cubic b-spline interpolation method for singular integral equations ...

More documents

Recommendations

Info

The method described here uses Langrange interpolation on “practical” abscissas xk = cos( kπ / n), k = 0( 1) n. The related results are given below: Let Ln ( f ; x) denote the Lagrange polynomial of degree n interpolating f (x) at = kπ n k = We may write x k cos( / ), 0( 1) n. n n ∑ k k = 0 L ( f ; x) = l ( x) f ( x ), (4) where the Langrange fundamental polynomials lk (x) are given by n lk ( x) ( 2bk / n) b jT j ( xk ) T j ( x), = ∑ j= 0 k k = 0( 1) n, 1 b = where 0 = n 2 (5) b and bk = 1, k = 1( 1) n −1. (x) T j are Chebyshev polynomials of the first kind. That lk ( x j ) = δ kj follows from the discrete orthogonality of the Chebyshev polynomials at the practical abscissas. For details, please see Chawla & Kumar (1979). For latest works on B-Spline, please see, Liu (1997), Prautzsch (2004) and Reif (2000). In past methods for solution of singular integral equations with logarithmic singularities have been also discussed in Alaylioglu & Lubinsky (1984), Atkinson & Sloan (1991), Atkinson (1988), Chakrabarti & Manam (2003), Saranen & Sloan (1992), Saranen (1991), Sloan & Burn (1992), Kress & Sloan (1993) and Manam (2003). 2. The Numerical Method we consider the singular integral equation of the first kind with a logarithmic singularity in its kernel. 1 ∫ −1 w( t) [ ln t − x + k( t, x) ] g( t) dt = f ( x), −1 ≤ x ≤ 1, where the kernel k( t, x) and the right hand side function f (x) are regular functions on −1, 1 and w(t) is the weight function [ ] 2 − 1 2 w ( t) = ( 1− t ) (7) In past a method for the integral equation (6) have been discussed in Ioakimidis (1981). In this chapter, we have described a cubic B-spline approximation method for the numerical solution of (6), using uniqueness condition of the form 1 ∫ −1 w ( t) g( t) dt = C (8) where C is a known constant . Here, we have evaluated the integrals 1 ( 4) ( 4) L( Bn, j ; x) w( t) Bn, j ( t) ln t − x dt, = ∫ −1 m = 1( 1) 4, j = 1( 1) n ( 4) using the representation of Bn , j ( t) given below: (6) (9)
B ( 4) n, j 3 ⎧ ( t − t j ) ⎪ , t j ≤ t ≤ t j+ 1, ⎪( t j+ 3 − t j )( t j+ 1 − t j )( t j+ 2 − t j ) ⎪ ⎪ ( t − t j ) ⎡ ( t − t j )( t j+ 2 − t) ( t j+ 3 − t)( t − t j+ 1) ⎤ ⎪ ⎢ + ⎥ ( t j+ 3 − t j ) ⎢( 2 )( 2 1) ( 3 1)( 2 1) ⎪ ⎣ t j+ − t j t j+ − t j+ t j+ − t j+ t j+ − t j+ ⎥⎦ ⎪ 2 (t j+ 4 − t)( t − t j+ 1) ⎪ + , t j+ 1 ≤ t ≤ t j+ 2 , ⎪ ( t j+ 4 − t j+ 1)( t j+ 2 − t j+ 1)( t j+ 3 − t j+ 1) ⎪ ⎪ ( t j+ 4 − t) ⎡ ( t − t j+ 1)( t j+ 3 − t) ( t j+ 4 − t)( t − t j+ 2 ) ⎤ ( t) = ⎨ ⎢ + ⎥ ⎪( t j+ 4 − t j+ 1) ⎢⎣ ( t j+ 3 − t j+ 1)( t j+ 3 − t j+ 2 ) ( t j+ 4 − t j+ 2 )( t j+ 3 − t j+ 2 ) ⎥⎦ ⎪ 2 ⎪ ( t − t j )( t j+ 3 − t) + , t j+ 2 ≤ t ≤ t j+ 3, ⎪ ( t j+ 3 − t j )( t j+ 3 − t j+ 1)( t j+ 3 − t j+ 2 ) ⎪ 3 ⎪ ( t j+ 4 − t) ⎪ , t j+ 3 ≤ t ≤ t j+ 4 , ⎪( t j+ 4 − t j+ 1)( t j+ 4 − t j+ 2 )( t j+ 4 − t j+ 3 ) ⎪ 0, otherwise. ⎪ ⎪ ⎩ (10) ≤ t ≤ t ≤ t ≤ ⋅⋅ ⋅ ≤ t ≤ 1. Thus ( 4) and replacing Bn , j ( t) by Lagrange interpolation at the “practical” abscissas xk = cos( kπ / n), k = 0( 1) n described in Chawla & Kumar (1979) and described here by equations (4) and (5). For derivation of (10), please see Prautzsch, Boehm & Paluszny [2002, pp. 61] & Phillips [2003, pp. 224]. The following results also have been used in the evaluation of singular integrals in (9): 1 ∫ ( 0 −1 2 − 1 2 1− t ) ln t − xT ( t) dt = −π ln 2 , (11) − 1 0 1 2 n+ 4 ( 4) { Bn, j ( t) } { } 4 n+ ( 4) t j and further B ( ) j= 0 n, j t over interval ( j , t j+ 4 ) is a cubic B-spline basis with knots is a cubic B-spline t . Now, we have approximated unknown function g (t) as: n g ∑ j n, j j= 0 ( 4) ( t) ≈ α B ( t) . (14) We get n equations from (6) using (14) n 1 ∑ ∫ − [ ] α j w( t) ln t − xi ( 4) + k( t, xi ) Bn, j ( t) dt 1 2 − 1 π 2 ∫ ( 1− t ) ln t − xT j ( t) dt = − T j ( x), j −1 j= 0 j ≥ 1, (12) by 1 choosing = f ( xi ), n collocation (15) points T j + 1( x) = 2xT j ( x) − T j−1 ( x); T0 = 1, T1 ( x) = x. where T j (x) are Chebyshev polynomials of the first kind. (13) −1 < x 0 < x1 < x2 < ⋅⋅ ⋅ < xn−1 < 1. Equation ( n + 1) th given below is obtained from equations (8) replacing g(t) by (14). Here, we have described a product integration method using cubic B-spline approximations for unknown function g(t). Here, we have taken n + 5 knots n 1 ( 4) ∑α j ∫ w( t) Bn, j ( t) dt = C . (16) j= 0 −1
Page 1: CUBIC B-SPLINE INTERPOLATION METHOD
Page 5: References [1] A. Alaylioglu and D.

cubic b-spline interpolation method for singular integral equations ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?