Singular Integral Equations arising in Water

1
Singular Integral Equations arising in Water
Wave Problems
Aloknath Chakrabarti
Department of Mathematics
Indian Institute of Science
Bangalore-560012, India
Emailalok_at_math.iisc.ernet.in
2
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.

3
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.

4
1.Solution of Abels Integral equation and its
generalization
Consider here the general form of Abels 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
5
(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 Abels integral equation (1.1)
completely.
6
Examples of Abels integral equation equation
(1.1)
Example-1.1
(1.7)
Solution
(1.8)
(1.9)
Example-1.2
(1.10)
Solution
7
Example 1.3
(1.11)
Solution
(1.12)
Here we have chosen
(1.13)
Example 1.4
Solution
(1.14)
8
A direct method of solution of Abels 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
9
(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)
10
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 Abels
integral equation which is given by the relation
(1.29)
where
(1.30)
The method of solution of the general Abels
integral equation (1.29) involves the theory of
functions of a complex variable leading to
Rieman Hilbert type boundary value problems.
11
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
xiy, 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).
12
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)
13
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-Sokhotski formulae (also referred to as
just the Plemelj formulae) involving the
Cauchy-type integrals ?(z). The formulae (A5)
can also be expressed as
14
(A7)
(Plemelj formulae)
15
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 Hölders condition in
16
(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.
17
Particular Example
Integral Equation
(G6)
Solution (by Gakhov 1996)
(G7)
where
(G8)
18
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)
19
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)
20
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
21
(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)
22
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)
23
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
24
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 lt x lt 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.
25
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 x0 but unbounded at x1
, with an integrable
singularity there. We select
(2.9)
(2.10)
so that we have
(2.11)
26
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
27
(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)
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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)
29
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)
30
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)
31
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
32
(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
33
(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).
34
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)
35
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.
36
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
gt 0, such that
(4.1)
with
(4.2)
(4.3)
(4.4)
(4.5)
with
and
37
(4.6)
In which Rjs are unknown constants to be
determined, along with the unknown functions
, and
(4.7)
38
The methods of solution
(4.8)
with j1,2 and
The unknown functions and the unknown
constants Rj are determined form the following
sets of dual integral equations
(4.9)
39
Existence of method of solutions

• By Ursell 1947

• By Williams 1966

40
(A). Ursells Method
The principal idea behind Ursells 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 Havelocks 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)
41
The consistency of relation (4.10) demands that
we must have
(4.13)
Then
(4.14)
Using Ursells 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.
42
Here again, using Ursells 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
Muskhelishvillis book and we easily deduce that
(4.21)
and
43
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
44
(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)
45
(B). Williamss Method
The major deviation in Williamss method from
Ursells 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
46
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
47
(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
.
48
(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 j1 there is no inconsistency as
.
49
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
50
(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
51
(4C.11)
and
(4C.12)
Then we obtain
(4C.13)
and
(4C.14)
after utilizing the standard results that
(4C.15)
and
(4C.16)
52
We finally obtain
(4C.17)
where relations (4C.3) is utilized.
We thus observe that the principal unknown
functions are determined
in the same forms as those derived in relations
(4.24), by employing Ursells method.
We observe, as expected, that the final values
of, and
obtained by this new method, agree completely
with the ones obtained earlier using Ursells
method.
53
References
1. Chakrabarti, A and Mandal, B. N., Derivation
of the solution of a simple Hypersingular
integral equation, 1998, Int. J. Math. Edcu.
Sci. Technol., 29, 47 53. 2.
Chakrabarti, A. A Survey on Two International
Methods used in Scattering of Surface Water
Waves, Advances in Fluid Mechanics, WIT
Press, Edited by B. N. Mandal, 1997, pp
232-253. 3. Chakrabarti, A. Solution of the
Generalized Abel Integral Equations, Jl.
Int. Eqns and Appl., 2006 (accepted). 4.
Chakrabarti, A. Solution of a Simple Hyper
Singular Integral Equation, Jl. Int. Eqns
and Appl., 2006 (accepted). 5. Grakhov, F.D.-
On new types of integral equations, soluble ion
closed form, Problems of Continuum
Mechanics, pp. 118-132, Published by the
Society of Industrial and Applied Mathematics,
(SIAM), Philadelphia, Pennsylvania, (1961).
54
References
6. Grakhov, F. D. Boundary Value Problems,
Perganon, Oxford, 1966. 7. Grakhov,
F.D.-Boundary Value Problems, pp. 531-535, Oxford
Press, London, Edinburgh, New York, Paris,
Frankfurt (1996). 8. Jones, D. S., 1982, The
theory of Generalized functions, Cambridge
University Press, Cambridge. 9. Lundgren, T.
and Chiang, D.- Solution of a class of singular
integral equations, Quart. Appl. Math. Vol. 24,
No. 4. (1967), pp. 301-313. 10. Mandal, B. N.
and Chakrabarti, A. Water Wave Scattering by
Bariers WIT Press, 2000. 11. Martin, P.
A., 1992, Exact solution of a simple hypersigular
integral equation, J. Integral Equation Appic.
4, 197 204. 12. Mikhlin, S. G Integral
Equations, Pergamon, New York, 1964.
55
References
13. Muskhelishvili, N. I. Singular Integral
equations, Noordhiff, Graringen,
1953 14. Muskhelishvili N. I., 1977, Singular
Integral Equations, (Groningen
Noordhoff). 15. Sakalyuk, K.D.- Abels
generalized integral equation, Dokl. Akad.
Nauk., SSSR, Vol. 131, No.4. (1960), pp.
748-751. 16. Sneddon, I. N. The use of
Integral Transforms, McGraw Hill, New York,
1972. 17. Ursell, F. The Effect of a
fixed Vertical Barier on Surface Waves in Deep
Water, Proc. Camb. Phil. Soc., 43, (1947),
pp 374 382. 18. Williams, W. E. A note on
scattering of water waves by a vertical barier
Proc. Camb. Phil. Soc., 62, (1966), pp 507
509.
56
Acknowledgement
I take this opportunity to thank
the University Grants Commission (UGC), New
Delhi, India for awarding me an Emeritus
Fellowship to carry out the research involving
this lecture.
57
Whilst most mathematicians like to enjoy
handling mathematical problems for their
complete solution, there exists a class of
mathematicians who enjoy creating mathematical
problems which can not be solved completely by
the aid of existing mathematical ideas.
58
THANK YOU FOR YOUR ATTENTION