Solving singular integral equations of the second kind using Chebyshev polynomials

A numerical developed technique to solve Fredholm integral equation of the second kind with separable singular kernel is proposed. This technique relies on the truncated expansion functions of the kernels in the finite series of the weighted Chebyshev polynomials of first, second, third, and fourth kinds. Three numerical examples are presented for verification and validation of the developed technique. The results showed that even with small n, the numerical results are accurate.


INTRODUCTION
Generally, Cauchy singular integral equations of the second kind can be expressed in the form: (1) where and Q(t, x) are real valued functions which satisfy the Hölder condition with respect to each of the independent variables, and are squareintegrable functions on the interval for any and is the solution to be determined.Equation 1 has singularity of the Cauchy-type.The Cauchy singular integral equations are encountered in a variety of mixed boundary value problems in mathematical physics such as fracture problems in solid mechanics (Ladopoulos, 2000), aerodynamics and plane elasticity (Kalandiya, 1975) and other related problems.
Several numerical techniques have been used to solve Cauchy singular integral equations including polynomials like the weighted Chebyshev polynomial of the second kind (Eshkuvatov et al., 2012), Bernstein polynomial method (Setia, 2014), using Legendre polynomial (Setia et al., 2015), the reproducing kernel Hilbert space method (Dezhbord et al., 2016), and the collocation technique based on the Bernstein polynomials (Seifi et al., 2017).
In this research work, we present a developed technique for solution of Cauchy singular integral equation by using the weighted Chebyshev polynomials of the first, second, third and fourth kinds.The used approximated method for solving Equation 1 stems from the work of Eshkuvatov et al. (2012) wherein approximate method has been developed to solve the case for k(t,x)=1 and Q(x,t)=0 using the weighted Chebyshev polynomial of the second kind only.

Consider
to be any of the four Chebyshev polynomials, which implies that .
The sequence is the set of weight functions of the Chebyshev polynomials of the first, second, third and fourth kinds, respectively and we denote these weight functions by  1  ,  2  ,  3  ,  (4) () respectively.Let the unknown function be written as (2) where ℎ() is some bounded function of on the interval and   ∈  1  ,  2  ,  3  ,  (4) () is the given weight function.Approximating in Equation 2 by using the Chebyshev polynomial gives: (3) where we can represent the unknown function as: (4) where are unknown coefficients to be determined.Substituting the approximate solution (Equation 4) for the unknown function into Equation 1, we obtain: (5) The integrals in Equation 5 can be calculated given that the kernel and can be expressed in the form (Dardery and Allan, 2013): Using the expressions (Equation 11) in Equation 9, and the defined relations (Equations 12, 13, 14 and 15), we can obtain an exact value for By multiplying each of the terms in Equation 10 by the Chebyshev polynomials of the first kind, and integrate from -1 to 1, we obtain the equation:

Proof
By making use of the last equation in relation (Equation 11) on the left-hand side of Equation 23, we get the expression on the right-hand side of Equation 23.

Lemma 2
For (24) where with reference to Equations 11 and 17, we have:

Proof
By making use of the last equation in relation (Equation 11) on the left-hand side of Equation 24, we get: We can now apply the relation defined in Equation 17to the last expression in the earlier stater equation and desired result is obtained.By using Lemma 1 and Lemma 2 in Equation 22 and then substitute Equation 22into equation 18, we obtain a system of linear equations to solve for the unknown coefficients . By substituting the values of into Equation 4, we obtain the numerical solution of the Equation 1.

NUMERICAL EXAMPLES
Here, we apply the numerical technique explained in the previously.

Solution
The analytical solution to Equation 25 Substituting the values of into Equation 27, the numerical solution of Equation 25 is obtained to be and are expressed in terms of the Chebyshev polynomials of the first kind, , of degree up to (Mason and Handscomb, 2003): (7)where the dash denotes that the term in the sum is to be halved if i is even and .By making use of Equation6, we can represent Equation 5 as: denotes that half of the first term in the sum has been considered.By making use of system (integrals on the right-hand side of Equation 23 can be calculated exactly by making use of Equation11and the relations (Equations 12, 13, 14 and 15).
Equation 25 using our developed technique, we set and the unknown function as: where is the Chebyshev polynomial of the second kind.The Chebyshev polynomial of the second kind is chosen since the function f(x) in Equation 25 has the weight function of Chebyshev polynomial of the second kind.By substituting Equation 27 into Equation 25properties of Chebyshev polynomials defined in Okecha and Onwukwe (2012) on the integrals in Equation 29 and simplify, we obtain: