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A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases

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A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases
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  REVIEW A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases Salih Yalc¸ınbas  * , Tug ˘c¸e Akkaya Department of Mathematics, Faculty of Science and Art, Celal Bayar University, Manisa, Turkey Received 14 December 2011; revised 14 February 2012; accepted 29 February 2012Available online 30 March 2012 KEYWORDS Mixed linear integro-differ-ential-difference equations;Collocations points;Matrix method;Boubaker series andpolynomials Abstract  In this paper, a new collocation method, which is based on Boubaker polynomials, isintroduced for the approximate solutions of mixed linear integro-differential-difference equationsunder the mixed conditions. The aim of this article is to present the applicability and validity of the technique and the comparisons are made with the existing results. The results demonstratethe accuracy and efficiency of the present work.   2012 Ain Shams University. Production and hosting by Elsevier B.V.All rights reserved. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542. Fundamental relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542.1. Matrix relations for the differential part  D ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1552.2. Matrix relations for the differential-difference part  F  ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562.3. Matrix relations for the integral part  I  ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562.4. Matrix relations for the mixed conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563. Method of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 *Corresponding author.E-mail addresses: salih.yalcinbas@bayar.edu.tr (S. Yalc¸inbas  ),tugce.akkaya@bayar.edu.tr (T. Akkaya).2090-4479    2012 Ain Shams University. Production and hosting byElsevier B.V. All rights reserved.Peer review under responsibility of Ain Shams University.http://dx.doi.org/10.1016/j.asej.2012.02.004 Production and hosting by Elsevier Ain Shams Engineering Journal (2012)  3 , 153–161 Ain Shams University Ain Shams Engineering Journal www.elsevier.com/locate/asejwww.sciencedirect.com  4. Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 1. Introduction In recent years, there has been a growing interest in the numer-ical treatment of the integro-differential-difference equations(IDDEs), which are combinations that the unknown functionsappear under the sign of integration and it also contain thederivatives and functional arguments of the unknown func-tions. The problem’s integral part can be classified into Fred-holm and Volterra integral equation. The upper bound of the region for integral part of Volterra type is variable, whileit is a fixed point number for that of Fredholm type. In thisstudy, we deal with Fredholm integro-differential-differenceequations, but all algorithms in our study can be adapted toVolterra integro-differential-difference equations with a littlemodification. Integro-differential-difference equations are sig-nificant branch of modern mathematics and arise frequentlyin many applied areas including engineering, mechanics, phys-ics, chemistry, astronomy, biology, economics, potential the-ory, electrostatics, etc. [1–12]. The mentioned IDDEs areusually difficult to solve analytically, so the approximate meth-ods and the numerical methods are required.In recent years, both mathematicians and physicists havedevoted considerable effort to the study of numerical solutionsof the integro-differential-difference equations. Many power-ful methods have been presented. For instance, Chebyshev col-location [13,14], Taylor collocation [15] and Bessel collocation [16], Tau method [17–20], Legendre wavelet [21], Sine–Cosine wavelets [22], finite difference method [23], Rationalized Haar functions method [24], Cas wavelet [25], differential transform [26], Homotopy perturbation [27], the Monte Carlo method [28] and the Direct method [29]. In this study, we will consider the following integro-differ-ential-difference equation, X mk ¼ 0 P k ð x Þ  y ð k Þ ð x Þ þ P nr ¼ 0 P  r ð x Þ  y ð r Þ ð x    s Þ¼  g ð x Þ þ Z   ba K  ð x ; t Þ  y ð t Þ dt ;  1  <  a 6 x ;  t 6 b 6 1 ð 1 Þ under the mixed conditions X m  1 k ¼ 0 ð a ik  y ð k Þ ð a Þ þ  b ik  y ð k Þ ð b Þ þ  c ik  y ð k Þ ð c ÞÞ ¼  k i  ; i   ¼  0 ; 1 ; . . . ; m    1 :  ð 2 Þ where the known functions  P k ð x Þ ; P  r ð x Þ ;  g ð x Þ ; K  ð x ; t Þ  are de-fined on interval  a 6 x ,  t 6 b ,  y ( x ) is an unknown functionand also  a ik ,  b ik ,  c ik  and  k i   are real or complex constants.The aim of this study is to get solution of the problem (1)and (2) as the truncated Boubaker series defined by  y ð x Þ ¼ X N n ¼ 0 b n B n ð x Þ ;  1  <  x 6 b  <  1 ð 3 Þ where  B n ( x ),  n  = 0, 1, 2,  . . .  denote the Boubaker polynomials; b n , 0 6 n 6 N are unknown Boubaker coefficients, and  N   ischosen any positive integer such that  N  P m . To find a nu-meric solution in the form (3) of the problem (1) and (2), weuse the collocation points defined by x i   ¼  a  þ  b    aN  i  ;  i   ¼  0 ; 1 ; . . . ; N  :  ð 4 Þ Here the standard Boubaker polynomials are defined by B 0 ð x Þ ¼  1 ; B 1 ð x Þ ¼  x ; B 2 ð x Þ ¼  x 2 þ  2 ; B m ð x Þ ¼  xB m  1 ð x Þ   B m  2 ð x Þ  for  :  m  >  2  and a monomial definition of these polynomials wasestablished by Labiadh and Boubaker [30]: B n ð x Þ ¼ X n ð n Þ  p ¼ 0 ð n    4  p Þð n    p Þ  C   pn   p    ð 1 Þ  p   x n  2  p ð 5 Þ where  n ð n Þ ¼  n 2    is denotes the floor function. 2. Fundamental relations Firstly, we can write the Boubaker polynomials B n ( x ) in thematrix form as follows  y ð x Þ ¼  B ð x Þ b  ð 6 Þ where B ð x Þ ¼ ½ B 0 ð x Þ B 1 ð x Þ B N  ð x Þ ;  b  ¼ ½ b 0 b 1   b N   T  : By using the expression (5) and taking  n  = 0, 1,  . . .  ,  N  , we findthe corresponding matrix relation as B ð x Þ ¼  X ð x Þ Z T  ð 7 Þ where X ð x Þ ¼ ½ 1 x  x N   and if   N   is odd, Z  ¼ u 0 ; 0  0 0 0    0 00  u 1 ; 0  0 0    0 0 u 2 ; 1  0  u 2 ; 0  0    0 0 ..................... u N   1 ; N   12 0  u N   1 ; N   32 0  . . .  u N   1 ; 0  00  u N  ; N   12 0  u N  ; N   32   0  u N  ; 0 26666666666643777777777775 if   N   is even, Z  ¼ u 0 ; 0  0 0 0    0 00  u 1 ; 0  0 0    0 0 u 2 ; 1  0  u 2 ; 0  0    0 0 ..................... 0  u N   1 ; N   22 0  u N   1 ; N   42 . . .  u N   1 ; 0  0 u N  ; N  2 0  u N  ; N   22 0    0  u N  ; 0 266666666664377777777775 154 S. Yalc¸ınbas  , T. Akkaya  where B n ð x Þ ¼ X n ð n Þ  p ¼ 0 u n ;  p x n  2  p ;  n  ¼  0 ; 1 ; . . . ; N  ;  p  ¼  0 ; 1 ; . . . ;  n 2 j k : u n ;  p  ¼ ð n    4  p Þð n    p Þ  C   pn   p   ð 1 Þ  p : Let us demonstrate Eq. (1) in the form D ð x Þ þ  F  ð x Þ ¼  g ð x Þ þ  I  ð x Þ ;  ð 8 Þ where D ð x Þ ¼ X mk ¼ 0 P k ð x Þ  y ð k Þ ð x Þ ; F  ð x Þ ¼ X nr ¼ 0 P  r ð x Þ  y ð r Þ ð x    s Þ ; I  ð x Þ¼ Z   ba K  ð x ; t Þ  y ð t Þ dt  ð 9 Þ We consider the solution  y ( x ) and its  k th derivative  y ( k ) ( x ) de-fined by a truncated Boubaker series (3). Then we can write its k th derivative in the matrix form  y ð k Þ ð x Þ ¼  B ð k Þ ð x Þ b ;  k  ¼  0 ; 1 ; 2 ; . . .  ð 10 Þ or from relations (6), (7) and (10), we obtain the following ma-trix forms  y ð x Þ ¼  X ð x Þ Z T  b  ð 11 Þ and  y ð k Þ ð x Þ ¼  X ð k Þ ð x Þ Z T  b :  ð 12 Þ To obtain the matrix  X ( k ) ( x ) in terms of the matrix  X ( x ), wecan use the following procedure: X ð 1 Þ ð x Þ ¼  X ð x Þ M 1 X ð 2 Þ ð x Þ ¼  X ð 1 Þ ð x Þ M  ¼  X ð x Þ M 2 X ð 3 Þ ð x Þ ¼  X ð 2 Þ ð x Þ M  ¼  X ð x Þ M 3 ... X ð k Þ ð x Þ ¼  X ð k  1 Þ ð x Þ M  ¼  X ð x Þ M k ð 13 Þ where M  ¼ 0 1 0    00 0 2    0 . . .    . . .    . . . 0 0 0    N  0 0 0    0 2666666437777775 : Consequently, by substituting the matrix form (12) and (13)into (10) we have the matrix relation  y ð k Þ ð x Þ ¼  X ð x Þ M k Z T  b :  ð 14 Þ By putting  x fi x    s  in the relation (6) we obtain the matrixform  y ð x    s Þ ¼  B ð x    s Þ b  ð 15 Þ Similar to (13), the relation between the matrix X( x    s ) and X ( x ) is X  ð x    s Þ ¼  X ð x Þ M  s  ð 16 Þ where M  s  ¼ 00   ð s Þ 0  10   ð s Þ 1  20   ð s Þ 2   N  0   ð s Þ N  011   ð s Þ 0  21   ð s Þ 1   N  1   ð s Þ N   1 0 022   ð s Þ 0   N  2   ð s Þ N   2 ............... 0 0 0    N N    ð s Þ 0 266666666666666664377777777777777775 : ð 17 Þ Differentiate both side of  (16) with respect to  x  and using therelation of  (13) we get X  ð r Þ ð x    s Þ ¼  X ð r Þ ð x Þ M  s  ¼  X ð x Þ M k M  s :  ð 18 Þ Thus, from (12) and (18), we can write  y ð r Þ ð x    s Þ ¼  X ð x Þ M k M  s Z T  b :  ð 19 Þ To obtain the desired solution, we can use Boubaker colloca-tion method; therefore, we substitute collocation points (4)into Eq. (8) and obtain the system D ð x i  Þ þ  F  ð x i  Þ ¼  g ð x i  Þ þ  I  ð x i  Þ :  ð 20 Þ So that D ð x i  Þ ¼ X mk ¼ 0 P k ð x i  Þ  y ð k Þ ð x i  Þ ;  F  ð x i  Þ¼ X nr ¼ 0 P  r ð x i  Þ  y ð r Þ ð x i     s Þ ;  I  ð x i  Þ ¼ Z   ba K  ð x i  ; t Þ  y ð t Þ dt  ð 21 Þ then we can write the system (20) in the matrix form D  þ  F  ¼  G  þ  I ;  ð 22 Þ where D  ¼ D ð x 0 Þ D ð x 1 Þ D ð x 2 Þ ... D ð x N  Þ 266666664377777775 ;  F  ¼ F  ð x 0 Þ F  ð x 1 Þ F  ð x 2 Þ ... F  ð x N  Þ 266666664377777775 ;  I  ¼ I  ð x 0 Þ I  ð x 1 Þ I  ð x 2 Þ ... I  ð x N  Þ 266666664377777775 ;  G  ¼  g ð x 0 Þ  g ð x 1 Þ  g ð x 2 Þ ...  g ð x N  Þ 266666664377777775 : ð 23 Þ 2.1. Matrix relations for the differential part D(x) To reduce the part  D ( x ) to the matrix form by means of thecollocation points (4), we first write the matrix  D  defined inEq. (23) as D  ¼ X mk ¼ 0 P k Y ð k Þ ð 24 Þ P k  ¼ P k ð x 0 Þ  0 0    00  P k ð x 1 Þ  0    00 0  P k ð x 2 Þ   0 ............... 0 0 0    P k ð x N  Þ 266666664377777775 ;  Y  ¼  y ð k Þ ð x 0 Þ  y ð k Þ ð x 1 Þ  y ð k Þ ð x 2 Þ ...  y ð k Þ ð x N  Þ 266666664377777775 :  ð 25 Þ By substituting collocation points (4) into Eq. (14), we have the matrix equation Y ð k Þ ¼  XM k Z T  b ;  ð 26 Þ A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases 155  where X  ¼ X  ð x 0 Þ X  ð x 1 Þ X  ð x 2 Þ ... X  ð x N  Þ 2666666437777775 ¼ 1  x 0  x 20    x N  0 1  x 1  x 21    x N  1 1  x 2  x 22    x N  2 ............... 1  x N   x 2 N     x N N  2666666437777775 :  ð 27 Þ Consequently, from the matrix forms (25) and (27), we obtainthe fundamental matrix relation for the differential part  D ( x ) D  ¼ X mk ¼ 0 PXM k Z T  b  ð 28 Þ 2.2. Matrix relations for the differential-difference part F(x) Secondly, Let us find the matrix  F  corresponding to the partF( x ). From (9) we can write F( x ) in matrix form F  ð x Þ ¼ X nr ¼ 0 P  r ð x Þ X ð x Þ M k M  s Z T  b :  ð 29 Þ or briefly F  ¼ X nr ¼ 0 P r XM k M  s Z T  b :  ð 30 Þ where P  r  ¼ P  r ð x 0 Þ  0 0    00  P  r ð x 1 Þ  0    00 0  P  r ð x 2 Þ   0 ............... 0 0 0    P  r ð x N  Þ 2666666437777775 :  ð 31 Þ 2.3. Matrix relations for the integral part I(x) Now we find the matrix  I  corresponding to the part  I  ( x ). Thekernel K( x ,  t ) can be expanded to the truncated Boubaker ser-ies in the form K  ð x ; t Þ ¼ X N r ¼ 0 h r ð x Þ B r ð t Þ¼  h 0 ð x Þ B 0 ð t Þ þ  h 1 ð x Þ B 1 ð t Þ þ  þ  h N  ð x Þ B N  ð t Þ ð 32 Þ Then the matrix representation of   K  ( x ,  t ) can be given as K  ð x ; t Þ ¼  H ð x Þ B T  ð t Þ ;  ð 33 Þ where H  ¼ ½ h 0 ð x Þ ; h 1 ð x Þ ; h 2 ð x Þ ; . . . ; h N  ð x Þ :  ð 34 Þ Substituting the matrix forms (33) and (6) corresponding to thefunctions  K  ( x ,  t ) =  H ( x ) B T  ( t ) and  y ( x ) =  X ( x ) Z T b  into theintegral part  I  ( x ), we get the matrix I  ð x Þ ¼ Z   ba H ð x Þ B T  ð t Þ B ð t Þ b dt ¼  H ð x Þ Z   ba B T  ð t Þ B ð t Þ dt   b  ¼  H ð x Þ Q b  ð 35 Þ in order to make easy the calculation of   I , we use the equationabove Q  ¼ Z   ba B T  ð t Þ B ð t Þ dt  ¼ ½ q ij   ;  i  ;  j   ¼  0 ; 1 ; 2 ; . . . ; N   ð 36 Þ where q ij   ¼  b i  þ  j  þ 1   a i  þ  j  þ 1 i   þ  j   þ  1  :  ð 37 Þ 2.4. Matrix relations for the mixed conditions We obtain the corresponding matrix forms for the conditions(2), by means of the relations (14), as X m  1 k ¼ 0 ½ a ik X ð a Þ þ  b ik X ð b Þ þ  c ik X ð c Þ M k Z T  b  ¼  k i  ; i   ¼  0 ; 1 ; . . . ; m    1  ð 38 Þ briefly, the matrix form for conditions (2) are U i  b  ¼ ½ k i   ;  ð 39 Þ where U i   ¼ X m  1 k ¼ 0 ½ a ik X ð a Þ þ  b ik X ð b Þ þ  c ik X ð c Þ M k Z T  ¼ ½ u i  0 ; u i  1 ; . . . ; u iN   ð 40 Þ 3. Method of solutions We now ready to construct the fundamental matrix equationcorresponding to Eq. (1). For this purpose, substituting thematrix relations (19), (29) and (35) into Eq. (1) and then sim-plifying, we obtain the fundamental matrix equation X mk ¼ 0 P k ð x Þ X ð x Þ M k Z T  þ X nr ¼ 0 P  r ð x Þ X ð x Þ M k M  s Z T    H ð x Þ Q ( ) b ¼  G : ð 41 Þ By plugging the collocation points  x i   defined by  x i   ¼  a þ b  aN   i  ;  i   ¼  0 ; 1 ; . . . ; N  :  We get the system of matrix equations X mk ¼ 0 P k ð x i  Þ X ð x i  Þ M k Z T  þ X nr ¼ 0 P  r ð x i  Þ X ð x i  Þ M k M  s Z T   H ð x i  Þ Q ( ) b ¼ G ;  i  ¼ 0 ; 1 ; ... ; N  or briefly the fundamental matrix equation X mk ¼ 0 PXM k Z T  þ X nr ¼ 0 P  r XM k M  s Z T    HQ ( ) b  ¼  G :  ð 42 Þ Hence, Eq. (42) can be written in the form W b  ¼  G  or  ½ W ; G  ;  W  ¼ ½ w  pq  ;  p ; q  ¼  0 ; 1 ; . . . ; n  ð 43 Þ where W  ¼ ½ w  pq  ¼ X mk ¼ 0 PXM k Z T  þ X nr ¼ 0 P  r XM k M  s Z T    HQ  ð 44 Þ Finally, to obtain the solution of Eq. (1) under the conditions(2), by replacing the row matrices (39) by the last m rows of the matrix (43), we have the new augmented matrix156 S. Yalc¸ınbas  , T. Akkaya  ½ W ; G  ¼ w 00  w 01    w 0 N   ;  g ð x 0 Þ w 10  w 11    w 1 N   ;  g ð x 1 Þ . . . . . . . . . ; . . . w N   m ; 0  w N   m ; 1    w N   m ; N   ;  g ð x N   m Þ u 00  u 01    u 0 N   ;  k 0 u 10  u 11    u 1 N   ;  k 1 . . . . . . . . . ; . . . u m  1 ; 0  u m  1 ; 1    u m  1 ; N   ;  k m  1 26666666666666643777777777777775 ð 45 Þ If,  rank W  ¼  rank ½ W ; G  ¼  N  þ  1  then we can write b  ¼ ð W Þ  1 G Thus the matrix  b  (thereby the coefficients  b 0 ,  b 1 ,  . . .  ,  b N  ) isuniquely determined. Also the Eq. (1) under the conditions(2) has a unique solution. This solution is given by truncatedBoubaker series (3). On the other hand, when  j W j ¼  0, if  rank W  ¼  rank ½ W ; G   <  N  þ  1 , then we may find a particularsolution. Otherwise if   rank W – rank ½ W ; G   <  N  þ  1 , then it isnot a solution. We can easily check the accuracy of the sug-gested method. Since the truncated Boubaker series (3) is anapproximate solution of Eq. (1), when the solution  y N ( x )and its derivatives are substituted in Eq. (1), the resulting equa-tion must be satisfied approximately; that is, for  x  =  x q  2  [0,1],  q  = 0, 1, 2,  . . . E  ð x q Þ ¼ X mk ¼ 0 P k ð x Þ  y ð k Þ ð x Þ þ X nr ¼ 0 P  r ð x Þ  y ð r Þ ð x    s Þ   Z   ba K  ð x ; t Þ  y ð t Þ dt    g ð x Þ  ffi  0  ð 46 Þ and  E  ð x q Þ 6 10  k q ( k q  positive integer). If max 10  k q ¼  10  k ( k positive integer) is prescribed, then the truncation limit  N   isincreased until the difference  E  ( x q ) at each of the points be-comes smaller than the prescribed 10  k . On the other hand,the error can be estimated by the function E  N  ð x Þ ¼ X mk ¼ 0 P k ð x Þ  y ð k Þ ð x Þ þ X nr ¼ 0 P  r ð x Þ  y ð r Þ ð x    s Þ Z   ba K  ð x ; t Þ  y ð t Þ dt    g ð x Þ :  ð 47 Þ If   E  N ( x ) fi 0, when  N   is sufficiently large enough, then the er-ror decreases. 4. Illustrative examples In this section, several numerical examples are given to illus-trate the properties of the method and also, shown that theabsolute error and its upper bound are consistent. All thenumerical computations have been done using Maple12. Example 1.  Let us first consider the Boubaker series solutionof second order linear integro-differential-difference equation[14] ð x  þ  4 Þ 2  y 00 ð x Þ  ð x  þ  4 Þ  y 0 ð x Þ þ  y ð x    1 Þ   y 0 ð x    1 Þ¼  ln ð x  þ  3 Þ   1 x þ 3  þ  3ln ð 3 Þ   5ln ð 5 Þ þ R  1  1  y ð t Þ dt y ð 0 Þ ¼  ln ð 4 Þ ;  y 0 ð 0 Þ ¼  1 = 4 8><>: We assume that the problem has a Boubaker polynomial solu-tion in the form  y ð x Þ ¼ X N n ¼ 0 b n B n ð x Þ where N   ¼  5 ;  P 0 ð x Þ ¼  0 ;  P 1 ð x Þ ¼ ð x  þ  4 Þ ;  P 2 ð x Þ¼ ð x  þ  4 Þ 2 ;  P  0 ð x Þ ¼  1 ;  P  1 ð x Þ ¼  1 ;  g ð x Þ ¼  ln ð x  þ  3 Þ   1 x þ 3  þ  3ln ð 3 Þ   5ln ð 5 Þ : The collocation points are computed as t 0  ¼  1 ; t 1  ¼  3 = 5 ; t 2  ¼  1 = 5 ; t 3  ¼  1 = 5 ; t 4  ¼  3 = 5 ; t 5  ¼  1 f g and from Eq. (42), the fundamental matrix equation of theproblem P 0 XZ T  þ  P 1 XMZ T  þ  P 2 XM 2 Z T  þ  P  0 XM  s Z T   þ P  1 XMM  s Z T    FQ  b  ¼  G  ð 48 Þ where the matrices are defined by X  ¼ 1   1 1   1 1   11   3 = 5 9 = 25   27 = 125 81 = 625   243 = 31251   1 = 5 1 = 25   1 = 125 1 = 625   1 = 31251 1 = 5 1 = 25 1 = 125 1 = 825 1 = 31251 3 = 5 9 = 25 27 = 125 81 = 625 243 = 31251 1 1 1 1 1 2666666666666666437777777777777775 ; P 0  ¼  0 ;  P 1  ¼ 3 0 0 0 0 00   3 : 2 0 0 0 00 0   3 : 4 0 0 00 0 0   3 : 6 0 00 0 0 0   3 : 8 00 0 0 0 0   4 2666666666666666437777777777777775 P 2  ¼ 9 0 0 0 0 00 10 : 24 0 0 0 00 0 11 : 56 0 0 00 0 0 12 : 96 0 00 0 0 0 14 : 44 00 0 0 0 0 16 26666666666643777777777775 ; P  0  ¼ 1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1 26666666666643777777777775 ; P  1  ¼ 1 0 0 0 0 00   1 0 0 0 00 0   1 0 0 00 0 0   1 0 00 0 0 0   1 00 0 0 0 0   1 26666666666643777777777775 A numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases 157
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