formulae giving the limiting values of the function (z), as z approaches a point on the cut (-1,1) from above and below respectively: (3.4)

The limiting values (3.4) can also be derived by utilizing the standard Plemelj formulae involving the limiting values of the Cauchy type integral (3.5) giving (3.6) and by the aid of the relation (3.7) along with the understanding that (3.8) Now, the two relations (3.4) can also be viewed as the following two equivalent relations (3.9)

By utilizing the first of the above two relations (3.9), we now rewrite the given hypersingular integral equation as (3.10) which represents a special Riemann-Hilbert type boundary value problem for the determination of the unknown function (z). If 0(z) represents a nontrivial solution of the homogeneous problem (3.10), satisfying (3.11) then we may rewrite the inhomogeneous problem (3.10) as (3.12) with (3.13) Thus, then second of the relations (3.9) suggests that we can determine the function (z) in the following form:

(3.14) where (3.15) Next, by utilizing the form (3.14) of the function  (z), along with the relation (3.13) and the second of the Plemelj-type formulae (3.9), we obtain the following result: (3.16) If we select (3.17) giving (3.18) we find that, because of the relations (3.13) and (3.14), we must select E0(z) to be equal to zero. Then, using the relation (3.18), along with the relation (3.15), we obtain from the relation (3.16), the following result:

(3.19) with (3.20) Finally, by integrating the relation (3.19), we can determine the solution of the given hypersingular integral equation, in the following form (3.21) where (3.22) This completes the method of solution f the hypersingular integral equation (3A), in principle, once the hypersingular integral occurring in the relation (3.20) is evaluated, for a given forcing function f(x).

We can derive the known form (3C) of the solution of the equation (3A), as obtained by Martin [1992], by using a procedure as described below: By integrating by parts, we obtain form the relation (3.22), that Another integration gives, because of the relation (3.20): (3.23) Ignoring an arbitrary constant [see(3.21)], when the following results are used: (3.24) and (3.25)

Special case: when (-1) = 0 = (1), the solution of the equation (3A), as given by the formulae (3.19) and (3.21) is obtained in the form (3.26) since we must have (3.27) The result (3.26) agrees with the form (3C), involving a weakly singular integral. The analysis presented above is believed to be self-contained and straightforward.

4. Problems of Fluid Mechanics (Water Waves) Mathematical Problem: Determination of the two-dimensional velocity potentials with i2 = -1, in the two-dimensional Cartesian xy coordinates, in the half – plane y > 0, such that (4.1) with (4.2) (4.3) (4.4) (4.5) with and

(4.6) In which Rj’s are unknown constants to be determined, along with the unknown functions , and (4.7)

The methods of solution (4.8) with j=1,2 and The unknown functions and the unknown constants Rj are determined form the following sets of dual integral equations: (4.9)

Existence of method of solutions • By Ursell [1947] • By Williams [1966]

(A). Ursell’s Method The principal idea behind Ursell’s method involves setting (4.10) Then we observe that because of the second of the relations (4.9) and that the unknown functions, for are singular at the turning points tj. Utilizing Havelock’s expansion theorem we find that we must have (4.11) Substituting from relations (4.11) in to the first of the dual relations (4.9), (4.12)

The consistency of relation (4.10) demands that we must have (4.13) Then (4.14) Using Ursell’s approach, we next operate both sides of equation (4.12), for each j by the operator formally and use the well-known identity (4.15) to obtain (4.16) Many researchers, including Ursell (1947), have studied the singular integral equations (4.16). The employment of various methods and solutions of such integral equations have become central in many important and interesting studies involving singular integral equations.

Here again, using Ursell’s idea, we first set (4.17) and Then obtain the following further reduced integral equations as given by (4.18) For the two reduced functions H1(y) and H2 (y) as defined by the relations: (4.19) The singular integral equations (4.18) are best solved by using the results available in Muskhelishvilli’s book and we easily deduce that (4.21) and

Then we find that (4.22) and (4.23) Substituting from relations (4.22) and (4.23) into relations (4.11), after integrating by parts we obtain (4.24) and where

(4.25) where J0 (x) and J1 (x) represent the standard Bessel functions of the first kind. Then, by using relations (4.22) and (4.23) in relations (4.14) and integrating by parts, we derive that (4.26)

(B). Williams’s Method The major deviation in Williams’s method from Ursell’s method lies in rewriting the basic dual integral equations (4.9) in the following alternative forms: (4.27) Then we must choose the constant Dj and Ej as follows: (4.28) and

If we now set (4.29) when the following identities are utilized and (4.30) with representing the standard modified Bessel functions. Finally we deduce that and (4.31) after using the following identities:

(4.32) and We can now easily determine the constants by using relations (4.26) and (4. 28), and we find that and (4.33) which are the most familiar results derived by Ursell [1947]. The full solutions of the two boundary value problems are thus completed when the relations (4.24) are substituted, in conjunction with relations (4.28) and (4.30), into the expressions (8), for the potentials .

(C). A New Method In this present approach, we start by rewriting the dual integral equations (4.9) in the alternative forms (4C.1) and Operating both sides of the equations by produces (4C.2) where (4C.3) with arbitrary constants, so that for the case j=1 there is no inconsistency as .

We set (4C.4) Then we easily derive the following equations for the determination of the two unknown functions and : (4C.5) and (4C.6) The above two equations (4C.5) and (4C.6) can easily be reduced to the following two Abel type integral equations

(4C.7) and (4C.8) by utilizing the following standard and elementary results: (4C.9) and (4C.10) The solutions of the two Abel equations (4C.7) and (4C.8) are immediate and we obtain

Singular Integral Equations arising in Water Wave Problems