10_304_Abdulwaki.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 3, 2009, (462-472) ISSN 1307-5543 – www.ejpam.com On the Semi-bounded Solution of Cauchy Type Sin- gular Integral Equations of the First Kind M. Abdulkawi∗, Z. K. Eshkuvatov, and N. M. A. Nik Long Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 Ser- dang, Selangor, Malaysia Abstract. This paper presents an efficient approximate method to obtain a numerical solu- tion, which is bounded at the end point x = −1, for Cauchy type singular integral equations of the first kind on the interval [−1,1]. The solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the third kind, and then com- puting the Cauchy singular integral which is obtained analytically. The known force function is interpolated using the Chebyshev polynomials of the fourth kind. The exactness of this approximate method is shown for characteristic equation when the force function is a cubic. Particular result is also given to show the exactness of this method. 2000 Mathematics Subject Classifications: 65R20, 45E05 Key Words and Phrases: Integral equations, Cauchy singular kernel, Chebyshev polynomials, Approximation. ∗Corresponding author. Email address: akawi�math.upm.edu.my (M. Abdulkawi) http://www.ejpam.com 462 c© 2009 EJPAM All rights reserved. M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 463 1. Introduction Let us consider the Cauchy type singular integral equations of the first kind ∫ 1 −1 ϕ(t) t − x d t + ∫ 1 −1 K(x , t)ϕ(t) d t = f (x), −1< x < 1, (1.1) where K and f are assumed to be real-valued functions belong to the class of Hölder continues functions on the sets [−1, 1]× [−1, 1] and [−1, 1], respectively. ϕ is un- known function to be determined. The singular integral equations have been widely used [1–4] in solving problems associated with aerodynamic, hydrodynamic and elas- ticity. The characteristic singular integral equation of equation (1.1) is of the form ∫ 1 −1 ϕ(t) t − x d t = f (x), −1< x < 1. (1.2) Eshkovatov et al. [5] discussed the efficient approximate method to solve charac- teristic equation (1.2) using Chebyshev polynomial approximations of the first, sec- ond, third, and fourth kinds with corresponding weight functions for four cases. The collocation points are chosen to be the zeros of Chebyshev polynomials. They showed that, the approximate method gives exact solution when the force function f is a linear. Abdulkawi et al. [6] presented a numerical solution of equation (1.1), which is bounded at the end points x ± 1. They used Chebyshev polynomials of the second kind with the corresponding weight function to approximate the density function and the Chebyshev polynomials of the first kind to approximate the force function. They showed that the numerical solution of characteristic equation is identical to the exact solution when the force function is a cubic. It is well known that the analytical solution of characteristic equation (1.2), which M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 464 is bounded at the end point x =−1, is given by the following formula ϕ(x) = − 1 π2 r 1+ x 1− x ∫ 1 −1 r 1− t 1+ t f (t) t − x d t . (1.3) By solving equation (1.1) with respect to its characteristic part, we will find that it is equivalent to the Fredholm equation type of the second kind [7] ϕ(t)+ ∫ 1 −1 N(t ,τ)ϕ(τ) dτ = F(t), N(t ,τ) = − 1 π2 r 1+ t 1− t ∫ 1 −1 r 1− x 1+ x K(x ,τ) x − t d x , F(t) =− 1 π2 r 1+ t 1− t ∫ 1 −1 r 1− x 1+ x f (x) x − t d x .                (1.4) in the sense of obtaining the solution which one can apply the Fredholm’s theorems. In this paper, we present an approximate solution for equation (1.1) which is bounded at the end point x = −1. 2. Approximate Solution of Equation (1.1) Guiding by the analytic solutions of characteristic equation given by (1.3), using the Chebyshev interpolation polynomials of third kind Vi and fourth kind Wi with corresponding weight functions ω1 and ω2 [8]; Vi(x) = cos � 2i + 1 2 cos−1 x � cos � 1 2 cos−1 x � , ω1(x) = r 1+ x 1− x , Wi(x) = sin � 2i + 1 2 cos−1 x � sin � 1 2 cos−1 x � , ω2(x) = r 1− x 1+ x .                , (2.1) and helping of the following important formula for singular integrals with the Cauchy kernel ∫ 1 −1 r 1+ t 1− t Vi(t) t − x d t = πWi(x), −1< x < 1, i = 0, 1, ..., n, (2.2) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 465 the approximate solution, which is bounded at the end point x = −1, of equation (1.1) is obtained. We will interpolate the known function f (x) by using the Chebyshev orthogonal polynomial of the fourth kind fn(x) of degree n as f (x)≈ fn(x) = n ∑ k=0 fk Wk(x) (2.3) where fk = 1 π ∫ 1 −1 r 1− t 1+ t f (t)Wk(t) d t . (2.4) Approximating the unknown function ϕ by ϕn which is defined as ϕn(x) = r 1+ x 1− x n ∑ j=0 a j Vj(x) (2.5) where the unknown coefficients ¦ a j ©n 0 are to be determined. Substituting (2.5) into (1.1) we obtain n ∑ j=0 a j ∫ 1 −1 r 1+ t 1− t Vj(t) t − x d t + n ∑ j=0 a j ∫ 1 −1 r 1+ t 1− t K(x , t)Vj(t) d t = f (x). (2.6) Using (2.2) into (2.6) we obtain π n ∑ j=0 a j Wj(x) + n ∑ j=0 a j ζ j(x) = f (x) (2.7) where ζ j(x) = ∫ 1 −1 r 1+ t 1− t K(x , t)Vj (t) d t . (2.8) Interpolating the function ζ j(x) by using the Chebyshev orthogonal polynomial of the fourth kind as follows ζ j(x)≈ n ∑ k=0 µ j,k Wk(x) (2.9) where µ j,k = 1 π ∫ 1 −1 r 1− x 1+ x ∫ 1 −1 r 1+ t 1− t K(x , t)Vj(t)Wk(x) d t d x . (2.10) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 466 Due to (2.3-2.4) and (2.9-2.10), equation (2.7) becomes n ∑ j=0 a j Wj(x) + 1 π n ∑ k=0 n ∑ j=0 a j µ j,k Wk(x) = 1 π n ∑ k=0 fk Wk(x). (2.11) The unknown coefficients ¦ a j ©n 0 are determined by solving the system of linear equations obtained by comparing the coefficients of Wj, j = 0, 1, 2, ..., n in both sides of equation (2.11) which is a0+ 1 π n ∑ j=0 a j µ j, 0 = 1 π f0, a1+ 1 π n ∑ j=0 a j µ j, 1 = 1 π f1, . . . . . . . . . ... ... ... . . . . . . . . . an+ 1 π n ∑ j=0 a j µ j, n = 1 π fn.                                  (2.12) where the coefficients � fk and ¦ µ j, k © are given by (2.4) and (2.10), respectively. 3. Approximate Solution of the Characteristic Equation (1.2) Theorem 3.1. If f (x) in characteristic equation (1.2) is a cubic function, then the approximate solution (2.5) is identical to the exact solution. Proof. Let us consider the characteristic singular integral equation ∫ 1 −1 ϕ(t) t − x d t = f (x), −1< x < 1. (3.1) Let f (x) in (3.1) be a cubic function i.e f (x) = c0+ c1 x + c2 x2+ c3 x3, −1 < x < 1. (3.2) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 467 Substituting (3.2) into (2.4) yields fk = 1 π ∫ 1 −1 r 1− t 1+ t � c0+ c1 t + c2 t2+ c3 t3 � Wk(t) d t . (3.3) Using the following Chebyshev recurrence relations of the third and fourth kinds, respectively, V0(x) = 1, V1(x) = 2x − 1, Vn(x) = 2x Vn−1(x)− Vn−2(x), n ≥ 2.    , (3.4) W0(x) = 1, W1(x) = 2x + 1, Wn(x) = 2x Wn−1(x)−Wn−2(x), n ≥ 2.    . (3.5) we have t3 = 1 8 � V3 (t) + V2 (t) + 3 (V1 (t) + V0 (t)) � = 1 8 � W3 (t)−W2 (t) + 3 (W1 (t)−W0 (t)) � , t2 = 1 4 � V2 (t) + V1 (t) + 2 V0 (t) � = 1 4 � W2 (t)− W1 (t) + 2W0 (t) � , t = 1 2 � V1 (t) + V0 (t) � = 1 2 � W1 (t)−W0 (t) � .                              (3.6) It is known that [8] ∫ 1 −1 r 1+ t 1− t Vm(t)Vn (t) d t =    0, n 6= m, π, n = m. (3.7) and ∫ 1 −1 r 1− t 1+ t Wm(t)Wn (t) d t =    0, n 6= m, π, n= m. (3.8) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 468 Due to (3.3), (3.6) and (3.8), we obtain f0 = c0 − c1− c2 2 − 3 c3 8 , f1 = 2 c1 − c2 4 + 3 c3 8 , f2 = 2 c2 − c3 8 , f3 = c3 8 .                  (3.9) From (2.12) when K(x , t) = 0, yields a j = 1 π f j, j = 0, ..., n. (3.10) The approximate solution (2.5) with n= 3 becomes ϕn(x) = 1 π r 1+ x 1− x � f0 + f1V1(x) + f2V2 (x) + f3V3 (x) � . (3.11) Substituting (3.9) and (3.10) into (3.11), we obtain the approximate solutions of characteristic equation (3.1) which is ϕn(x) = 1 π r 1+ x 1− x p (x), p(x) = c0− c1 + 1 2 (c2 − c3) + (c1 − c2+ 1 2 c3)x + (c2− c3)x 2+ c3 x3.      (3.12) In order to obtain the exact solution of equation (3.1), we substitute (3.2) into (1.3) which gives ϕ(x) = − 1 π2 r 1+ x 1− x ∫ 1 −1 r 1− t 1+ t c0 + c1 t + c2 t2+ c3 t3 t − x d t . (3.13) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 469 It is easy to see that ∫ 1 −1 r 1− t 1+ t 1 t − x d t =−π, ∫ 1 −1 r 1− t 1+ t t t − x d t =−π(x − 1), ∫ 1 −1 r 1− t 1+ t t2 t − x d t =−π(x2− x + 0.5), ∫ 1 −1 r 1− t 1+ t t3 t − x d t =−π(x3− x2+ 0.5 x − 0.5).                          (3.14) Using (3.14) into (3.13), we obtain the exact solution of equation (3.1) which is identical to the approximate solutions (3.12). 4. Particular Result Let us consider the integral equation ∫ 1 −1 ϕ(t) t − x d t + ∫ 1 −1 (x3+ t3)ϕ(t) d t = 3x3+ 2x2 + x , −1< x < 1 (4.1) and we seek the solution of this equation which is bounded at x =−1. From (2.4) and (3.6) yields fk = 1 π ∫ 1 −1 r 1− x 1+ x � 3t3+ 2t2+ t � Wk(t) d t = 1 π ∫ 1 −1 r 1− x 1+ x � 3 8 W3(t) + 1 8 W2(t)+ 9 8 W1(t)− 5 8 W0(t) � Wk(t)d t .        (4.2) Using (3.8) into (4.2), we have � f0 =− 5 8 , f1 = 9 8 , f2 = 1 8 , f3 = 3 8 � . (4.3) Due to (2.10) we get µ j,k = 1 π ∫ 1 −1 r 1− x 1+ x ∫ 1 −1 r 1+ t 1− t � x3+ t3 � Vj(t)Wk(x) d t d x . (4.4) M. Abdulkawi, Z. Eshkuvatov, and N. Nik Long / Eur. J. Pure Appl. Math, 2 (2009), (462-472) 470 Using orthogonal property (3.7) into (4.4) we obtain µ0,k = 1 π ∫ 1 −1 r 1− x 1+ x  πx3+ ∫ 1 −1 r 1+ t 1− t t3 V0 d t  Wk(x) d x (4.5) and µ j,k = 1 π ∫ 1 −1 r 1− x 1+ x   ∫ 1 −1 r 1+ t 1− t t3 Vj d t  Wk(x) d x , j = 1, 2, ..., n. (4.6) Due to (3.6-3.7), equation (4.5) becomes µ0,k = ∫ 1 −1 r 1− x 1+ x � x3+ 3 8 � Wk(x) d x (4.7) which gives µ0,0 = 0, µ0,1 = 3π 8 , µ0,2 = − π 8 , µ0,3 = π 8 . � (4.8) From (4.6) with help of (3.6-3.7), yields µ1,k = 3 8 ∫ 1 −1 r 1− x 1+ x Wk(x) d x (4.9) which gives µ1,0 = 3π 8 , µ1,k = 0, k = 1, 2, 3. � (4.10) Similarly, we obtain µ2,0 = µ3,0 = π 8 , µ2,k = µ3,k = 0, k = 1, 2, 3. ª (4.11) Due to (2.12), (4.3) and (4.8, 4.10-4.11) we have the following system of linear equations ak + 1 π 3 ∑ j=0 a j µ j,k = 1 π fk, k = 0, 1, 2, 3. (4.12) It is not difficult to see that the solution of the system (4.12) is a0 = − 71 55π , a1 = 177 110π , a2 =− 2 55π , a3 = 59 110π . � (4.13) REFERENCES 471 Substituting the values of the coefficients ¦ a j ©3 0 into (2.5) yields the approximate solution of equation (4.1) ϕn(x) = 1 55π r 1+ x 1− x � 236 x3 − 126 x2 + 63 x − 128 � (4.14) which is identical to the exact solution. 5. Conclusion The Chebyshev orthogonal polynomials of the third and fourth kinds are used to approximate the unknown density function which is bounded at the end point x = −1, and the known force function, respectively, for solving the Cauchy type sin- gular integral equation of the first kind. Theorem 3.1 shows the exactness of the approximate method presented for characteristic equation when the force function is a cubic. Particular result also shows that this approximate method does not only give the exact solution for characteristic equation but also for other Cauchy type singular integral equations of the first kind. ACKNOWLEDGEMENTS This work was supported by University Putra Malaysia un- der Graduate Research Fellowship (GRF). References [1] Gakhov, F.D.: Boundary value problems, Translation edited by Sneddon, I.N, Pergamon Press Ltd(1963). [2] Ladopoulos, E.G.: Singular integral equations, Linear and Non-Linear , Theory and its applications in Science and Engineering ,Springer-Verlag (2000). REFERENCES 472 [3] Martin, P.A. and Rizzo, F.J.: On boundary integral equations for crack problems, Proc. Roy. Soc. A, 421, 341-345 (1989). [4] Muskhelishvili, N.I.: Singular Integral Equations, Edited by J.R.M. Radok, Noordhoff In- ternational publishing Leyden (1977). [5] Z.K. Eshkuvatov, , N.M.A. Nik Long, M. Abdulkawi. Approximate solution of singular in- tegral equations of the first kind with Cauchy kernel. Appl. Math. Lett, 22, 651-657(2009). [6] M. Abdulkawi, Z.K. Eshkuvatov, N.M.A. Nik Long. A Note on the Numerical Solution of Singular Integral Equations of Cauchy type. IJAMC. 5: 2, 90-93(2009). [7] Lifanov, I. K.: Singular Integral Equation and Discrete Vortices, VSO, The Netherlands (1996). [8] Mason, J.C. and Handscomb, D.C.: Chebyshev polynomials, CRC Press LLC (2003).