© Hindawi Publishing Corp. DUAL INTEGRAL EQUATIONS—REVISITED

Dual integral equations with trigonometric kernel are reinvestigated here for a solution. The behaviour of one of the integrals at the end points of the interval complementary to the one in which it is defined plays the key role in determining the solution of the dual integral equations. The solution of the dual integral equations is then applied to find an exact solution of the water wave scattering problems. 2000 Mathematics Subject Classification: 45F10.


Introduction.
Boundary value problems with mixed boundary conditions arising in different branches of mathematical physics can be reduced to dual integral equations.A mixed boundary condition is the one in which one condition is prescribed at one part of the boundary while some other condition is prescribed at the remaining part of the boundary.The solution of the dual integral equations essentially depends on the behaviour of one of the integrals at the end points of the interval complementary to the one in which it is defined [1,4].This behaviour is dictated by the physics of the problem.
In the present paper, we consider the following dual integral equations: where A(k) is an unknown function, and R is an unknown constant.This integral equation arises in the well-known problem of scattering water waves by a vertical barrier under the assumption of linearised theory [5,6,7,8].The vertical barrier may be (i) partially immersed in deep water, (ii) completely submerged and extending infinitely downwards in deep water, (iii) a vertical wall with a gap, or (iv) a submerged plate.The solution of (1.1) has been obtained here by noting the behaviour of the second equation of (1.1) at the end points of the interval G j , which can be determined from physical consideration.Equation (1.1) was then reduced to a singular integral equation whose kernel involves Cauchy and logarithmic type singularity.The solution of this singular integral equation is known (cf.[3,4,6,8]).The solution of (1.1) was then obtained by utilizing the solution of aforesaid singular integral equation.Knowing the solution of (1.1), the solution of the corresponding scattering problems was obtained in a closed form.In Section 2, we consider the genesis of dual integral equation (1.1), and in Section 3, we find the solution of (1.1) and hence the solution of the corresponding scattering problems.

Genesis of the dual integral equations.
The two-dimensional problem of the scattering of surface waves by a vertical barrier present in deep water under the assumption of linearised theory consists in solving mixed two-dimensional boundary value problem given as follows: φ j satisfies the free surface condition the condition on the barrier, ∂φ j ∂x = 0 on x = 0 y ∈ B j , j = 1, 2, 3, 4. (2.3) Here, B j represents the vertical barrier.(i) For j = 1, the barrier is partially immersed to a depth a 1 below the mean free surface y = 0 so that B 1 = (0,a 1 ).
(ii) For j = 2, the vertical barrier is completely submerged and extends infinitely downwards, so B 2 = (a 2 , ∞). (iii) For j = 3, the vertical barrier is in the form of a wall with a gap, so B 3 = (0,a 3 ) + (a 4 , ∞). (iv) For j = 4, the barrier is in the form of a plate submerged in deep water, so B 4 = (a 5 ,a 6 ).The bottom condition is given by At the sharp edges of the barrier, we must have where r denotes the distance from sharp edges a j of the barrier, j = 1,...,6 where T j , R j are unknown complex constant.The function φ j , j = 1, 2, 3, 4, represents the velocity potential for two-dimensional irrotational motion corresponding to various scattering problems.The function exp(−Ky +iKx) (dropping the time dependent factor exp(−iσ t) where σ is the circular frequency K = σ 2 /g, g being acceleration due to gravity) represents the wave propagating from the negative x-direction incident upon the barrier B j .The complex constants R j and T j are the reflection and transmission coefficients, respectively.

The solution of (1.1).
Let where h j (y) is the unknown function.In view of (2.9), (2.3), and (2.4), as y → a 2 , bounded as y → 0, (3.3) ) as y → a i , i = 5, 6, → 0 a s y → ∞, bounded as y → 0. (3.5) By Havelocks' expansion theorem [8], we have from (3.1) ) Substituting A j (k) from (3.7) into (2.11),we have Simplifying (3.8) and applying (d/dy + K), we have This is a singular integral equation in h j (t), whose kernel involves a combination of Cauchy type and logarithmic singularity.An appropriate solution of (3.9) can be obtained by considering the behaviour of h j (t) at the end points of G j , which is given in (3.2), (3.3), (3.4), and (3.5) for various configurations of the barrier.Hence (3.6) and (3.7) show that the behaviour of h j (t) at the end points of G j plays the key role in determining the solution of (1.1).Now, considering (3.2), (3.3), (3.4), and (3.5), we find h j (t) for j = 1, 2, 3, 4 and hence A j (k) and R j for j = 1, 2, 3, 4.
(1) Knowing (3.2), h 1 (t) is given by (cf.[8]) where C 1 is a constant.Substituting h 1 (t) in (3.6) and (3.7), we have To find C 1 , A 1 (k) and R 1 are substituted in the first equation of (1.1) to get where C 2 is a constant.Substituting in (3.6) and (3.7) The constant C 2 is determined by substituting A 2 (k), R 2 in first equation of (1.1).On simplification, this gives where and hence (3.6) and (3.7) give ( To find C 3 , substitute A 3 (k) and R 3 in the first equation of (1.1) to get where where C 4 and d 2 0 are constants, and (3.6) and (3.7) give To determine C 4 and d 2 0 , we substitute A 4 (k) in the first equation of (1.1) to get the relations which yield This determines d 2 0 .Equating (3.26) and (3.28), we have where (3.32) Thus, knowing A j (k) and R j , the corresponding φ j (x, y) for j = 1, 2, 3, 4 are known from (2.7).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning )P (u)du , y < a 6 , (

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation