Solving integral equations based on radial basis function interpolation

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1 Ð 3 Sept Communiction on Applied Mthemtics nd Computtion Vol.31 No.3 DOI /j.issn ÀÆ Þµ Å Đ Ò Î Í Ï ( ÄÇ ÏÁ ±Å Ø Ä ) Â ÕÉ ¹ (rdil bsis function, RBF) º Æ Þ ÊÆ ¹ ±Ý» RBF Á Á ß»Á Á Á Æ Ï ÊÆ¹ Ð ±Ý RBF ß º Þ Ø ÜÃËµ (multiqudric, MQ) ¹ Ñ Í «Ð ÒÒ ÚÀ RBF Ü» ¹ ½ «½ ºÖ Û Ú ² «½ Æ RBF ºÚ² Ã ³ «ÉÆ ºÆ Ø Ô½ Ü½ Fredholm ¼ Volterr Æ ±¹Å ßÞ ØÔ ¼ÒÒ «ßÔ Ð Æ Ï ß Ü Á Á ÕÉ ¹ (rdil bsis function, RBF) º Ø ÜÃËµ (multi-qudric, MQ); º ß 2010 Ö ß 45A05; 45G10 È¾Ö ß O ÌÇ A ÃËß (2017) Solving integrl equtions bsed on rdil bsis function interpoltion ZHANG Huiqing, CHEN Yu, NIE Xin (The Stte Key Lbortory of Trnsmission Equipment nd System Sfety nd Electricl New Technology, Chongqing University, Chongqing , Chin) Abstrct In order to cquire numericl method with high precision chrcteristic nd multi-dimensionl dptbility for solving integrl equtions, the rdil bsis function (RBF) is introduced. Specificlly, the unknown function is firstly expressed s liner combintion of RBF, then the integrl eqution is trnsformed to discrete liner or nonliner equtions through the colloction method, nd finlly, n pproximte representtion for the unknown function is obtined fter determining the weights of RBF. The multi-qudric (MQ) function is dopted due to its superior interpoltion performnce. In ddition, solving three or higher dimensionl integrl equtions re esily implemented becuse the RBF is function of distnce nd the only chnging is the distnce formul. The Guss qudrture formul, the geometric splitting nd numericl integrl methods re employed ³Ú ; Ð ²Æ Ç½ ( ) ¼ ÌÄ Â ĐÊ À»Ä E-mil: zhnghuiqing@cqu.edu.cn

2 Ð in clculting the integrl in RBF interpoltion mtrix. Numericl exmples s Fredholm or Volterr equtions in one or two-dimension hve been crried out. The results show tht the proposed method is esily to implement, nd it hs the dvntge of high precision nd convenience. Therefore, it is new nd suitble method for solving integrl equtions. Key words liner integrl eqution; nonliner integrl eqution; rdil bsis function (RBF) interpoltion; multi-qudric (MQ); numericl integrtion method 2010 Mthemtics Subject Clssifiction 45A05; 45G10 Chinese Librry Clssifiction O » º Ð º Ð ÂÔ Ð Ð Ï ¹ ¹ ÓÐ ¹ Â Ð ÂÔ Ë ¹ ßÙ Ï ÅÑÑ ¹ Ð Ó Úº f(p) = µ k(p, Q)F[f(Q)]dQ + g(p), P, Q Ω, (1) Ω Ú g(p) º Æ k(p, Q) ºº F[f(Q)] ºÉÅ Ö Ω º È µ ºÖ ¼ Û F[f(Q)] Ú ºÀ» À º Fredholm» Volterr ¹ Þ É Þ ¹Å Þ Â Þ Ô º Þ ¼ ÎÐÞ ¹Ð Ð ¹Å È Þ Þ ¾ Hr Ëµ Å Ó À Fredholm Ð [1] ; Hr ËµÅ Ó¼ [2]»Û¼ [3] À Ð ÌÂ Æ«Dubechie Ëµ ¾È ÞÅ Û À Volterr Ð [4] ; Đ Sinc ½ÞÅ Ó¼ Ó À [5]» À Fredholm Ð [6] ; Å± ÞÅ Ó Volterr Ð [7], Å±» ¾ÞÅ À Fredholm-Volterr Ð [8] ; Chebyshev ½ÐÞÅ Û¼À Ð [9] Đ Å± ÞÅ Û¼ Fredholm-Volterr Ð [10] ; ÆÚ [11] «Å º Û Fredholm Å» Ú Chebyshev ÆÚ Ë µ Ì ÆÚ È Ø ¹ «Ëµ ¹ Â Ô Æ Û¼ ¼ ¾ ¹ ßÁ ¾ ÑÓ ¹ÑÑÅ ¾ ¼ ÔÈ (rdil bsis function, RBF) Â Þ 1968 Hrdy ¼ Û ¹ Ô È¼ ¹ÑÑ Å ßÙ ¹ Ï Đ [12] Kns [13] ÂÏ RBF Đ Ð Å ¹ Ó µ Þ ¼ RBF ¹ Ç RBF Å Ð ÓÝ»³ Ù Ï RBF Ð Ï Ð Å Ù [14-15] Ï Đ Å Ó¼ Û À Ð À - Ð [16] À

3 3 Á Å ÕÉ ¹ º Æ 277 Volterr-Fredholm-Hmmerstein Ð [17] À Ð [18-20]. Ò» Å Å Û¼ È Û Fredholm Ð [21]. Ð Ö ¾Ë Î ßÁ ĐÏ RBF ÐÞÐ Ï Ð Å RBF ¹ RBF Å À À Ð ÐÞ Ó Â ÐÆ µó¼ Û¼ ¹ ÐÞ Â 1 RBF ÆÁ RBF φ: R + R ( Û º R d ) Þ ÕÆ ß r = x x j [22]. RBF Û Ð φ(r) = r 2 lnr Å φ(r) = e βr2 Hrdy ÛÂÊ (multi-qudric, MQ) Frnke 1982 µ MQ ÐÞÞ 29 Û ¹ÐÞ ¾ Â ßÁ MQ Åº ¹ ÅÚº φ(x) = x c 2 + α 2, (2) Ú c Þ α º ¼ Ó «ÈßÂ Í Õ Ì α = βh, h º Èß β Ù ¼ RBF ¹ØÞ RBF À À¾ Ü¾ Ô ÓÀ Û RBF Üº (x m, f(x m )), m = 1, 2,, N, Ï ¹È Ú f(x) = λ i φ i (x) = λ i (x ci ) 2 + α 2, (3) f(x m ) = λ i φ i (x m ) = λ i (xm c i ) 2 + α 2, (4) Ù± Üº [Φ d ][λ] = [f d ], (5) Ú [f d ] = [f(x 1 ), f(x 2 ),, f(x N )] T º Û ¹ [λ] = [λ 1, λ 2,, λ N ] T ºÉ [Φ d ] º ¹Ù± Ù± ÆÂ Úº Φ mn = (x m c n ) 2 + α 2. Å Ú (5) ßÁ Ú x ¹ f(x) º [λ] = [Φ d ] 1 [f d ]. (6) f(x) = [Φ(x)][λ] = [Φ(x)][Φ d ] 1 [f d ]. (7)

4 Ð 2 RBF ÓÐ Å ØÓ Á RBF Å Ð ÅÅ RBF ÞÅ Ð ÉÅ RBF À À¾ Ü È Ð Đ ½ÐÞ Î Ù ¾ Ð ½Ð Ù± ÆÞ RBF Â Ú Ù Þ Ð RBF ØÝ À» À Ð RBF Å µ 2.1 ÙÔ RBF Ó¼À Ð Ó Úº f(x) = µ b x k(x, t)f(t)dt + g(x), x [, b], (8) Ú b x Ü º b º x, ²ÔĐ Fredholm Volterr Ð b º Í» RBF Õ ÏÉÅ f(x) Üº RBF À¾ À ¾ [,b]»ó x m À Ð Íº b λ i φ i (x m ) µ λ i k(x m, t)φ i (t)dt = g(x m ), x m [, b]. (9) Ì M Æ Ð Ù± Úº [Φ µk][λ] = [G], (10) Ú Φ mn = (x m c n ) 2 + α 2, K mn = b k(x m, t) (t c n ) 2 + α 2 dt, [G] = [g(x 1 ), g(x 2 ),, g(x M )] T, x m, c n ²ÔĐº ½» RBF ½ x m» c n Ì Ó ßÁ Æ M» Æ N Â Φ Ù±ºÐ± Ù± K mn º» Å Å Ú 1 1 h(ξ)dξ Q A q h(ξ q ). (11) q=1 µ t=(b + )/2+(b )ξ/2=p(ξ) ÈÈ Â Úº K mn = b b 2 k(x m, t)φ n (t)dt = b k[x m, p(ξ)]φ n [p(ξ)]dξ Q A q k[x m, p(ξ q )]φ n [p(ξ q )]. (12) q=1

5 3 Á Å ÕÉ ¹ º Æ ÙÔ RBF Û¼À Ð Ó Úº f(x, y)=µ k(x, y, t, s)f(s, t)dsdt + g(x, y), (x, y) Ω, (13) RBF ¹ Úº λ i φ i (x m, y m ) = µ λ i Ω Ω k(x m, y m, s, t)φ i (s, t)dsdt + g(x m, y m ), (14) Ì M Æ ½ ¹Ù± [Φ µk][λ] = [G]. È² K mn = Ω k(x m, y m, s, t)φ n (s, t)dsdt ºÛ¼ Ú ² Ù» È Â ÐÆ (i) Ù È Â ÍÙ È Ω º [,b] [c,d] Õ Đ Û¼Å Å Ú µ t=(b+)/2+(b )ξ/2=p(ξ), s=(d + c)/2+(d c)η/2=q(η) ÈÈ K mn = d b c (b )(d c) 4 k(x m, y m, s, t)φ n (s, t)dsdt n 1 n 2 A i A j k[x m, y m, p(ξ i ), q(η j )]φ n [p(ξ i ), q(η j )]. (15) j=1 (ii) È Â Í È Ω º È Õ» Å Ú» ¹Å ÐÞ ³ Þ Óµ» ¹ ÐÞ Ï È À½ Â ²Æ È Å» ¹ µ K mn = k(x m, y m, s, t)φ n (t, s)dsdt = k(x m, y m, s, t)φ n (t, s)dsdt I i, (16) Ω i Ω i i Ú Ω i Ü i Æ È Û¼ È Ë Đ ßÙ» Í È Ô i Æ Ë Đ ÛÂ ¹ Ú I i = S 3 i k(x m, y m, x icj, y icj )φ n (x icj, y icj ), (17) 3 j=1 Ú Æ (x ij,y ij )(j = 1, 2, 3) Â Ë S i Æ (x icj, y icj ). 2.3 Õ ÙÔ RBF Ó¼ À Ð Ó Úº f(x) = µ b x k(x, t)f[f(t)]dt + g(x), x [, b]. (18)

6 Ð b º» RBF ÏÉÅ f(x) Üº RBF À¾ λ i φ i (x) = µ b N ] k(x, t)f λ i φ i (t) dt + g(x), x [, b]. (19) [, b]»ó x m º Ú Üº λ i φ i (x m ) µ b N ] k(x m, t)f λ i φ i (t) dt g(x m ) = 0, x m [, b]. (20) Ì M Æ À Ð À Ù±Úº [Φ][λ] µ[k] [G] = 0, (21) Ë K m = [ b ] k(x m, t)f λ i φ i (t) dt = b h(t)dt Þ Î [λ] À» ÈÐÞÅ Ô È ²Ó¹ ÛÎ [λ] = [λ 1, λ 2,, λ N ] T Í ¹ Ú (21) ÆÐ Â Å K m Å Ú K m = b 2 b q=1 2.4 Õ ÙÔ RBF N ] k[x m, p(ξ)]f λ i φ i (p(ξ)) dξ Q N ] A q k[x m, p(ξ)]f λ i φ i (p(ξ)). (22) Û¼ À Ð Ó Úº f(x, y) = µ k(x, y, t, s)f[f(t, s)]dtds + g(x, y), (x, y) Ω, (23) RBF ¹ Úº Ω λ i φ i (x m, y m ) = µ k(x m, y m, t, s)f Ω [ N ] λ i φ i (t, s) dtds + g(x m, y m ). (24) Ì M Æ ½ À Ð À Ù± Ú [Φ][λ] µ[k] [G] = 0, K m Â Ô Ù È Đ Û¼Å Å Úº K m (b )(d c) 4 n 1 n 2 N A i A j k[x m, y m, p(ξ i ), q(η j )]F j=1 ] λ i φ i (p (ξ i ), q(η j )). (25)

7 3 Á Å ÕÉ ¹ º Æ 281 Ô È» ¹Å ÐÆ ÄÙ» K m = Ω = i N ] k(x m, y m, t, s)f λ i φ i (t, s) dtds N ] k(x m, y m, t, s)f λ i φ i (t, s) dtds, (26) Ω i N ] k(x m, y m, t, s)f λ i φ i (t, s) dtds Ω i = S i 3 3 N ] k(x m, y m, x icj, y icj )F λ i φ i (x icj, y icj ). (27) j=1 2.5 RBF «ÙÔ ± ÏÉ Ï RBF Å À Ð Ó¹ ¾ (i) RBF ½Ò½ ÉÅ Û Ò½ RBF c, ¼ α ½ x m ; (ii) Î Ð À Ï RBF ¹ ÚÈ Ð Đ Í Ð À À Ð Õ Ù± Úº [Φ µk][λ] = [G], À ÕÂĐº [Φ][λ] µ[k] [G] = 0; (iii) Â Ù± Æ Ô Ó¼ Û¼Ù È Đ Å ÞÂ «ÙÔ Û¼È È Â È Å»«(iv) Â Î ÔÀ Ð Đ SVD Å ¹Ù±Ð [Φ µk][λ] = [G], Ù Ô À Ð Đ Ö ÈÞÅ [Φ][λ] µ[k] [G] = 0; (v) Â ÉÅ f(x) RBF Ú f(x) f(x) = [Φ(x)][λ]. 3 Æ ¹ Ø ¹ Ó¼Ã Fredholm Ð» Volterr Ð Û¼Ù» È Ð º ÐÞ ÑÑ» Ð ¼ Ò f e (x k )» f cl (x k ) ² Ü x k ¹»Â ¹ Ð Ûº φ RMS = 1 [f e (x k ) f cl (x k )] N 2. (28) 3.1 ÙÔ «º 1 Ó¼ Volterr À Ð f(x) + x 0 k=1 (x t)f(t)dt = 1, 0 x 1,

8 Ð Ð º f(x)=cos(x).» Ì RBF Èß h=0.1, ¼ β=10, º 11; ¹ Â» 60 Å Ú«ÌÈß h t = Æ¾ ÐÞ Â ³º ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º Ù [15] ÌÂ ¼» Æ¾ Ð º Ë ÐÞ Ð Ì Å ÑÑ [23] Ó¼À Ð x f(x) + e xs f(s)ds = g(x), 1 g(x) = e 4x + 1 x + 4 (ex(x+4) e (x+4) ), 1 x 1, º f(x) = e 4x. Å È [ 1,1] Èß h = 1 3 Ì ¼ α=8.8, ¹ Â» 7 Å Ú ¾Èßº h t =0.2. ÐÞ Â ³º ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º 1.563, ¹Ù± Í Èß h=0.25 Ì» ¼ α=4.2,» 9 Å Ú ¾Èßº h t =0.2. ÐÞ Â ³º ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º , ¹ Ù± Í º Ë ÐÞ ( ¹» 7 9 Å )» [23] ÐÞ (Legendre- Guss-Lobtto ) ÂÆ Ô Ó» 6 8 Å Æ Ú Ù RBF ¹ ÓÉ Ø ± ÏÅ¼ Í RBF N=10 Õ Ð Å Í 1 [23] Ñ É Ñ Ý Ê N=6 N=8 ß [23] ß Ó¼ Fredholm À Ð 1 e (x+1)t f(t)dt = g(x), 0 g(x) = 1 0 x 1, ex+1 (x + 1) 2 + ex+1 x + 1, Ð º f(x) = x. Ì RBF Èß h=0.1, ¼ β=20, Æ º 11;» 20 Å Â ³ 2 Ü [1]» Hr ËµÐÞ Í Ñ j=4

9 b3b 9I ^? qq?- 4.\B ~ N 283 q }: , JN~p: , JNJp: ; U j=6 q J , / u 3}z[`'p , / U 2 48 [1] mk Q4mk yir K do# Z T*{ (j=4) T*{ (j=6) 4~{ w0" P H A }? Fredholm Z 1Z f (x, y) = u(x, y) + 1 (t sin(s) + 1)f (s, t)dsdt, 0 0 u(x, y) = x cos(y) sin(1)(3 + sin(1))/6, 0 6 x, y 6 1, }?[?Yb. f (x,y) = xcos(y). #3 T RBF 3X P{ h=0.1, [>0 β=35, ; /3X Æ. 121 w0 A 3 x, y }P T 20 Æ A 2 P{. h =0.05, #`'$, 2 [JN~p: , } : , Jp :6JN b[:6r(/jp:6r(`$ 1 x 3E [3] Hr S)}z (α=4, m=32) q[jn~p: t \ ;i6\ 2 1

10 Ð 4 Û¼ È Volterr Ð f(x, y) (t sin(s) + 1)f(s, t)dsdt = xcos(y) , Ω Ω = {(x, y) R 2 : 0 y 1, 0.25 (y 0.5) 2 x } 1 4(y 0.5) 2, Ð º f(x,y) = xcos(y). Ì RBF» 2 Ü º 35, ¼ α=βh=2.5. Ω º È Đ ÍÏ º Æ Ë» Æ Ì Ë Åº¾ Ù± Æ Â È ¹» ÛÂ Ú ³ ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º Ù [20] Ì 33 Â Õ Ð º ,» Å Å Ð º ¾ 2 RBF È ¾² 3.2 Õ Ô «º 5 Ó¼ Fredholm À Ð f(x) = x 1 Ð º f(x) = 2 x 2. Ì RBF Èß h= t f(t)dt (2 2 1)x x 2, 0 x 1,, ¼ β=8, º 10; Â» 10 Å Ú«Ì¾Èß h t =0.05. Ð ÞÂ ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º ; Ù [6] ÐÞ» ÐÞ ¾ Ô ³Ë 3. Ë RBF ÐÞ Ð Ì Å ÑÑ ÐÞ Å Æ

11 3 Á Å ÕÉ ¹ º Æ 285 Í 3 [6]» Ñ É Ñ Ý x t [6], N=10 ß ÃÇĐÕ Ó¼ Volterr À Ð f(x) = e 2x x 0 [f 2 (t)+f(t)]dt, 0 x 1, Ð º f(x) = e x. Ì ¼ β=7,» 10 Å º» [2] ³Ä ÕÂÌ RBF Èß h= 1 3, h=1 7, h= 1 15» h= 1 31, ¾Èß h t= Ì«À Æ Â ³ 3 Þ» ÐÞ ÂÆ Ô ³ Ô º» [8] ÄÔ Ì RBF Èß h = 1 5, Â ³ 5 Ü 3 4» 5 Ë Ç Æ ÅÂ ÑÑ RBF ÐÞ ÑÑ³ Í 4 ÊÎ½ Û» Ý N=4 N=8 N=16 N=32 RMS ÃÇĐÕ ÃÇÃÕ ¾ 3 [2]» Ñ É Ñ Ë ¾

12 Ð Í 5 [8]» Ñ É Ñ Ý Ê x t [8], M=3, N=6 ß N= [23] À Ð x f(x) + e x 3s [f 2 (s)]ds = g(x), 1 1 g(x)= 2(1+36π 2 ) [(1+36π2 )e x e x cos(6πx)+6πe x sin(6πx) 36eπ 2 ]e x +e x sin(3πx), Ð º f(x) = e x sin(3πx). Å È [ 1, 1] Èß h = 1 3 Ì» «¼ α=0.5; ¹ Â ²» 7 Å Ú«¾Èßº h t =0.1; ÐÞ¾ ÂÆ Ì» ¼ α=1.3,» 9 Ô º Èß h = 1 4 Å Ú ¾Èßº h t =0.1, ÐÞ¾ ÂÆ Ô º Í 6 É Ñ [23] Ñ Ë Ê N=6 N=8 ß [23] ß ÐÞ ÂÆ Ô À Ð Å Õ [23] Ñ ß Å RBF À Ð ÈÐÞÐ Æ ÄÙ Â ÑÑ Å Ù» MATLAB ÑÑ Û ÜÅ Ð 7 Û¼ È Volterr À Ð y x f(x, y) = g(x, y) + (x + y t s)f 2 (t, s)dtds, 0 0 g(x, y) = x + y 1 12 xy(x3 + 4x 2 y + 4xy 2 + y 3 ), Ð º f(x, y) = x + y. 0 x, y 1, Ì RBF Èß h=0.2, ¼ β=5, Æ º 36 x, y ÐÈ Ì 10 ÆÅ ¾Èßº h t =0.1, º 121. ³ ¾

13 3 Á Å ÕÉ ¹ º Æ 287 ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º ¹ Ô Ê»ÂÔ Ê 4 Ü [3] Hr ËµÐÞ (α = 4, m = 32) Õ ÂÆ Ô º , ÐÞ ÑÑ ¾ 4 ¾ 8 Û¼ È Volterr À Ð x f(x, y) Ω 1 + y (1 + t + s)f2 (t, s)dtds = g(x, y), x g(x, y) =, (1 + x + y) y Ð º f(x, y) = 1/(1 + x + y) 2. È Ω 5 Ü Ì RBF» Æ º 38, ¼ α = βh = Đ ÍÏ È º 288 Æ Ë» 185 Æ Ó¹Ñ Ï 657 Æ Åº¾ Â Ô Ë È ¹» ÛÂ Ú ³ ¾ ÂÆ Ô º , Ð º , ÂÔ ÂÆ¹º ¾ 5 RBF È ¾² 4 Ï RBF Ð Ï Ð ½ RBF ÐÞÅ Ð Ô ½ Ð À Ù± Æ Â ÐÆ Ð ÀÅ ÐÞ Ì

14 Ð ¾Ô RBF ÌÐÞ Þ Å ÐÆ ÑÑ «¹ Þ Ä Ö Ì (i) Ê RBF Å Ð ÅÅ Ï RBF Ð Ï Ð Å RBF À À¾ ÉÅ Û ÔĐÀ À È Ð À ²Đ SVD ÈÐÞ Î Ù Ð (ii) Ð ÀÔĐ Ù± ÆÂ Ô Ó¼»Û¼Ù È» Å Å ÐÆ«ÙÔ È» ¹ ÐÆ Ó¼ Û¼ À» À Ð Å ³ ÐÞÝ ÓÐ»ÑÑ Å Þ ¾ Ð Å ÎÐÞ (iii) MQ ¹ Ý Ð Ì Å ÑÑ ßÙÂ Ë Ó ¹ RBF ÐÞ Ð Õ Ð ÅÏ 10 5 ÑÑ Ù RBF Þß ¼ Å¼ ¹Õ ß Â Ú Þ Åº ßÙ ± ÏÅ¼ Ð Å [1] Rbbni M, Mleknejd K, Aghzdeh N, Mollpoursl R. Computtionl projection methods for solving Fredholm integrl eqution [J]. Applied Mthemtics nd Computtion, 2007, 19: [2] Aziz I, Sirj-ul-Islm. New lgorithms for the numericl solution of nonliner Fredholm nd Volterr integrl equtions using Hr wvelets [J]. Journl of Computtionl nd Applied Mthemtics, 2013, 239: [3] Bbolin E, Bzm S, Lim P. Numericl solution of nonliner two-dimensionl integrl equtions using rtionlized Hr functions [J]. Commun Nonliner Sci Numer Simult, 2011, 16: [4] Sberi-Ndjfi J, Mehrbinezhd M, Akbri H. Solving Volterr integrl equtions of the second kind by wvelet-glerkin scheme [J]. Computers nd Mthemtics with Applictions, 2012, 63: [5] Mleknejd K, Mollpoursl R, Alizdeh M. Convergence nlysis for numericl solution of Fredholm integrl eqution by Sinc pproximtion [J]. Commun Nonliner Sci Numer Simult, 2011, 16: [6] Mleknejd K, Nedisl K. Appliction of Sinc-colloction method for solving clss of nonliner Fredholm integrl equtions [J]. Computers nd Mthemtics with Applictions, 2011, 62: [7] Mleknejd K, Rhimi B. Modifiction of block pulse functions nd their ppliction to solve numericlly Volterr integrl eqution of the first kind [J]. Commun Nonliner Sci Numer Simult, 2011, 16: [8] Mirzee F, Hoseini A A. Numericl solution of nonliner Volterr-Fredholm integrl equtions using hybrid of block-pulse functions nd Tylor series [J]. Alexndri Engineering Journl,

15 3 Á Å ÕÉ ¹ º Æ , 52(3): [9] Bbolin E, Abbsbndy S, Ftthzdeh F. A numericl method for solving clss of functionl nd two dimensionl integrl equtions [J]. Applied Mthemtics nd Computtion, 2008, 198: [10] Bbolin E, Mleknejd K, Mordd M, Rhimi B. A numericl method for solving Fredholm- Volterr integrl equtions in two-dimensionl spces using block pulse functions nd n opertionl mtrix [J]. Journl of Computtionl nd Applied Mthemtics, 2011, 235: [11] Xie W J, Lin F R. A fst numericl solution method for two dimensionl Fredholm integrl equtions of the second kind [J]. Applied Numericl Mthemtics, 2009, 59: [12] Hrdy R L. Multiqudric equtions of topogrphy nd other irregulr surfces [J]. Journl of Geophysicl Reserch, 1971, 76: [13] Kns E J. Multiqudrics A scttered dt pproximtion scheme with pplictions to computtionl fluid dynmics I. Surfce pproximtions nd prtil derivtive estimtes [J]. Computers nd Mthemtics with Applictions, 1990, 19(8/9): [14] Golbbi A, Seifollhi S. Numericl solution of the second kind integrl equtions using rdil bsis function networks [J]. Applied Mthemtics nd Computtion, 2006, 174: [15] Golbbi A, Seifollhi S. An itertive solution for the second kind integrl equtions using rdil bsis functions[j]. Applied Mthemtics nd Computtion, 2006, 181: [16] Golbbi A, Seifollhi S. Rdil bsis function networks in the numericl solution of liner integro-differentil equtions [J]. Applied Mthemtics nd Computtion, 2007, 188: [17] Prnd K, Rd J A. Numericl solution of nonliner Volterr-Fredholm-Hmmerstein integrl equtions vi colloction method bsed on rdil bsis functions [J]. Applied Mthemtics nd Computtion, 2012, 218: [18] Alipnh A, Dehghn M. Numericl solution of the nonliner Fredholm integrl equtions by positive definite functions [J]. Applied Mthemtics nd Computtion, 2007, 190: [19] Golbbi A, Mmmdov M, Seifollhi S. Solving system of nonliner integrl equtions by n RBF network [J]. Computers nd Mthemtics with Applictions, 2009, 57: [20] Assri P, Adibi H, Dehghn M. A numericl method for solving liner integrl equtions of the second kind on the non-rectngulr domins bsed on the meshless method [J]. Applied Mthemticl Modelling, 2013, 22(37): [21] Alipnh A, Esmeili S. Numericl solution of the two-dimensionl Fredholm integrl equtions using Gussin rdil bsis function [J]. Journl of Computtionl nd Applied Mthemtics, 2011, 235: [22] ÕÉ ¹ Ü ¹ º [J]. 2002, 19(2): [23] Tng T, Xu X, Cheng J. On spectrl methods for Volterr integrl equtions nd the convergence nlysis [J]. J Comput Mth, 2008, 26(6):