Semiorthogonal Quadratic B-Spline Wavelet Approximation for Integral Equations

Transcription

1 Mathematical Sciences Vol. 3, No. (29) 99- Semiorthogonal Quadratic B-Spline Wavelet Approximation for Integral Equations Mohsen Rabbani a,, Nasser Aghazadeh b a Department of Mathematics, Islamic Azad University (Sari Branch), Sari, Iran b Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz , Iran. aghazadeh@iust.ac.ir Abstract Compactly supported quadratic semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of Fredholm integral equations of the second kind. First, the quadratic B-spline wavelets and their properties are presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Keywords: Wavelets, Semiorthogonal, Quadratic B-spline, Fredholm Integral equation, Second kind. c 29 Published by Islamic Azad University-Karaj Branch. Introduction In the present paper, we apply compactly supported quadratic semiorthogonal (SO) B-spline wavelets, specially constructed for the bounded interval to solve the second kind linear Fredholm integral equations of the form: y(x) k(x, t)y(t)dt = g(x), x () Corresponding Author. Address: mrabbani@iust.ac.ir

2 Mathematical Sciences Vol. 3, No. (29) where k(x, t) and g(x) are known functions and y(x) is unknown function to be determind. In recent years, the application of methods based on wavelets have influenced many areas of applied mathematics. In areas such as the numerical analysis of differential equations, wavelets are recognized as a powerful tool. Another area in which the wavelet is gaining considerable attention is the study of integral equations. Wavelets can be separated into two distinct types; orthogonal and semiorthogonal 2]. In 8], the two categories of wavelets, orthogonal and semiorthogonal, are compared, and it is shown that semiorthogonal wavelets are best suited for integral equation applications. Our method consist of reducing () to a set of algebraic equations by expanding the unknown function as quadratic B-spline wavelets with unknown coefficients. The properties of these wavelets are then utilized to evaluate the unknown coefficients. 2 B-spline scaling functions and wavelets on, ] When semiorthogonal wavelets are constructed from B-spline of order m, the lowest octave level j = j is determined in 3] by 2 j 2m, (2) so as to give a minimum of one complete wavelet on the interval, ]. In this paper, we will use a wavelet generated by a quadratic B-spline, m = 3 cardinal B-spline function. From (2), the third-order B-spline lowest level, which must be an integer, is determined to j = 3. This constrains all octave levels to j 3. As in the case with all semiorthogonal wavelets, the third-order B-spline also serve

3 M. Rabbani and N. Aghazadeh as scaling functions. The third-order B-spline scaling functions are given by φ j,k (x) = 2 (x j k) 2, k x j k + ; 3 4 ((x j k) 3 2 )2, k + x j k + 2; 2 ((x j k) 3) 2, k + 2 x j < k + 3, k =,..., 2 j 3;, otherwise. (3) with the respective left and right hand side boundary scaling functions φ j,k (x) = 2 (x j k) 2, x j ; k = 2, otherwise. (4) φ j,k (x) = 3 4 ((x j k) 3 2 )2, k + x j k + 2; 2 ((x j k) 3) 2, k + 2 x j k + 3, k = ;, otherwise. (5) φ j,k (x) = 2 (x j k) 2, k x j k + ; 3 4 ((x j k) 3 2 )2, k + x j k + 2; k = 2 j 2;, otherwise. (6) φ j,k (x) = 2 ((x j k) 3) 2, k + 2 x j k + 3, k = 2 j ;, otherwise. (7) The actual coordinate position x is related to x j, according to x j = 2 j x. (also, you can

4 2 Mathematical Sciences Vol. 3, No. (29) see ]) The third-order B-spline wavelets are given by ψ j,k (x) = 48 2(x j k) 2, k x j k + 2 ; (x j k) 62(x j k) 2, k + 2 x j k + ; (x j k) + 42(x j k) 2, k + x j k ; (x j k) 252(x j k) 2, k x j k + 2; 6( (x j k) + 352(x j k) 2 ), k + 2 x j k ; 6( (x j k) + 352(x j k) 2 ), k x j k + 3; 2( (x j k) + 626(x j k) 2 ), k + 3 x j k ; (x j k) 42(x j k) 2, k x j k + 4; (x j k) + 62(x j k) 2, k + 4 x j k ; 2((x j k) 5) 2, k x j k + 5. (8) for k =,,, 2 j 5. And the respective left and right hand side boundary wavelets are: ψ j,k (x) = 48 6( (x j k) + 352(x j k) 2 ), k + 2 x j k ; 6( (x j k) + 352(x j k) 2 ), k x j k + 3; 2( (x j k) + 626(x j k) 2 ), k + 3 x j k ; (x j k) 42(x j k) 2, k x j k + 4; (x j k) + 62(x j k) 2, k + 4 x j k ; 2((x j k) 5) 2, k x j k + 5. (9)

5 M. Rabbani and N. Aghazadeh 3 for k = 2, ψ j,k (x) = 48 for k =, ψ j,k (x) = 48 for k = 2 j 4 and ψ j,k (x) = (x j k) + 42(x j k) 2, k + x j k ; (x j k) 252(x j k) 2, k x j k + 2; 6( (x j k) + 352(x j k) 2 ), k + 2 x j k ; 6( (x j k) + 352(x j k) 2 ), k x j k + 3; 2( (x j k) + 626(x j k) 2 ), k + 3 x j k ; (x j k) 42(x j k) 2, k x j k + 4; (x j k) + 62(x j k) 2, k + 4 x j k ; 2((x j k) 5) 2, k x j k + 5. () 2(x j k) 2, k x j k + 2 ; (x j k) 62(x j k) 2, k + 2 x j k + ; (x j k) + 42(x j k) 2, k + x j k ; (x j k) 252(x j k) 2, k x j k + 2; 6( (x j k) + 352(x j k) 2 ), k + 2 x j k ; 6( (x j k) + 352(x j k) 2 ), k x j k + 3; 2( (x j k) + 626(x j k) 2 ), k + 3 x j k ; (x j k) 42(x j k) 2, k x j k + 4; () 2(x j k) 2, k x j k + 2 ; (x j k) 62(x j k) 2, k + 2 x j k + ; (x j k) + 42(x j k) 2, k + x j k ; (x j k) 252(x j k) 2, k x j k + 2; 6( (x j k) + 352(x j k) 2 ), k + 2 x j k ; 6( (x j k) + 352(x j k) 2 ), k x j k + 3; (2)

6 4 Mathematical Sciences Vol. 3, No. (29) for k = 2 j 3. 3 Function approximation For any fixed positive integer M, a function f(x) defined over, ] may be presented by B-spline scaling functions as where f(x) = 2 M k= 2 s k φ M,k = S T Φ M (3) with s k = S = s 2, s,..., s 2 ] M, ] Φ M = φ M, 2, φ M,,..., φ M,2 M, (4) f(x) φ M,k (x)dx, k = 2,,..., 2 M, (5) where φ M,k (x) are dual functions of φ M,k (x). These can be obtained by linear combinations of φ M,k (x), k = 2,,..., 2 M as follows. Let Φ M be the dual functions of Φ M given by Using (4) and (6), we get Φ M = φm, 2, φ M,,..., φ M,2 M ]. (6) where I is (2 M + 2) (2 M + 2) identity matrix. Let P M = Φ M Φ T Mdx = I, (7) The entry (P M ) i,j of the matrix P M in (8) is calculated from From (7) and (8), we get Φ M Φ T Mdx. (8) φ M,i (x)φ M,j (x)dx. (9) Φ M = (P M ) Φ M. (2)

7 M. Rabbani and N. Aghazadeh 5 Furthermore, a function f(x) defined over, ] may be presented by B-spline wavelets as f(x) = 7 k= 2 c k φ 3,k (x) + 2 i 3 i=3 j= 2 d i,j ψ i,j (x). (2) If the infinite series in (2) is truncated at M, then (2) can be written as f(x) 7 k= 2 c k φ 3,k (x) + M 2 i 3 i=3 j= 2 d i,j ψ i,j (x) = C T Ψ(x), (22) where φ 3,k and ψ i,j are scaling and wavelets functions, respectively, and C and Ψ are (2 M+ + 2) vectors given by C = c 2, c,..., c 7, d 3, 2, d 3,,..., d 3,5,..., d M, M+,..., d M,2 M 3] T, (23) Ψ = φ 3, 2, φ 3,,..., φ 3,7, ψ 3, 2, ψ 3,,..., ψ 3,5,..., ψ M, M+,..., ψ M,2 M 3] T, (24) with c k = f(x) φ 3,k (x)dx, k = 2,,..., 7, (25) d i,j = f(x) ψ i,j (x)dx, i = 3, 4,..., M, j = 2,,..., 2 i 3, (26) where φ 3,k (x) and ψ i,j (x) are dual functions of φ 3,k (x) and ψ i,j (x) respectively. These can be obtained by linear combinations of φ 3,k (x), k = 2,..., 7 and ψ i,j (x), i = 3, 4,..., M, j = 2,,..., 2 i 3, as follows. Let Φ = φ 3, 2 (x), φ 3, (x),..., φ 3,7 (x)] T, (27) Ψ = ψ 3, 2 (x), ψ 3, (x),..., ψ M,2 M 3(x)] T. (28)

8 6 Mathematical Sciences Vol. 3, No. (29) Using (3)-(7) and (27) we get ΦΦ T dx = P 3 = and from (8)-(2) and (28) we have (29) Ψ Ψ T dx = P 2 (3) where P 3 and P 2 are and 2 M 2 M matrices, respectively. For M = 3 the P 2 matrix is: P 2 = Suppose Φ and Ψ are the dual functions of Φ and Ψ respectively, given by Φ = φ3, 2 (x), φ 3, (x),..., φ ] T 3,7 (x), (3)

9 M. Rabbani and N. Aghazadeh 7 Ψ = ψ3, 2 (x), ψ 3, (x),..., ψ ] T M,2 M 3(x). (32) Using (25)-(28),(3) and (32) we have ΦΦ T dx = I, Ψ Ψ T dx = I 2 (33) where I and I 2 are and 2 M 2 M identity matrices, respectively. Then (29), (3) and (33) give Φ = P3 Φ, Ψ = P 2 Ψ (34) 4 Second kind Fredholm integral equations In this section, we solve linear Fredholm integral equation of the second kind of the form () by using B-spline wavelets. For this we use (22) to approximate y(x) y(x) = C T 2 Ψ(x), (35) where Ψ(x) is defined in (24), and C 2 is (2 M+ +2) unknown vector defined similarly to C in (23). We also expand g(x) and k(x, t) by B-spline dual wavelets Ψ defined as in (3) and (32) as where From (35) and (36) we get g(x) = C T Ψ, k(x, t) = Ψ T (t)θ Ψ(x), (36) Θ (i,j) = k(x, t)y(t)dt = ] k(x, t)ψ i (t)dt Ψ j (x)dx. (37) Ψ T (t)θ Ψ(x)C T 2 Ψ(t)dt = C T 2 Θ Ψ(x) (38) By applying (35)-(38) in equation () we have: C T 2 Ψ(x) C T 2 Θ Ψ(x) = C T Ψ(x) (39)

10 8 Mathematical Sciences Vol. 3, No. (29) By multiplying (39) in Ψ T (x) and integrating respect to x we get: C T 2 P C T 2 Θ = C T, (4) in which P is a (2 M+ + 2) (2 M+ + 2) square matrix given by P = Ψ(x)Ψ T (x)dx = P 3 (4) P 2 and so C T 2 = CT (P Θ). Now we can calculate the solution with y(x) = C T 2 Ψ(x). 5 Conclusion In the present work, a technique has been developed for solving linear Fredholm integral equation of the second kind. The method is based upon compactly supported Quadratic semiorthogonal B-spline wavelets. The problem has been reduced to solving a system of linear algebraic equations that can be solved with any of usual methods. References ] Aghazadeh N., Maleknejad K. (27) Using quadratic B-spline scaling functions for solving integral equations, International Journal: Mathematical Manuscripts,, No., -6. 2] Chui C.K., An introduction to wavelets, wavelet analysis and its applications, Vol, Academic Press, Massachusetts, ] Goswami J.C., Chan A.K., Chui C.K. (995) On solving first-kind integral equations using wavelets on a bounded interval, IEEE Transactions on antennas and propagation, 43, ] Koro K., Abe K. (2) Non-orthogonal spline wavelets for boundary element analysis, Engineering Analysis with Boundary Elements, 25,

11 M. Rabbani and N. Aghazadeh 9 5] Lakestani M., Razzaghi M., Dehghan M. (25) Solution of nonlinear Fredholm- Hammerstein integral equations by using semiorthogonal spline wavelets, Mathematical Problems in Engineering, ] Lakestani M., Razzaghi M., Dehghan M. (26) Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Mathematical Problems in Engineering, Article ID 9684, 2 pages, 26. 7] Lakestani M., Razzaghi M., Dehghan M. (26) Numerical solution of the controlled Duffing oscillator by semi-orthogonal spline wavelets, Physics Scripta, 74, ] Nevels R.D., Goswami J.C., Tehrani H. (997) Semiorthogonal versus orthogonal wavelet basis sets for solving integral equations, IEEE Trans. Antennas Propagat, 45, No. 9,

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