approximate solutions for mixed nonlinear volterra fredholm type integral equations via modified block pulse functions

Journal of the Association of Arab Universities for Basic and Applied Sciences (2012) 12, 65–73 University of Bahrain Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com ORIGINAL ARTICLE Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations via modified block-pulse functions Farshid Mirzaee *, Elham Hadadiyan Department of Mathematics, Faculty of Science, Malayer University, Malayer 65719-95863, Iran Available online 10 July 2012 KEYWORDS Mixed nonlinear Volterra– Fredholm type integral equations; Block-pulse functions; Operational matrix Abstract In this article a robust approach for solving mixed nonlinear Volterra–Fredholm type integral equations of the first kind is investigated By using the modified two-dimensional blockpulse functions (M2D-BFs) and their operational matrix of integration, first kind mixed nonlinear Volterra–Fredholm type integral equations can by reduced to a nonlinear system of equations The coefficients matrix of this system is a block matrix with lower triangular blocks Some theorems are included to show the convergence and advantage of this method Numerical results show that the approximate solutions have a good degree of accuracy ª 2012 University of Bahrain Production and hosting by Elsevier B.V All rights reserved Introduction In this paper we applied the direct method for solving mixed nonlinear Volterra–Fredholm type integral equations of the first Z Zkind of the form: x Gx; y; s; t; us; tịịdtds ẳ fx; yị; x; yị ẵ0; 1ị X; X ð1Þ * Corresponding author Tel./fax: +98 8513339944 E-mail addresses: f.mirzaee@malayeru.ac.ir, mirzaee@mail.iust.ac.ir (F Mirzaee) 1815-3852 ª 2012 University of Bahrain Production and hosting by Elsevier B.V All rights reserved Peer review under responsibility of University of Bahrain http://dx.doi.org/10.1016/j.jaubas.2012.05.001 Production and hosting by Elsevier where u(s,t) is an unknown function, f(x,y) and G(x,y,s,t,u(s,t)) are analytical function on [0,1) · X and [0,1) · X4, respectively, where X is a close subset on Rd d ẳ 1; 2; 3ị Existence and uniqueness results for Eq (1) may be found in (Diekmann, 1978; Pachpatte, 1978; Thieme, 1977) Equation of type (1) often arise from the mathematical modeling of the spreading, in space and time, of some contagious disease in a population living in a habitat X (Diekmann, 1978; Thieme, 1977), in the theory of nonlinear parabolic boundary value problems (Pachpatte, 1978), and in many physical and biological models The literature on numerical methods for solving Eq (1) mainly consists of projection methods, collocation methods, the trapezoidal Nystroăm method, Adomain decomposition method, Hes homotopy perturbation method and the twodimensional block-pulse functions (Adomian, 1990, 1994; Adomian and Rach, 1992; Biazar et al., 2011; Brunner, 1990; Cardone et al., 2006; Cherruault et al., 1992; Guoqiang, 1995; Hacia, 1996; Kauthen, 1989; Maleknejad and Fadaei Yami, 2006; Maleknejad and Hadizadeh, 1999; Maleknejad and Mahdiani, 2011; Wazwaz, 2006; Yee, 1993) 66 F Mirzaee, E Hadadiyan Assume now that: Gðx; y; s; t; us; tịị ẳ kx; y; s; tịẵus; tފp ; ð2Þ where p is a positive integer In the present paper, we apply a modification of block-pulse functions (Maleknejad and Rahimi, 2011), to solve the mixed nonlinear Volterra– Fredholm type integral Eq (1) with Eq (2) M2D-BFs and their properties Definition An (m + 1)2-set of M2D-BFs consists of (m + 1)2 functions which are defined over district D = [0,1) · [0,1) as follows: & /i1 ;i2 x; yị ẳ x; yị Di1 ;i2 ; otherwise: Now suppose that X be a (m + 1)2-vector Hence by using Eq (9) we obtain: e m;e x; yị; Um;e x; yịUTm;e x; yịX ẳ XU 10ị e ẳ diagXị is a (m + 1) · (m + 1) diagonal matrix where X 2 2.2 M2D-BFs expansions A function f(x,y) defined over district L2(D) may be expanded by the M2D-BFs as: m X m X fx; yị fm;e x; yị ẳ fi1 ;i2 /i1 ;i2 x; yị i1 ẳ0 i2 ẳ0 ẳ ẳ UTm;e ðx; yÞFm;e ; FTm;e Um;e ðx; yÞ ð11Þ i1 ; i2 ẳ 01ịm; 3ị where Fm,e is an (m + 1) · vector given by Fm;e ¼ ½f0;0 ; ; f0;m ; ; fm;0 ; ; fm;m ŠT ; where Di1 ;i2 ẳ fx; yịjx Ii1 ;e ; y Ii2 ;e g, and a ¼ 0; > < ẵ0; h eị Ia;e ẳ ẵah e; a ỵ 1ịh eị a ẳ 11ịm; > : ẵ1 e; 1ị a ẳ m: 4ị m where m is an arbitrary positive integer, and h ¼ Since, each M2D-BF takes only one value in its subregion, the M2D-BFs can be expressed by the two modified onedimensional block-pulse functions (M1D-BFs): /i1 ;i2 ðx; yÞ ¼ /i1 ðxÞ/i2 ðyÞ; ð5Þ where /i1 ðxÞ and /i2 ðyÞ are the M1D-BFs related to variables x and y, respectively The M2D-BFs are disjointed with each other: & /i1 ;i2 x; yị i1 ẳ j1 ; i2 ẳ j2 ; 6ị /i1 ;i2 x; yị/j1 ;j2 x; yị ẳ otherwise: and are orthogonal with each other: Z 1Z /i1 ;i2 ðx; yÞ/j1 ;j2 ðx; yÞdydx &0 MIi1 ;e ịMIi2 ;e ị i1 ẳ j1 ; i2 ¼ j2 ; ¼ otherwise: ð12Þ and Um,e(x,y) is defined in Eq (8), and fi1 ;i2 , are obtained as: Z Z fx; yịdydx: 13ị fi1 ;i2 ẳ MðIi1 ;e ÞMðIi2 ;e Þ Ii1 ;e Ii2 ;e Similarly a function of four variables, k(x,y,s,t), on district L2(D · D) may be approximated with respect to M2D-BFs such as: kðx; y; s; tÞ ’ UTm;e ðx; yÞKm;e Um;e ðs; tÞ; ð14Þ where Um,e(x,y) and Um,e(s,t) are M2D-BFs vector of dimension (m + 1)2, and Km,e is the (m + 1)2 · (m + 1)2 M2D-BFs coefficients matrix Convergence analysis In this section, we show that the given method in the previous À1Á sections, is convergent and its order of convergence is O km For our purposes we will need the following theorems Theorem Let ð7Þ fm;e x; yị ẳ m X m X fi1 ;i2 /i1 ;i2 x; yị; i1 ẳ0 i2 ẳ0 where (x,y) D, i1,i2,j1,j2 = 0(1)m and MðIi1 ;e Þ and MðIi2 ;e Þ are length of intervals Ii1 ;e and Ii2 ;e , respectively and fi1 ;i2 ¼ 2.1 Vector forms MðIi1 ;e ÞMðIi2 ;e Þ Z Z fðx; yÞ/i1 ;i2 ðx; yÞdxdy; i1 ; i2 ¼ 0ð1ÞðmÞ: Consider the first (m + 1) terms of M2D-BFs and write them concisely as (m + 1)2-vector: Um;e x; yị ẳ ẵ/0;0 x; yị; ; /0;m ðx; yÞ; ; /m;0 ðx; yÞ; ; T /m;m ðx; yފ ; ðx; yÞ D: ð8Þ B B B T Um;e x; yịUm;e x; yị ẳ B B @ 15ị achieves its minimum value and also we have Whence Eqs (6) and (8) implies that: Then the following equation Z 1Z ðfðx; yÞ À fm;e ðx; yÞÞ2 dxdy; /0;0 ðx; yÞ 0 /0;1 ðx; yÞ 0 /m;m ðx;yÞ Z C C C C C A : mỵ1ị2 mỵ1ị2 9ị Z f2 x; yịdxdy ẳ X X f2i1 ;i2 k/i1 ;i2 x; yịk2 : 16ị i1 ẳ0 i2 ¼0 Proof It is an immediate consequence of theorem which was proved by Jiang and Schaufelberger (1992) h Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations Theorem Assume f(x,y) is continuous and is differentiable over district [Àh,1 + h] · [Àh,1 + h], and fm;ei ðx; yÞ; ei ¼ ihk ; for i = 0(1)(k À 1), are correspondingly M2D-BFs(e0) = 2DBFs, M2D-BFs(e1), , M2D-BFs(ekÀ1) expansions of f(x, y) based on (m + 1)2 M2D-BFs over district D and fm;k x; yị ẳ k iẳ0 lim fm x; yị ẳ lim fm;e0 x; yị ẳ fx; yị: m!ỵ1 Proof See (Maleknejad et al., 2010) @f @f Proof consider ðx; yÞ and @y ðx; yÞ in the district @x i1 iỵ1We i1 iỵ1 which are approximately equal to con; ; m m m m stants n1 and n2, respectively, where m is so large Also, we use n1xÁ+ n2y + b instead of f(x,y) in the district i1 page z= iỵ1 i1 iỵ1 i i ; ỵ e1 m m  im ;i m ÂÁ m ; m Now in the district ; ỵ e1 we have: m m    kÀ1 kÀ1 &  X 1X i lh i lh n1 ỵ n2 fm;el ¼ k l¼0 m k m k l¼0     i lh i ỵ lh ỵ b ỵ n1 ỵ n2 m k m k     i ỵ lh i lh ỵ n2 ỵ b ỵ n1 m k m k     ' i ỵ lh i ỵ lh ỵ n2 þb þ b þ n1 m k m k  i iỵ1 ỵ n ỵ n ịhk 1ị ; 17ị ẳ n1 ỵ n2 ị m m ỵ b 2k fm;k x; yị ẳ k Theorems and conclude that error estimation for M2D1 BFs is kex; yịk ẳ O km If we assume E1 and E2 are errors between f(x,y) and its 2D-BFs and M2D-BFs expansions, respectively, from Theorem we have Ep2ffiffi k1 E1 , and from (Maleknejad et al., 2010) we have E1 m2M, where M is bounded of iDf(x,y)i and m shows number of 2D-BFs So, we have p 2M E2 ẳ kex; yịk ; ð21Þ km where k is times of modifications of the M2D-BFs series Assume now that f(x,y) is approximated by fm;ei x; yị ẳ m X m X fi1 ;i2 /i1 ;i2 x; yị; i1 ẳ0 i2 ẳ0 whereas, fi1 ;i2 are the approximation of fi1 ;i2 and fm;ei ðx; yÞ ¼ m X m X fi1 ;i2 /i1 ;i2 ðx; yị; i1 ẳ0 i2 ẳ0 18ị kfi1 ;i2 /i1 ;i2 x; yị fx; yịk ẳ kfi1 ;i2 /i1 ;i2 ðx; yÞ À fðx; yÞ À fi1 ;i2 /i1 ;i2 x; yị ỵ fi1 ;i2 /i1 ;i2 x; yịk In other words: max x;y2ẵmi ;mi ỵe1 ị kfi1 ;i2 /i1 ;i2 x; yị fx; yịk ỵ kfi ;i / ðx; yÞ jfðx; yÞ À fm;k ðx; yÞj i i max jn1 x ỵ n2 y þ b À fm;k ðx; yÞj K jn1 þ n2 i i m m x;y2ẵm;mỵe1 ị  ỵ b fm;k x; yịj n1 ỵ n2 ịh ; 2k so, we have: ẳ max jfx; yị max kfx; yị fm;ei ðx; yÞk1 P ei ei ðx; yÞ D ðx; yÞ D0 i i À fm;ei ðx; yÞj jn1 ỵ n2 ỵ b m m &   i i i i n1 ỵ n2 ỵ b ỵ n1 ỵ n2 ỵh m m m m     i i i ỵ h ỵ n2 ỵ b ỵ n1 ỵh ỵ bỵn1 m m m   '  n1 ỵ n2 ịh i ỵ n2 ỵ h ỵ b  ẳ ; m  Á  Á where D0 ¼ mi ; mi ỵ h mi ; mi ỵ h By using Eqs (19) and (20) the proof is completed h then for ðx; yÞ Di1 ;i2 we have ẳ mi ỵ h and Eq (17) can be reformulated as: i n1 ỵ n2 ịh : fm;k x; yị ẳ n1 ỵ n2 ị ỵ b ỵ m 2k ex; yị ẳ fx; yị fm x; yị: m!ỵ1 kfx; yị fm;k x; yịk1 K maxkfx; yị fm;ei x; yịk1 : k ei iỵ1 m Theorem Let the representation error between f(x,y) and its two-dimensional block-pulse functions, fm x; yị ẳ fm;e0 x; yị (M2D-BFse0 ị ẳ 2D BFsị, over the district D, as follows: Then kex; yịk ẳ Om1 ị and kÀ1 X fm;ei ðx; yÞ; then for sufficient large m we have: but 67 i1 ;i2 À fi1 ;i2 /i1 ;i2 ðx; yÞk: ð22Þ We have ð19Þ kfi1 ;i2 /i1 ;i2 ðx; yÞ À fi1 ;i2 /i1 ;i2 ðx; yÞk ¼ !12  ðfi1 ;i2 /i1 ;i2 ðx; yÞ À fi1 ;i2 /i1 ;i2 ðx; yÞÞ dydx Z Z Ii1 ;ei Ii2 ;ei ¼ jfi1 ;i2 À fi1 ;i2 j !12 Z Z dydx Ii1 ;ei Ii2 ;ei ¼ MðIi1 ;ei ÞMðIi2 ;ei Þjfi1 ;i2 À fi1 ;i2 j MðIi ;e ÞMðIi ;e Þkfm À fk : i ð20Þ i ð23Þ Consequently by using Eqs (21)(23), the following error bound is obtained: p 2M ỵ MðIi1 ;ei ÞMðIi2 ;ei Þkfm À fk1 : ð24Þ kfi1 ;i2 /i1 ;i2 À fðx; yÞk km Moreover Eq (24) implies that: h lim fm;ei x; yị ẳ fx; yị: m!ỵ1 25ị 68 F Mirzaee, E Hadadiyan ẵux; yịpỵ1 ẳ ux; yịẵux; yịp Method of solution ẳ UTm;e Um;e ðx; yÞUTm;e ðx; yÞUm;e;p In this section, we solve mixed nonlinear Volterra–Fredholm type integral equations of the first kind of the form Eq (1) with Eq (2) by using M2D-BFs We now approximate functions u(x,y),f(x,y),[u(x,y)]p and k(x,y,s,t) with respect to M2D-BFs by manipulation as Section 2: uðx; yÞ ’ UTm;e ðx; yÞUm;e ; > > > > < fðx; yÞ ’ UT ðx; yÞFm;e ; m;e > ðuðx; yÞÞp ’ UTm;e ðx; yÞUm;e;p ; > > > : kðx; y; s; tÞ ’ UTm;e ðx; yÞKm;e Um;e ðs; tị; e m;e;p Um;e x; yị: ẳ UTm;e U 29ị Now by using Eq (28) we obtain h iT pỵ1 pỵ1 pỵ1 e m;e;p ẳ upỵ1 ; UTm;e U 0;0 ; ; u0;m ; ; um;0 ; ; um;m ð30Þ therefore Eq (28) holds for (p + 1), and the lemma is established h To approximate the integral part in Eq (1) with Eq (2), from Eq (26) we get Z xZ kx; y; s; tịẵus; tịp dtds 26ị where Um,e(x,y) is defined in Eq (8), the vectors Um,e, Fm,e, Um,e,p, and matrix Km,e are M2D-BFs coefficients of u(x,y),f(x,y), [u(x,y)]p and k(x,y,s,t) respectively ’ Lemma Let (m + 1)2-vectors Um,e and Um,e,p be M2D-BFs coefficients of u(x,y) and [u(x,y)]p, respectively If Um;e ẳ ẵu0;0 ; ; u0;m ; ; um;0 ; ; um;m ŠT ; Z ¼ x Z UTm;e ðx; yÞKm;e Um;e ðs; tÞUTm;e ðs; tÞUm;e;p dtds UTm;e ðx; yÞKm;e Z x Z  Um;e ðs; tÞUTm;e ðs; tÞdtds Um;e;p : 31ị 27ị then we have: h iT Um;e;p ẳ up0;0 ; ; up0;m ; ; upm;0 ; ; upm;m ; Now by using Eqs (5) and (9), denoting Rj for the (j + 1)th row of the conventional integration operational matrix Pm,e ((Pm,e)(m+1)·(m+1) is operational matrix of 1D-BFs defined overR [0,1), see Maleknejad and Mahdiani, 2011) and consider1 ing /i tịdt ẳ MIi;e ị follows: 28ị where p P 1, is a positive integer Proof (By induction) When p = 1, Eq (28) follows at once from [u(x,y)]p = u(x,y) Suppose that Eq (28) holds for p, Z x Z Um;e ðs; tÞUTm;e ðs; tÞdtds 0Rx R1 B B B B B ¼B B B B @ 0 /0 ðsÞ/0 ðtÞdtds 0 B B B B B B B B B ¼B B B B B B B B @ ðh À eÞR0 Um;e ðxÞ Rx R1 / ðsÞ/ m ðtÞdtds 0 hR0 Um;e ðxÞ 0 C C C C C C C C C A /m sị/m tịdtds mỵ1ị2 mỵ1ị2 ðh À eÞRm Um;e ðxÞ C C C C C C C C C C C C C C C C C A eR0 Um;e ðxÞ Rx R1 we shall deduce it for (p + 1) Since [u(x,y)]p+1 = u(x,y) [u(x,y)]p, from Eqs (26) and (10) it follows that eRm Um;e xị 32ị : mỵ1ị2 mỵ1ị2 Also by using Eq (5), Eq (8) can be reformulated as: Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations /0 ðxÞ B B B B B B B B Um;e x; yị ẳ B B B B B B @ /0 ðxÞ /m ðxÞ 69 C C C C C C C C : ẵ/0 yị; ; /m ðyÞ; ; /0 ðyÞ; ; /m yịTmỵ1ị2 : C C C C C C A /m xị mỵ1ị2 mỵ1ị2 33ị So, we have UTm;e x; yịKm;e ẳ ẵ/0 yị; ; /m ðyÞ; ; /0 yị; ; /m yịmỵ1ị2 k1;1 /0 xị k1;mỵ1ị /0 xị k1;mmỵ1ị /0 ðxÞ B B B B B kmỵ1ị;1 / xị kmỵ1ị;mỵ1ị / xị kmỵ1ị;mmỵ1ị / xị 0 B B B B B B B kmmỵ1ị;1 /m xị kmmỵ1ị;mỵ1ị /m xị kmmỵ1ị;mmỵ1ị /m xị B B B @ kmỵ1ị2 ;1 /m xị kmỵ1ị2 ;mỵ1ị /m xị kmỵ1ị2 ;mmỵ1ị /m xị k1;mỵ1ị2 /0 xị C C C C kmỵ1ị;mỵ1ị2 /0 ðxÞ C C C C : C C C kmmỵ1ị;mỵ1ị2 /m xị C C C C A kmỵ1ị2 ;mỵ1ị2 /m xị mỵ1ị2 mỵ1ị2 ð34Þ Also, we have: heị > < /0 xị ỵ h eị/1 xị ỵ ỵ h eị/m xị; Ri Uxị ẳ h2 /i xị ỵ h/iỵ1 xị ỵ ỵ h/m xị; > :e / xị; m and & /i xị; i ẳ j : /i xị/j xị ẳ 0; otherwise iẳ0 i ẳ 11ịm 1ị ; iẳm 35ị By using Eqs (32), (34) and (35), Eq (31) can be reformulated as: ẵ/0 yị; ; /m ðyÞ; ; /0 yị; ; /m yịmỵ1ị2 1 A00 0 B C C B A10 A11 B C ð36Þ B C :B A20 A21 A22 :Um;e;p ; C B C B C @ A Am0 Am1 Am2 Amm mỵ1ị2 mỵ1ị2 where Ai;j ẳ MI ị j;e > < MðIr;e Þklz /i ðxÞ; > : MðIj;e ÞMðIr;e Þklz /i ðxÞ; and is a zero matrix Also A00 0 B C B A10 A11 C B C B A20 A21 A22 C B C B C B C @ A Am0 Am1 Am2 Amm mỵ1ị2 mỵ1ị2 /0 xị B B B B B /0 ðxÞ B B B ¼ B B B /m ðxÞ B B B @ 0 /m ðxÞ C C C C C C C C C C C C C C A :Q; mỵ1ị2 mỵ1ị2 38ị iẳj ; otherwise where l ẳ m ỵ 1ịi ỵ 1ị1ịm ỵ 1ịi ỵ 1ịị; z ẳ m ỵ 1ịj ỵ 1ị1ịm ỵ 1ịj ỵ 1ịị; ! z ; r ẳ z m þ 1Þ ðm þ 1Þ ð37Þ where Q00 BQ B 10 B B Q ¼ B Q20 B B @ Q11 Q21 0 Q22 0 Qm0 Qm1 Qm2 Qmm C C C C C C C A mỵ1ị2 mỵ1ị2 ; 39ị 70 F Mirzaee, E Hadadiyan Table Numerical results of Example with M2D-BFs Table Nodes ðx; yÞ Error for m ẳ x; yị ẳ 2l k=1 k=2 k=3 l=1 l=2 l=3 l=4 0.03748936 0.05090571 0.02574872 0.04112879 0.02837304 0.03943282 0.01813532 0.02553757 0.02522153 0.03190341 0.01391462 0.02222506 MðIj;e Þ MðIr;e Þklz ; MIj;e ịMIr;e ịklz ; iẳj : otherwise Nodes x; yị Present method x; yị ẳ 2l m = and k = Method of Maleknejad and Mahdiani (2011) m = 16 l=1 l=2 l=3 l=4 0.02837304 0.03943282 0.01813532 0.02553757 0.0288649 0.0398778 0.0310669 0.0277814 So, we have : Z xZ kx; y; s; yịẵus; tịp dtds UTm;e ðx; yÞQUm;e;p : ð40Þ 0 Error(x,y) 0.15 0.1 0.05 0.8 0.6 y 0.4 0.2 0.2 0.4 0.6 0.8 x (a) m = and k = 0.12 0.09 Error(x,y) Qi;j ¼ 0.06 0.03 0.8 0.6 y 0.4 0.2 0 0.2 0.6 0.4 0.8 x (b) m = and k = 0.06 Error(x,y) ( Error results for Example 0.04 0.02 0.8 0.6 y 0.4 0.2 0 0.2 0.4 0.6 x (c) m = and k = Figure Absolute value of error, Example with m = and k = 1,2,3 0.8 ð41Þ Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations um;e x; yị ẳ UTm;e Um;e x; yÞ: Numerical results of Example with M2D-BFs Table Nodes x; yị x; yị ẳ k=1 k=2 k=3 l=1 l=2 l=3 l=4 0.04289052 0.06470576 0.03803418 0.03592263 0.03148406 0.04585369 0.02437943 0.02836157 0.02743874 0.04071913 0.02249845 0.02045372 uðx; yÞ ’ um;k x; yị ẳ ẳ UTm;e x; yịQUm;e;p ) Fm;e ẳ QUm;e;p : k1 1X um;ei x; yị; k iẳ0 44ị where ei ẳ ihk ; i ẳ 01ịk 1ị is the estimation of the solution of mixed nonlinear Volterra–Fredholm type integral equation of the first kind Substituting Eqs (26) and (41) into Eq (1) with Eq (2) gives: UTm;e ðx; yÞFm;e ð43Þ Then Error for m = Àl 71 Numerical examples ð42Þ In this section to demonstrate the effectiveness of our approach several examples are presented All results are computed by using a program written in the Matlab The After solving the above nonlinear system by using Newton– Raphson method, we can find Um,e and then Error(x,y) 0.15 0.1 0.05 0.8 0.6 y 0.4 0.2 0 0.2 0.4 0.6 0.8 x (a) m=8 and k = 0.08 Error(x,y) 0.06 0.04 0.02 0.8 0.6 0.4 y 0.2 0 0.2 0.4 0.6 0.8 x (b) m = and k = 0.05 Error(x,y) 0.04 0.03 0.02 0.01 0.8 0.6 y 0.4 0.2 0 0.2 0.4 0.6 x (c) m = and k = Figure Absolute value of error, Example with m = and k = 1,2,3 0.8 72 F Mirzaee, E Hadadiyan Table Error results for Example Nodes x; yị Present method x; yị ẳ 2l m = and k = Method of Maleknejad and Mahdiani (2011) m = 16 l=1 l=2 l=3 l=4 0.03148406 0.04585369 0.02437943 0.02836157 0.04289006 0.04589757 0.04034072 0.04312157 answer is the coefficients of M2D-BFs expansion of the solution of mixed nonlinear Volterra–Fredholm type integral equation Also, we have shown that À our Á approach is convergent and its order of convergence is O km This method can be easily extended and applied to mixed nonlinear Volterra–Fredholm type integral equations of the second kind and nonlinear system of the mixed Volterra–Fredholm type integral equations References numerical experiments are carried our for the selected grid point which are proposed as (2Àl; l = 1,2,3,4) and m terms and k times of modifications of the M2D-BFs series The following problems have been tested Example Consider the following mixed linear Volterra– Fredholm type integral equation (Maleknejad and Mahdiani, 2011): Z x Z cosy tịesx us; tịdtds ẳ fx; yị; x; yị ẵ0; 1ị X; 45ị where fx; yị ẳ xex cosyị ỵ sin2 yị þ sinðyÞÞ: ð46Þ The exact solution is u(x,y) = eÀxcos(y) Table and Fig illustrate the numerical results for this example The error results for proposed method besides the error for method of Maleknejad and Mahdiani (2011) are tabulated in Table Example Consider the following mixed nonlinear Volterra– Fredholm type integral equation (Maleknejad and Mahdiani, 2011): Z x Z t ỵ yịe2sx u2 s; tịdtds ẳ fx; yị; x; yị ẵ0; 1ị X; ð47Þ where 1 fðx; yÞ ẳ xyex ỵ xex xyex2 xex2 : 4 ð48Þ The exact solution is u(x,y) = eÀxÀy Table and Fig illustrate the numerical results for this example The error results for proposed method besides the error for method of Maleknejad and Mahdiani (2011) are tabulated in Table Conclusion In this paper a computational method for approximate solution of mixed nonlinear Volterra–Fredholm type integral equations of the first kind, based on the expansion of the solution as series of M2D-BFs was presented This method 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