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Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier
Applied Mathematics Letters 58 (2016) 19–25
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier R. Gayen a,∗ , Sourav Gupta a , A. Chakrabarti b a b
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
article
info
Article history: Received 28 December 2015 Accepted 26 January 2016 Available online 9 February 2016 Keywords: First kind integral equation Direct polynomial approximation Chebyshev polynomials
abstract The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Weakly singular integral equations of the first kind arise frequently in the research problems of various fields of mathematical physics [1–3]. Several analytical and numerical methods have been devised to determine solutions to this type of integral equations [4–7]. Here we propose two numerical methods to solve a weakly singular integral equation connected with a special mixed boundary value problem arising in the study of water wave scattering by a thin partially immersed barrier in deep water. This problem was first solved analytically by Ursell [8] by reducing it to a first kind singular integral equation of the form ∞ y + u 1 1 − f (u) Kln − du = 0, a < y < ∞ (1) − y−u y+u y−u a where K is a constant. ∗ Corresponding author. E-mail address: [email protected] (R. Gayen).
http://dx.doi.org/10.1016/j.aml.2016.01.018 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
20
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
That the integral is interpreted in the sense of Cauchy principal value, is represented by ‘−’ sign on the integral. The unknown function f (y) which represents the fluid velocity along the line below the thin plate 1 behaves as f (t) ∼ O (|t − a|− 2 ) as t → a. The integral Eq. (1) was solved by Ursell [8] analytically and the reflection and the transmission coefficients were determined in terms of Bessel functions. This problem was also solved by Williams [9] through a weakly singular integral equation approach. Variants of Williams’ method were developed by Chakrabarti [10,11]. If the water depth is finite, it is not possible to obtain the exact solution and researchers employed different methods to obtain numerical estimates for the physical quantities. Some of the notable research work in this direction are found in Parsons and Martin [12], Abul-Azm [13], Porter and Evans [14]. As the number of problems with exact solutions is quite limited, with the advent of high speed computers, researchers are involved in developing algorithms with faster rate of convergence. In our present work, we make an effort to solve Ursell’s problem by reducing it to a non-homogeneous first kind integral equation and solving it by polynomial approximations of the unknown function together with collocation at suitable points. In order to achieve this, we first reduce the governing boundary value problem in terms of a first kind non-homogeneous integral equation for determining the velocity function across the vertical line below the thin barrier. Using the behaviour of the velocity at the end point of the plate and at infinity, the unknown velocity is represented as a product of a known elementary function and an unknown smooth function. Two different ways of approximating the smooth function give rise to two different methods. In the first approach the function is expressed to an unknown polynomial of degree N . This polynomial is substituted into the integral equation and the free variable is collocated at the finite number of points by the zeros of Chebyshev polynomial of the second kind. The second approach consists of approximating the unknown function by an unknown series of Chebyshev polynomials. Both the procedures yield systems of linear algebraic equations when collocated at finite number of points. The linear systems are solved to determine the discrete numerical values of the unknown function. Using these values the reflection coefficient is determined numerically. Graphical representation of the reflection curve reveals that the results exactly coincide with those of Ursell [8]. Thus the current analysis presents two simple techniques for solving a first kind integral equation in which the unknown function has prescribed end behaviour. As a test, the methods have been employed to solve a problem whose exact solution is well established in the literature. The methods are quite general and can be employed to problems for which analytical solutions do not exist. 2. Mathematical formulation Making the usual assumptions of linearized theory of water waves, the boundary value problem for the governing velocity potential φ(x, y) is described by the following equations (cf. Mandal and Chakrabarti [15]) ∇2 φ = 0
in the fluid region,
(2)
along with the free surface boundary condition Kφ +
∂φ =0 ∂y
on y = 0,
(3)
the condition on the barrier ∂φ = 0, ∂x
(4)
and the boundary condition on the bottom ▽φ → 0
as y → ∞.
(5)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
21
Near the submerged edge of the barrier φ behaves as ∂φ −1 ∼ O(| y − a | 2 ) as x → 0, y → a. ∂x The radiation condition is given by (1 − R)φinc (x, y) as x → ∞, φ(x, y) ∼ φinc (x, y) + Rφinc (−x, y) as x → −∞
(6)
(7)
where R is the reflection coefficient and is to be determined. 3. Reduction to the first kind integral equation Using Havelock’s expansion of water wave potential, a representation of φ(x, y) satisfying the equations (2), (3), (5) and (7) is given by (cf. Chakrabarti [11]) ∞ inc A(ξ)L(ξ, y)e−ξx , x > 0, (1 − R)φ (x, y) + 0 ∞ φ(x, y) = (8) φinc (x, y) + Rφinc (−x, y) − A(ξ)L(ξ, y)eξx , x < 0 0
where L(ξ, y) = ξ cos ξy − K sin ξy and A(ξ) is an unknown function to be determined. We denote by f (y) the unknown velocity function along the vertical line below the origin. Thus f (y) =
∂φ (0, y), ∂x
0 < y < ∞.
(9)
Also, if the potential difference across line x = 0 is denoted by g(y), then g(y) = φ(+0, y) − φ(−0, y),
0 < y < ∞.
(10)
Since the velocity vanishes on the plate and the pressure is continuous across the line below the plate, we have that y ∈ L = [0, a),
(11)
y ∈ L = (a, ∞).
(12)
f (y) = 0, and g(y) = 0,
In order to obtain an integral equation for f (y), we first use the representation of φ(x, y) given by the relation (8). Taking the derivative of both sides of the relation (8) and then substituting it into the relation (9), f (y) is found as ∞ f (y) = iK(1 − R)e−Ky − ξA(ξ)L(ξ, y)dξ. (13) 0
The unknown function A(ξ) and the reflection coefficient R are determined in terms of the velocity f (y) using Havelock’s inversion formula (cf. Mandal and Chakrabarti [15]). Thus A(ξ) and R are found to be given by the following relations ∞ 2 1 A(ξ) = − f (t)L(ξ, t)dt, (14) π ξ(ξ 2 + K 2 ) a and R = 1 + 2i a
∞
f (t)e−Kt dt.
(15)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
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Next using the relations (8), (10) and (12) we obtain that ∞ f (t)M (y, t)dt = −Re−Ky
a ≤ y < ∞,
(16)
a
where the kernel M (y, t) has the following form [cf. Ursell [8])]: 2 ∞ (ξ cos ξy − K sin ξy)(ξ cos ξt − K sin ξt) M (y, t) = dξ π 0 ξ(ξ 2 + K 2 ) K(y+t) υ e 1 y + t −K(y+t) ln dυ . = − 2e π y−t υ −∞ Elimination of R from the equations (15) and (16), gives rise to a first kind non-homogeneous integral equation of the form ∞ f (t)K(y, t)dt = −e−Ky , a ≤ y < ∞ (17) a
where K(y, t) = M (y, t) + 2ie−K(y+t) .
(18)
An analysis of the flow close to the edge of the barrier reveals that 1 t → a. f (t) ∼ O √ t−a Also, since f (t) → 0 as t → ∞ like O( t12 ), f (t) can be expressed as f (t) =
1 h(t) √ t t−a 3 2
where h(t) ∈ C ∞ . Replacing the expression of f (t) given by the relation (19) in the equation (17) we obtain that ∞ 1 h(t)K(y, t)dt = e−Ky , a ≤ y < ∞. 3√ t2 t − a a A suitable change of variables yields the following integral equation valid in the range [−1, 1]: 1 2a √ 2a 2Ka 2a 1 1 √ √ , du = e− 1−v ; v ∈ [−1, 1]. h 1 − uK 1−v 1−u 2a −1 1 − u2 1 − u
(19)
(20)
(21)
To solve the integral equation (21), we employ two methods described in the next two sections. 4. Direct polynomial approximation 2a In our first approach we approximate h 1−u by using a polynomial of degree N as given by the relation N 2a h = a 1 − u2 cj uj 1−u j=0
(22)
where cj (j = 0, 1, . . . , N )’s are unknown constants to be determined. Substituting the expansion (22) into the relation (21) we obtain that N j=0
2Ka
cj Aj (v) = −e− 1−v
(23)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
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where 1 Aj (v) = √ 2
1
2a √ 1 1 2a uj 1 − u M + 2ie−2Ka( 1−v + 1−u ) du. , 1−v 1−u −1
(24)
To find the N + 1 constants cj , we collocate at N + 1 points v = vi (i = 0, 1, . . . , N ), where vi ’s are chosen as the zeros of the Chebyshev polynomial of the second kind as given by i+1 π, i = 0, 1, . . . , N . N +2 This yields the following linear system of algebraic equations: vi = cos
N
2Ka − 1−v
cj Aj (vi ) = −e
i
,
i = 0, 1, . . . , N.
(25)
(26)
j=0
Equation (26) is a linear system of (N + 1) algebraic equations with (N + 1) unknowns, which can be solved by any standard method to determine cj ’s numerically. The Reflection Coefficient In order to determine the unknown reflection coefficient R, we substitute the form of f (t) into the relation (15). This produces an integral representation of R in terms of the unknown function h. Transforming the range of the integration to [−1, 1] and using the approximation of h as given by the relation (22), the computable form of R is obtained as 1 N √ √ 2Ka cj uj 1 − ue− 1−u du. (27) R = 1 + 2i −1
j=0
Since the unknowns are determined by solving the linear system of algebraic equations (26) numerically, the integral on the right hand side of equation (27) can be evaluated using standard Gauss quadrature formula. Here we have used 16 point Gauss quadrature. Choosing a suitable N , we solve the linear system of algebraic equations (26) by standard numerical method and obtain cj ’s (j = 0, 1, . . . , N ). These are substituted into equation (27) to find approximate discrete values of R. 5. Approximation through chebyshev polynomials 2a appearing in the equation (20) by the Chebyshev polynomials Approximation of the function h 1−u forms the basis of our second method. Here we approximate h as N 2a h =a dj Tj (u) 1−u j=0
(28)
where Tj (u) is the jth order Chebyshev polynomial of the first kind and dj ’s (j = 0, 1, . . . , N ) are unknown constants to be determined. Substituting the expansion of h from the relation (28) into the relation (21) and collocating at N + 1 points v = vi , we obtain the following system of algebraic equations for finding dj ’s: N
2Ka − 1−v
dj Bj (vi ) = e
i
i = 0, 1, . . . , N
(29)
j=0
where vi ∈ [−1, 1] and are given in (25). The known coefficients Bj (vi ) involves an integration which is carried out following Conte and De Boor [16]. We express Bj (vi ) as 1 g (v , u) √j i Bj (vi ) = du (30) 1 − u2 −1
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
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Table 1 Ka = 0.50
Ka = 1.30
N
Ursell [8]
Method 1 (Abs error)
Method 2 (Abs error)
N
Ursell [8]
Method 1 (Abs error)
Method 2 (Abs error)
151
0.43938
0.98912
0.439378
25
0.989120
201
0.439378
51
0.989120
205
0.439378
0.44171 (2.3 × 10−3 ) 0.441455 (2.1 × 10−3 ) 0.441167 (1.8 × 10−3 ) 0.441133 (1.7 × 10−3 )
11
171
0.44108 (1.7 × 10−3 ) 0.439859 (4.8 × 10−4 ) 0.439321 (5.7 × 10−5 ) 0.439338 (4.0 × 10−5 )
75
0.989120
0.98872 (4.0 × 10−4 ) 0.988980 (1.4 × 10−4 ) 0.989037 (8.3 × 10−5 ) 0.989056 (6.4 × 10−5 )
0.988339 (7.8 × 10−4 ) 0.988718 (4.0 × 10−4 ) 0.988953 (1.7 × 10−4 ) 0.989011 (1.1 × 10−4 )
where 2a √ 1 1 1 2a −2Ka( 1−v + 1−u ) i + + 2ie gj (vi , u) = √ Tj (u) 1 − u M . 1 − vi 1−u 2 The integral appearing in the relation (30) is evaluated as
1
−1
N g (v , u) π √j i gj (vi , ξk ) du = N +1 1 − u2 k=0
where ξk ’s (k = 0, 1, . . . , N ) are the zeros of the Chebyshev polynomials of the first kind. The system of linear algebraic equations (29) is next solved to determine the unknown constants dj ’s. These are used to compute the reflection coefficient by using the following relation: N √ dj R = 1 + 2i j=0
1
−1
√ Tj (u) 1 − u − 2Ka √ e 1−u du. 1 − u2
(31)
6. Discussion In the Table 1, we display the numerical results for |R| obtained by the Method 1 and Method 2 respectively and compare the data with those computed from Ursell’s [8] expression for R. The absolute errors are displayed in parenthesis. The table shows the convergence of the methods with truncation size N of the finite series (22) and (28) for two dimensionless wave numbers Ka = 0.50 and Ka = 1.30. It is observed from this table that accuracies of order 10−5 and 10−3 are achieved at N = 201 through the Method 1 and Method 2 respectively for Ka = 0.50. However, for higher value of wave number (Ka = 1.30) accuracies of order 10−5 and 10−4 could be attained only at N = 51. In Fig. 1, we plot the data for |R| as computed by the method of Ursell, Method 1 and Method 2 of the present analysis. The data are represented by a solid line, ‘∗’ and ‘+’ respectively. The figure and the table reveal the excellent agreement of Ursell’s results with those obtained by both the algorithms of the current work. Thus we have been successful to provide two simple straightforward and efficient methods for solving a particular first kind weakly singular integral equation arising in a water wave problem. Application of these methods, to more complicated problems of water waves, is in progress. Acknowledgement A. Chakrabarti is grateful to NASI for financial support in the forms of NASI-Senior Scientist Platinum Jubilee Fellow and NASI Honorary-Scientist (Reference Numbers: NAS/322/12/2010-11 and NAS/1022/1/2015-16).
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
25
Fig. 1. |R| vs. dimensionless wave number.
References [1] K. Mei, J. Van Bladel, Low-frequency scattering by rectangular cylinders, IEEE Trans. Antennas Propag. 11 (1963) 292–304. [2] L. Pyl, D. Clouteau, G. Degrande, A weakly singular boundary integral equation in elastodynamics for heterogeneous domains mitigating fictitious eigenfrequencies, Eng. Anal. Bound. Elem. 28 (2004) 1493–1513. [3] S. Banerjea, B.N. Mandal, Solution of a singular integral equation in a double interval arising in the theory of water waves, Appl. Math. Lett. 6 (1993) 81–84. [4] A. Chakrabarti, S.R. Manam, Solution of a logarithmic singular integral equation, Appl. Math. Lett. 16 (2003) 369–373. [5] S. Banerjea, B. Dutta, On a weakly singular integral equation and its application, Appl. Math. Lett. 21 (2008) 729–734. [6] Y. Cao, M. Huang, L. Liu, Y. Xu, Hybrid collocation methods for Fredholm integral equations with weakly singular kernels, Appl. Numer. Math. 57 (2007) 549–561. [7] T. Okayama, T. Matsuo, M. Sugihara, Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind, J. Comput. Appl. Math. 234 (2010) 1211–1227. [8] F. Ursell, The effect of a fixed vertical barrier on surface waves in deep water, Proc. Cambridge Philos. Soc. 43 (1947) 374–382. [9] W.E. Williams, Note on the scattering of water by a vertical barrier, Proc. Cambridge Philos. Soc. 62 (1966) 507–509. [10] A. Chakrabarti, Solution of two singular integral equations arising in water wave problems, ZAMM 69 (1989) 457–459. [11] A. Chakrabarti, A survey on two mathematical methods used in scattering of surface water waves, in: B.N. Mandal (Ed.), Advances in Fluid Mechanics, Mathematical Techniques for Water Waves, Computational Mechanics Publications, Southampton, Boston, MA, 1997, pp. 231–249. [12] N.F. Parsons, P.A. Martin, Scattering of water waves by submerged plate using hypersingular integral equation, Appl. Ocean Res. 14 (1992) 415–423. [13] A.G. Abul-Azm, Wave diffraction through submerged breakwaters, J. Waterw, Port C. ASCE 119 (1993) 587–605. [14] R. Porter, D.V. Evans, Complementary approximations to wave scattering by vertical barriers, J. Fluid Mech. 294 (1995) 155–180. [15] B.N. Mandal, A. Chakrabarti, Water wave scattering by barriers, Wit Pr./Computational Mechanics, 2000. [16] S.D. Conte, C. de Boor, Elementary numerical analysis-An algorithmic approach, MacGraw-Hill, 1980.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier R. Gayen a,∗ , Sourav Gupta a , A. Chakrabarti b a b
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
article
info
Article history: Received 28 December 2015 Accepted 26 January 2016 Available online 9 February 2016 Keywords: First kind integral equation Direct polynomial approximation Chebyshev polynomials
abstract The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Weakly singular integral equations of the first kind arise frequently in the research problems of various fields of mathematical physics [1–3]. Several analytical and numerical methods have been devised to determine solutions to this type of integral equations [4–7]. Here we propose two numerical methods to solve a weakly singular integral equation connected with a special mixed boundary value problem arising in the study of water wave scattering by a thin partially immersed barrier in deep water. This problem was first solved analytically by Ursell [8] by reducing it to a first kind singular integral equation of the form ∞ y + u 1 1 − f (u) Kln − du = 0, a < y < ∞ (1) − y−u y+u y−u a where K is a constant. ∗ Corresponding author. E-mail address: [email protected] (R. Gayen).
http://dx.doi.org/10.1016/j.aml.2016.01.018 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
20
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
That the integral is interpreted in the sense of Cauchy principal value, is represented by ‘−’ sign on the integral. The unknown function f (y) which represents the fluid velocity along the line below the thin plate 1 behaves as f (t) ∼ O (|t − a|− 2 ) as t → a. The integral Eq. (1) was solved by Ursell [8] analytically and the reflection and the transmission coefficients were determined in terms of Bessel functions. This problem was also solved by Williams [9] through a weakly singular integral equation approach. Variants of Williams’ method were developed by Chakrabarti [10,11]. If the water depth is finite, it is not possible to obtain the exact solution and researchers employed different methods to obtain numerical estimates for the physical quantities. Some of the notable research work in this direction are found in Parsons and Martin [12], Abul-Azm [13], Porter and Evans [14]. As the number of problems with exact solutions is quite limited, with the advent of high speed computers, researchers are involved in developing algorithms with faster rate of convergence. In our present work, we make an effort to solve Ursell’s problem by reducing it to a non-homogeneous first kind integral equation and solving it by polynomial approximations of the unknown function together with collocation at suitable points. In order to achieve this, we first reduce the governing boundary value problem in terms of a first kind non-homogeneous integral equation for determining the velocity function across the vertical line below the thin barrier. Using the behaviour of the velocity at the end point of the plate and at infinity, the unknown velocity is represented as a product of a known elementary function and an unknown smooth function. Two different ways of approximating the smooth function give rise to two different methods. In the first approach the function is expressed to an unknown polynomial of degree N . This polynomial is substituted into the integral equation and the free variable is collocated at the finite number of points by the zeros of Chebyshev polynomial of the second kind. The second approach consists of approximating the unknown function by an unknown series of Chebyshev polynomials. Both the procedures yield systems of linear algebraic equations when collocated at finite number of points. The linear systems are solved to determine the discrete numerical values of the unknown function. Using these values the reflection coefficient is determined numerically. Graphical representation of the reflection curve reveals that the results exactly coincide with those of Ursell [8]. Thus the current analysis presents two simple techniques for solving a first kind integral equation in which the unknown function has prescribed end behaviour. As a test, the methods have been employed to solve a problem whose exact solution is well established in the literature. The methods are quite general and can be employed to problems for which analytical solutions do not exist. 2. Mathematical formulation Making the usual assumptions of linearized theory of water waves, the boundary value problem for the governing velocity potential φ(x, y) is described by the following equations (cf. Mandal and Chakrabarti [15]) ∇2 φ = 0
in the fluid region,
(2)
along with the free surface boundary condition Kφ +
∂φ =0 ∂y
on y = 0,
(3)
the condition on the barrier ∂φ = 0, ∂x
(4)
and the boundary condition on the bottom ▽φ → 0
as y → ∞.
(5)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
21
Near the submerged edge of the barrier φ behaves as ∂φ −1 ∼ O(| y − a | 2 ) as x → 0, y → a. ∂x The radiation condition is given by (1 − R)φinc (x, y) as x → ∞, φ(x, y) ∼ φinc (x, y) + Rφinc (−x, y) as x → −∞
(6)
(7)
where R is the reflection coefficient and is to be determined. 3. Reduction to the first kind integral equation Using Havelock’s expansion of water wave potential, a representation of φ(x, y) satisfying the equations (2), (3), (5) and (7) is given by (cf. Chakrabarti [11]) ∞ inc A(ξ)L(ξ, y)e−ξx , x > 0, (1 − R)φ (x, y) + 0 ∞ φ(x, y) = (8) φinc (x, y) + Rφinc (−x, y) − A(ξ)L(ξ, y)eξx , x < 0 0
where L(ξ, y) = ξ cos ξy − K sin ξy and A(ξ) is an unknown function to be determined. We denote by f (y) the unknown velocity function along the vertical line below the origin. Thus f (y) =
∂φ (0, y), ∂x
0 < y < ∞.
(9)
Also, if the potential difference across line x = 0 is denoted by g(y), then g(y) = φ(+0, y) − φ(−0, y),
0 < y < ∞.
(10)
Since the velocity vanishes on the plate and the pressure is continuous across the line below the plate, we have that y ∈ L = [0, a),
(11)
y ∈ L = (a, ∞).
(12)
f (y) = 0, and g(y) = 0,
In order to obtain an integral equation for f (y), we first use the representation of φ(x, y) given by the relation (8). Taking the derivative of both sides of the relation (8) and then substituting it into the relation (9), f (y) is found as ∞ f (y) = iK(1 − R)e−Ky − ξA(ξ)L(ξ, y)dξ. (13) 0
The unknown function A(ξ) and the reflection coefficient R are determined in terms of the velocity f (y) using Havelock’s inversion formula (cf. Mandal and Chakrabarti [15]). Thus A(ξ) and R are found to be given by the following relations ∞ 2 1 A(ξ) = − f (t)L(ξ, t)dt, (14) π ξ(ξ 2 + K 2 ) a and R = 1 + 2i a
∞
f (t)e−Kt dt.
(15)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
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Next using the relations (8), (10) and (12) we obtain that ∞ f (t)M (y, t)dt = −Re−Ky
a ≤ y < ∞,
(16)
a
where the kernel M (y, t) has the following form [cf. Ursell [8])]: 2 ∞ (ξ cos ξy − K sin ξy)(ξ cos ξt − K sin ξt) M (y, t) = dξ π 0 ξ(ξ 2 + K 2 ) K(y+t) υ e 1 y + t −K(y+t) ln dυ . = − 2e π y−t υ −∞ Elimination of R from the equations (15) and (16), gives rise to a first kind non-homogeneous integral equation of the form ∞ f (t)K(y, t)dt = −e−Ky , a ≤ y < ∞ (17) a
where K(y, t) = M (y, t) + 2ie−K(y+t) .
(18)
An analysis of the flow close to the edge of the barrier reveals that 1 t → a. f (t) ∼ O √ t−a Also, since f (t) → 0 as t → ∞ like O( t12 ), f (t) can be expressed as f (t) =
1 h(t) √ t t−a 3 2
where h(t) ∈ C ∞ . Replacing the expression of f (t) given by the relation (19) in the equation (17) we obtain that ∞ 1 h(t)K(y, t)dt = e−Ky , a ≤ y < ∞. 3√ t2 t − a a A suitable change of variables yields the following integral equation valid in the range [−1, 1]: 1 2a √ 2a 2Ka 2a 1 1 √ √ , du = e− 1−v ; v ∈ [−1, 1]. h 1 − uK 1−v 1−u 2a −1 1 − u2 1 − u
(19)
(20)
(21)
To solve the integral equation (21), we employ two methods described in the next two sections. 4. Direct polynomial approximation 2a In our first approach we approximate h 1−u by using a polynomial of degree N as given by the relation N 2a h = a 1 − u2 cj uj 1−u j=0
(22)
where cj (j = 0, 1, . . . , N )’s are unknown constants to be determined. Substituting the expansion (22) into the relation (21) we obtain that N j=0
2Ka
cj Aj (v) = −e− 1−v
(23)
R. Gayen et al. / Applied Mathematics Letters 58 (2016) 19–25
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where 1 Aj (v) = √ 2
1
2a √ 1 1 2a uj 1 − u M + 2ie−2Ka( 1−v + 1−u ) du. , 1−v 1−u −1
(24)
To find the N + 1 constants cj , we collocate at N + 1 points v = vi (i = 0, 1, . . . , N ), where vi ’s are chosen as the zeros of the Chebyshev polynomial of the second kind as given by i+1 π, i = 0, 1, . . . , N . N +2 This yields the following linear system of algebraic equations: vi = cos
N
2Ka − 1−v
cj Aj (vi ) = −e
i
,
i = 0, 1, . . . , N.
(25)
(26)
j=0
Equation (26) is a linear system of (N + 1) algebraic equations with (N + 1) unknowns, which can be solved by any standard method to determine cj ’s numerically. The Reflection Coefficient In order to determine the unknown reflection coefficient R, we substitute the form of f (t) into the relation (15). This produces an integral representation of R in terms of the unknown function h. Transforming the range of the integration to [−1, 1] and using the approximation of h as given by the relation (22), the computable form of R is obtained as 1 N √ √ 2Ka cj uj 1 − ue− 1−u du. (27) R = 1 + 2i −1
j=0
Since the unknowns are determined by solving the linear system of algebraic equations (26) numerically, the integral on the right hand side of equation (27) can be evaluated using standard Gauss quadrature formula. Here we have used 16 point Gauss quadrature. Choosing a suitable N , we solve the linear system of algebraic equations (26) by standard numerical method and obtain cj ’s (j = 0, 1, . . . , N ). These are substituted into equation (27) to find approximate discrete values of R. 5. Approximation through chebyshev polynomials 2a appearing in the equation (20) by the Chebyshev polynomials Approximation of the function h 1−u forms the basis of our second method. Here we approximate h as N 2a h =a dj Tj (u) 1−u j=0
(28)
where Tj (u) is the jth order Chebyshev polynomial of the first kind and dj ’s (j = 0, 1, . . . , N ) are unknown constants to be determined. Substituting the expansion of h from the relation (28) into the relation (21) and collocating at N + 1 points v = vi , we obtain the following system of algebraic equations for finding dj ’s: N
2Ka − 1−v
dj Bj (vi ) = e
i
i = 0, 1, . . . , N
(29)
j=0
where vi ∈ [−1, 1] and are given in (25). The known coefficients Bj (vi ) involves an integration which is carried out following Conte and De Boor [16]. We express Bj (vi ) as 1 g (v , u) √j i Bj (vi ) = du (30) 1 − u2 −1
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Table 1 Ka = 0.50
Ka = 1.30
N
Ursell [8]
Method 1 (Abs error)
Method 2 (Abs error)
N
Ursell [8]
Method 1 (Abs error)
Method 2 (Abs error)
151
0.43938
0.98912
0.439378
25
0.989120
201
0.439378
51
0.989120
205
0.439378
0.44171 (2.3 × 10−3 ) 0.441455 (2.1 × 10−3 ) 0.441167 (1.8 × 10−3 ) 0.441133 (1.7 × 10−3 )
11
171
0.44108 (1.7 × 10−3 ) 0.439859 (4.8 × 10−4 ) 0.439321 (5.7 × 10−5 ) 0.439338 (4.0 × 10−5 )
75
0.989120
0.98872 (4.0 × 10−4 ) 0.988980 (1.4 × 10−4 ) 0.989037 (8.3 × 10−5 ) 0.989056 (6.4 × 10−5 )
0.988339 (7.8 × 10−4 ) 0.988718 (4.0 × 10−4 ) 0.988953 (1.7 × 10−4 ) 0.989011 (1.1 × 10−4 )
where 2a √ 1 1 1 2a −2Ka( 1−v + 1−u ) i + + 2ie gj (vi , u) = √ Tj (u) 1 − u M . 1 − vi 1−u 2 The integral appearing in the relation (30) is evaluated as
1
−1
N g (v , u) π √j i gj (vi , ξk ) du = N +1 1 − u2 k=0
where ξk ’s (k = 0, 1, . . . , N ) are the zeros of the Chebyshev polynomials of the first kind. The system of linear algebraic equations (29) is next solved to determine the unknown constants dj ’s. These are used to compute the reflection coefficient by using the following relation: N √ dj R = 1 + 2i j=0
1
−1
√ Tj (u) 1 − u − 2Ka √ e 1−u du. 1 − u2
(31)
6. Discussion In the Table 1, we display the numerical results for |R| obtained by the Method 1 and Method 2 respectively and compare the data with those computed from Ursell’s [8] expression for R. The absolute errors are displayed in parenthesis. The table shows the convergence of the methods with truncation size N of the finite series (22) and (28) for two dimensionless wave numbers Ka = 0.50 and Ka = 1.30. It is observed from this table that accuracies of order 10−5 and 10−3 are achieved at N = 201 through the Method 1 and Method 2 respectively for Ka = 0.50. However, for higher value of wave number (Ka = 1.30) accuracies of order 10−5 and 10−4 could be attained only at N = 51. In Fig. 1, we plot the data for |R| as computed by the method of Ursell, Method 1 and Method 2 of the present analysis. The data are represented by a solid line, ‘∗’ and ‘+’ respectively. The figure and the table reveal the excellent agreement of Ursell’s results with those obtained by both the algorithms of the current work. Thus we have been successful to provide two simple straightforward and efficient methods for solving a particular first kind weakly singular integral equation arising in a water wave problem. Application of these methods, to more complicated problems of water waves, is in progress. Acknowledgement A. Chakrabarti is grateful to NASI for financial support in the forms of NASI-Senior Scientist Platinum Jubilee Fellow and NASI Honorary-Scientist (Reference Numbers: NAS/322/12/2010-11 and NAS/1022/1/2015-16).
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Fig. 1. |R| vs. dimensionless wave number.
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