Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 DOI 10.1186/s13662-016-0753-2 R E S E A R C H Open Access A new generlized Jcobi Glerkin opertionl mtrix of derivtives: two lgorithms for solving fourth-order boundry vlue problems Wleed M Abd-Elhmeed 1,2*,HnyMAhmed 3,4 nd Youssri H Youssri 2 * Correspondence: wlee_9@yhoo.com 1 Deprtment of Mthemtics, Fculty of Science, University of Jeddh, Jeddh, Sudi Arbi 2 Deprtment of Mthemtics, Fculty of Science, Ciro University, Giz, Egypt Full list of uthor informtion is vilble t the end of the rticle Abstrct This pper reports novel Glerkin opertionl mtrix of derivtives of some generlized Jcobi polynomils. This mtrix is utilized for solving fourth-order liner nd nonliner boundry vlue problems. Two lgorithms bsed on pplying Glerkin nd colloction spectrl methods re developed for obtining new pproximte solutions of liner nd nonliner fourth-order two point boundry vlue problems. In fct, the key ide for the two proposed lgorithms is to convert the differentil equtions with their boundry conditions to systems of liner or nonliner lgebric equtions which cn be efficiently solved by suitble numericl solvers. The convergence nlysis of the suggested generlized Jcobi expnsion is crefully discussed. Some illustrtive exmples re given for the ske of indicting the high ccurcy nd effectiveness of the two proposed lgorithms. The resulting pproximte solutions re very close to the nlyticl solutions nd they re more ccurte thn those obtined by other existing techniques in the literture. MSC: 65M70; 65N35; 35C10; 42C10 Keywords: generlized Jcobi polynomils; Legendre polynomils; opertionl mtrix; fourth-order boundry vlue problems; Glerkin nd colloction methods 1 Introduction Spectrl methods re globl methods. The min ide behind spectrl methods is to pproximte solutions of differentil equtions by mens of truncted series of orthogonl polynomils. The spectrl methods ply prominent roles in vrious pplictions such s fluid dynmics. The three most used versions of spectrl methods re: tu, colloction, nd Glerkin methods (see for exmple [1 8]). The choice of the suitble used spectrl method suggested for solving the given eqution depends certinly on the type of the differentil eqution nd lso on the type of the boundry conditions governed by it. In the colloction pproch, the test functions re the Dirc delt functions centered t specil colloction points. This pproch requires the differentil eqution to be stisfied exctly t the colloction points. The tu-method is synonym for expnding the residul function s series of orthogonl polynomils nd then pplying the boundry conditions s constrints. The tu pproch hs n dvntge tht it cn be pplied to problems with 2016 Abd-Elhmeed et l. This rticle is distributed under the terms of the Cretive Commons Attribution 4.0 Interntionl License (http://cretivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 2 of 16 complicted boundry conditions. In the Glerkin method, the test functions re chosen in wy such tht ech member of them stisfies the underlying boundry conditions of the given differentil eqution. There is extensive work in the literture on the numericl solutions of high-order boundry vlue problems (BVPs). The gret interest in such problems is due to their importnce in vrious fields of pplied science. For exmple, lrge number of problems in physics nd fluid dynmics re described by problems of this kind. In this respect, there is huge number of rticles hndling both high odd- nd high even-order BVPs. For exmple, in thesequenceofppers,[5, 9 11],the uthors hve obtined numericl solutions for evenorder BVPs by pplying the Glerkin method. The min ide for obtining these solutions is to construct suitble bsis functions stisfying the underlying boundry conditions on the given differentil eqution, then pplying Glerkin method to convert ech eqution to system of lgebric equtions. The suggested lgorithms in these rticles re suitble for hndling one nd two dimensionl liner even-order BVPs. The Glerkin nd Petrov- Glerkin methods hve the dvntge tht their pplictions on liner problems enble one to investigte crefully the resulting systems, especilly their complexities nd condition numbers. There re mny lgorithms in the literture which re pplied for hndling fourth-order boundry vlue problems. For exmple, Bernrdi et l. in [12] suggested some spectrl pproximtions for hndling two dimensionl fourth-order problems. In the two leding rticles of Shen [13, 14], the uthor developed direct solutions of fourth-order two point boundry vlue problems. The suggested lgorithms in these rticles re bsed on constructing compct combintions of Legendre nd Chebyshev polynomils together with the ppliction of the Glerkin method. Mny other techniques were used for solving fourth-order BVPs, for exmple, vritionl itertion method is pplied in [15], nonpolynomil sextic spline method in [16], quintic non-polynomil spline method in [17], nd the Glerkin method (see [18, 19]). Theorems which list the conditions for the existence nd uniqueness of solution of such problems re thoroughly discussed in the importnt book of Agrwl [20]. The pproch of employing opertionl mtrices of differentition nd integrtion is considered n importnt technique for solving vrious kinds of differentil nd integrl equtions. The min dvntge of this pproch is its simplicity in ppliction nd its cpbility for hndling liner differentil equtions s well s nonliner differentil equtions. There re lrge number of rticles in the literture in this direction. For exmple, the uthors in [6], employed the tu opertionl mtrices of derivtives of Chebyshev polynomils of the second kind for hndling the singulr Lne-Emden type equtions. Some other studies in [21, 22] employ tu opertionl mtrices of derivtives for solving the sme type of equtions. The opertionl mtrices of shifted Chebyshev, shifted Jcobi, nd generlized Lguerre polynomils nd other kinds of polynomils re employed for solving some frctionl problems (see for exmple, [23 27]). In ddition, recently in the two ppers of Abd-Elhmeed [28, 29] one introduced nd used two Glerkin opertionl mtrices for solving, respectively, the sixth-order two point BVPs nd Lne-Emden equtions. In this pper, our min im is fourfold: Estblishing novel Glerkin opertionl mtrix of derivtives of some generlized Jcobi polynomils. Investigting the convergence nlysis of the suggested generlized Jcobi expnsion.
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 3 of 16 Employing the introduced opertionl mtrix of derivtives to numericlly solve liner fourth-order BVPs bsed on the ppliction of Glerkin method. Employing the introduced opertionl mtrix of derivtives for solving the nonliner fourth-order BVPs bsed on the ppliction of colloction method. The contents of the pper is orgnized s follows. Section 2 is devoted to presenting n overview on clssicl Jcobi nd generlized Jcobi polynomils. Section 3 is concerned with deriving the Glerkin opertionl mtrix of derivtives of some generlized Jcobi polynomils. In Section 4, we implement nd present two numericl lgorithms for the ske of hndling liner nd nonliner fourth-order BVPs bsed on the ppliction of generlized Jcobi-Glerkin opertionl mtrix method (GJGOMM) for liner problems nd generlized Jcobi colloction opertionl mtrix method (GJCOMM) for nonliner problems. Convergence nlysis of the generlized Jcobi expnsion is discussed in detil in Section 5. Numericl exmples including some discussions nd comprisons re given in Section 6 for the ske of testing the efficiency, ccurcy, nd pplicbility of the suggested lgorithms. Finlly, conclusions re reported in Section 7. 2 An overview on clssicl Jcobi nd generlized Jcobi polynomils The clssicl Jcobi polynomils P n (α,β) (x) ssocited with the rel prmeters (α > 1, β > 1)(see[30] nd[31]) re sequence of polynomils defined on [ 1, 1]. Define the normlized orthogonl polynomils R n (α,β) (x)(see[32]) R (α,β) n (x) n (1), (x)= P(α,β) n P (α,β) nd define the shifted normlized Jcobi polynomils on [, b]s R n (α,β) (x)=r n (α,β) ( 2x b b ). (1) The polynomils R n (α,β) (x) re orthogonl on [, b] with respect to the weight function (b x) α (x ) β, in the sense tht (b x) α (x ) β R m (α,β) (x) R n (α,β) 0, m n, (x) dx = (2) h n α,β, m = n, where h α,β n = (b )λ n!ɣ(n + β +1)[Ɣ(α +1)] 2, λ = α + β +1. (3) (2n + λ)ɣ(n + λ)ɣ(n + α +1) It should be noted here tht the Legendre polynomils re prticulr polynomils of Jcobi polynomils. In fct, R (0,0) n (x)=l n (x), where L n (x) is the stndrd Legendre polynomil of degree n. Let w α,β (x)=(b x) α (x ) β.wedenotebyl 2 (, b)theweightedl 2 spce with inner w α,β product: (u, v) w α,β (x):= u(x)v(x)w α,β (x) dx,
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 4 of 16 2 nd the ssocited norm u w α,β =(u, u) 1. Now, the definition of the shifted Jcobi polynomils will be extended to include the cses in which α nd/or β 1. Now ssume tht w α,β l, m Z,nddefine (b x) l (x ) m R ( l, m) i+l+m (x), l, m 1, J (l,m) (b x) l R ( l,m) i+l (x), l 1, m > 1, i (x)= (x ) m R (l, m) i+m (x), l > 1,m 1, R (l,m) i (x), l, m > 1. (4) It is worthy to note here tht in the cse of [, b] = [ 1, 1], the polynomils defined in (4) re the so-clled generlized Jcobi polynomils (J (l,m) i (x)), which re defined by Guo et l. in [33]. Now, the symmetric generlized Jcobi polynomils J ( n, n) i (x) cnbeexpressed explicitly in terms of the Legendre polynomils, while the symmetric shifted generlized Jcobi polynomils J ( n, n) i (x) cn be expressed in terms of the shifted Legendre polynomils. These results re given in the following two lemms. Lemm 1 For every nonnegtive integer n, nd for ll i 2n, one hs J ( n, n) i (x)=n! nd in prticulr, n j=0 ( 1) j( n) j (i 2n +2j + 1 2 )Ɣ(i 2n + j + 1 2 ) Ɣ(i n + j + 3 2 ) L i+2j 2n (x), (5) [ ] J ( 2, 2) 8 2(2i 3) i (x)= L i 4 (x) (2i 5)(2i 3) 2i 1 L 2i 5 i 2(x)+ 2i 1 L i(x). (6) Proof FortheproofofLemm1,see[9]. Now, Lemm 2 is direct consequence of Lemm 1. Lemm 2 For every nonnegtive integer n, for ll i 2n, one hs ( ) b 2n n J ( n, n) i (x)= n! 2 nd in prticulr, j=0 ( 1) j( ) n j (i 2n +2j + 1 2 )Ɣ(i 2n + j + 1 2 ) Ɣ(i n + j + 3 2 ) L i+2j 2n (x), (7) J ( 2, 2) (b ) 4 [ ] i (x)= L 3) 5 i 4 (x) 2(2i 2(2i 5)(2i 3) 2i 1 L i 2 (x)+2i 2i 1 L i (x). (8) The following lemm is lso of interest in the sequel. Lemm 3 The following integrl formul holds: J ( 2, 2) j+4 (x) dx = L j 1(x) (j + 1 2 ) + 3 3L j+1 (x) 3L j+3 (j + 1 2 )(j + 5 2 ) 2 (j + 9 2 )(j + 3 2 ) 2 + L j+5(x) (j + 5 2 ). (9) 3
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 5 of 16 Proof Lemm 3 follows if we integrte eqution (8) (for the cse[, b] = [ 1, 1]) together with the id of the following integrl formul: L j (x) dx = L j+1(x) L j 1 (x). (10) 2j +1 3 Generlized Jcobi Glerkin opertionl mtrix of derivtives In this section, novel opertionl mtrix of derivtives will be developed. For this purpose, we choose the following set of bsis functions: φ i (x)= J ( 2, 2) i+4 (x)=(x ) 2 (b x) 2 R (2,2) i (x), i =0,1,2,... (11) It is esy to see tht, the set of polynomils {φ i (x):i =0,1,2,...} is linerly independent set. Moreover, they re orthogonl on [, b] with respect to the weight function w(x) = 1, in the sense tht (x ) 2 (b x) 2 φ i (x)φ j (x) dx (x ) 2 (b x) = 0, i j, 2 4(b ) 5 (2i+5)(i+1) 4, i = j. Let us denote Hw r (I) (r = 0,1,2,...), s the weighted Sobolev spces, whose inner products nd norms re denoted by (, ) r,w nd r,w,respectively(see[4]). To ccount for homogeneous boundry conditions, we define H 2 0,w (I)={ v H 2 w (I):v()=v(b)=v ()=v (b)=0 }, where I =(, b). Define the following subspce of H 2 0,w (I): V N = spn { φ 0 (x), φ 1 (x),...,φ N (x) }. Any function f (x) H 2 0,w (I)cnbeexpndeds f (x)= c i φ i (x), (12) i=0 where c i = (2i +5)(i +1) 4 4(b ) 5 f (x)φ i (x) dx. (13) (x ) 2 (b x) 2 Assume tht f (x)in eqution(12) cn be pproximteds f (x) f N (x)= N c i φ i (x)=c T (x), (14) i=0 where C T =[c 0, c 1,...,c N ], (x)= [ φ 0 (x), φ 1 (x),...,φ N (x) ] T. (15)
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 6 of 16 Now, we re going to stte nd prove the min theorem, from which novel Glerkin opertionl mtrix of derivtives will be introduced. Theorem 1 If the polynomils φ i (x) re selected s in (11), then for ll i 1, one hs Dφ i (x)= 2 b i 1 j=0 (i+j) odd (2j +5)φ j (x)+η i (x), (16) where η i (x) is given by (b x)( + b 2x), ieven, η i (x)=2(x ) ( b)(b x), iodd. (17) Proof The key ide is to prove Theorem 1 on [ 1, 1], nd hence the proof on the generl intervl [, b] cn esily be trnsported. Now, we intend to prove the reltion Dψ i (x)= i 1 j=0 (i+j) odd (2j +5)ψ j (x)+δ i (x), (18) where ψ i (x)=j ( 2, 2) i+4 (x), nd δ i (x)=4 ( x 2 1 ) x, i even, 1, i odd. To prove (18), it is sufficient to prove tht the following identity holds, up to constnt: Q i (x) dx = ψ i (x), (19) where Indeed Q i (x)= i 1 j=0 (i+j) odd (2j +5)ψ j (x)+δ i (x). Q i (x) dx = i 1 j=0 (i+j) odd (2j +5) J ( 2, 2) j+4 (x) dx + δ i (x) dx. (20)
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 7 of 16 If we mke use of Lemm 3, then the ltter eqution-fter performing some mnipultionsis turned into the reltion where μ i = Q i (x) dx =8 i 1 j=1 (i+j) odd + { 1, i odd, 0, i even. [ L j 1 (x) (2j + 1)(2j +3) + 3L j+1 (x) (2j + 1)(2j +7) 3L j+3 (x) (2j + 3)(2j +9) L j+5 (x) (2j + 7)(2j +9) ] +5μ i ψ 0 (x) dx + δ i (x) dx, (21) After performing some rther lengthy mnipultions on the right hnd side of (21), eqution (19)isobtined. Now, if x in (18)isreplcedby 2x b, then fter performing some mnipultions, we get b Dφ i (x)= 2 b i 1 j=0 (i+j) odd (2j +5)φ j (x)+η i (x), (22) where η i (x)isgivenby (b x)( + b 2x), i even, η i (x)=2(x ) ( b)(b x), i odd, nd this completes the proof of Theorem 1. Now, with the id of Theorem 1, the first derivtive of the vector (x) definedin(15) cn be expressed in mtrix form: d (x) dx = H (x)+η(x), (23) where η(x) =(η 0 (x), η 1 (x),...,η N (x)) T,ndH =(h ij ) 0 i,j N is n (N +1) (N +1)mtrix whose nonzero elements cn be given explicitly from eqution (16)by 2 (2j +5), i > j,(i + j)odd, b h ij = 0, otherwise. For exmple, for N = 5, the opertionl mtrix M is the following (6 6) mtrix: 0 0 0 0 0 0 5 0 0 0 0 0 H = 2 0 7 0 0 0 0 b 5 0 9 0 0 0 0 7 0 11 0 0 5 0 9 0 13 0 6 6.
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 8 of 16 Corollry 1 The second-, third- nd fourth-order derivtives of the vector (x) re given, respectively, by d 2 (x) = H 2 (x) + Hη(x)+η (1) (x), dx 2 (24) d 3 (x) = H 3 (x) + H 2 η(x)+hη (1) (x)+η (2) (x), dx 3 (25) d 4 (x) = H 4 (x) + H 3 η(x)+h 2 η (1) (x)+hη (2) (x)+η (3) (x). dx 4 (26) 4 Two lgorithms for fourth-order two point BVPs In this section, we re interested in developing two numericl lgorithms for solving both of the liner nd nonliner fourth-order two point BVPs. The Glerkin opertionl mtrix of derivtives tht introduced in Section 3 is employed for this purpose. The liner equtions re hndled by the ppliction of the Glerkin method, while the nonliner equtions re hndled by the ppliction of the typicl colloction method. 4.1 Liner fourth-order BVPs Consider the liner fourth-order boundry vlue problem u (4) (x)+f 3 (x)u (3) (x)+f 2 (x)u (2) (x)+f 1 (x)u (1) (x)+f 0 (x)u(x)=g(x), x (, b), (27) subject to the homogeneous boundry conditions u()=u(b)=u ()=u (b)=0. (28) If u(x) is pproximted s u(x) u N (x)= N c k φ k (x)=c T (x), (29) k=0 then mking use of equtions (23)-(26), the following pproximtions for y (l) (x), 1 l 4, re obtined: where u (1) (x) C T( H (x)+η ), u (2) (x) C T( H 2 (x) + θ 2 (x) ), (30) u (3) (x) C T( H 3 (x) + θ 3 (x) ), u (4) (x) C T( H 4 (x) + θ 4 (x) ), (31) θ 2 (x)=hη(x)+η (1) (x), θ 3 (x)=h 2 η(x)+hη (1) (x)+η (2) (x), θ 4 (x)=h 3 η(x)+h 2 η (1) (x)+hη (2) (x)+η (3) (x). If we substitute equtions (29)-(31)intoeqution(27), then the residul, r(x),of this eqution cn be written r(x)=c T( H 4 (x) + θ 4 (x) ) + f 3 (x)c T( H 3 (x) + θ 3 (x) ) + f 2 (x) ( H 2 (x) + θ 2 (x) ) + f 1 (x)c T( H (x)+η(x) ) + f 0 (x)c T (x) g(x). (32)
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 9 of 16 The ppliction of the Glerkin method (see [4]) yields the following (N +1)linerequ- tions in the unknown expnsion coefficients, c i,nmely w(x)r(x)φ i (x) dx = r(x) R (2,2) i (x) dx =0, i =0,1,...,N. (33) Thus eqution (33) genertes set of (N + 1) liner equtions which cn be solved for the unknown components of the vector C, nd hence the pproximte spectrl solution u N (x) given in (29) cn be obtined. Remrk 1 It should be noted tht the problem (27), governed by the nonhomogeneous boundry conditions u()=γ 1, u(b)=γ 2, u ()=ζ 1, u (b)=ζ 2, (34) cnesilybetrnsformedtoproblemsimilrto(27)-(28)(see[10]). 4.2 Solution of nonliner fourth-order two point BVPs Consider the following nonliner fourth-order boundry vlue problem: u (4) (x)=f ( x, u(x), u (1) (x), u (2) (x), u (3) (x) ), (35) governed by the homogeneous boundry conditions u()=u(b)=u ()=u (b)=0. (36) If u (l) (x), 0 l 4, re pproximted s in (29)-(31), then the following nonliner equtions in the unknown vector C cn be obtined: C T( H 4 (x)+θ 4 (x) ) F ( x, C T (x), C T( H (x)+η(x) ), C T( H 2 (x)+θ 2 (x) ), C T( M 3 (x)+θ 3 (x) )). (37) An pproximte solution u N (x) cn be obtined by employing the typicl colloction method. For this purpose, eqution (37) is collocted t (N + 1) points. These points my be tken to be the zeros of the polynomil R (2,2) N+1 (x), or by ny other choice. Hence, set of (N + 1) nonliner equtions is generted in the expnsion coefficients, c i. This nonliner system cn be solved with the id of suitble solver, such s the well-known Newton itertive method. Therefore, the corresponding pproximte solution u N (x) cn be obtined. 5 Convergence nlysis of the pproximte expnsion In this section, the convergence nlysis of the suggested generlized Jcobi pproximte solution will be investigted. We will stte nd prove theorem in which the expnsion in (12) of function f (x)=(x ) 2 (b x) 2 G(x) H0,w 2 (I), where G(x)isofboundedfourth derivtive, converges uniformly to f (x).
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 10 of 16 Theorem 2 A function f (x) =(x ) 2 (b x) 2 G(x) H0,w 2 (I), w(x) = 1 with (x ) 2 (b x) 2 G (4) (x) M, cn be expnded s n infinite sum of the bsis given in (12). This series converges uniformly to f (x),nd the coefficients in (12) stisfy the inequlity c i < M(b )4 2π 8i, i 6. (38) i 4 Proof From eqution (13), one hs c i = (2i +5)(i +1) 4 4(b ) 5 f (x)φ i (x) (x ) 2 (b x) 2 dx, nd with the id of eqution (11), the coefficients c i my be written lterntively in the form c i = (2i +5)(i +1) 4 4(b ) 5 J ( 2, 2) i+4 (x)g(x) dx. (39) Mking use of Lemm 2,thepolynomils J ( 2, 2) i (x) cn be expnded in terms of the shifted Legendre polynomils, nd so the coefficients c i tke the form c i = (i +1) 4 8(b )(2i +3) [ ] L +5) 2i +3 i (x) 2(2i 2i +7 L i+2 + 2i +7 L i+4 G(x) dx. If the lst reltion is integrted by prts four times, then the repeted ppliction of eqution (10)yields c i = (b )3 (i +1) 4 8(2i +3) where I (4) (x)isgivenby I (4) (x)= I (4) (x)g (4) (x) dx, i 4, (40) L i 4 (x) 16(2i 5)(2i 3)(2i 1)(2i +1) 3L i 2 (x) 8(2i 5)(2i 1)(2i + 1)(2i +7) + + + which cn be written s I (4) (x)= 15L i (x) 16(2i 3)(2i 1)(2i + 7)(2i +9) 5(2i +5)L i+2 (x) 4(2i 1)(2i + 1)(2i + 7)(2i + 9)(2i +11) 15L i+4 (x) 16(2i + 1)(2i + 7)(2i + 11)(2i +13) 3L i+6 (x) 8(2i + 7)(2i + 9)(2i + 11)(2i +15) (2i +3)L i+8 (x) 16(2i + 7)(2i + 9)(2i + 11)(2i + 13)(2i +15), (41) 6 m=0 B m L i+2m 4 (x), sy, nd then the coefficients c i tke the form c i = (b { )3 (i +1) b 6 4 B m L i+2m 4 }G 8(2i +3) (x) (4) (x) dx. (42) m=0
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 11 of 16 Now, mking use of the substitution 2x b = cos θ enbles one to put the coefficients c b i in the form c i = (b { )4 (i +1) π 6 } 4 B m L i+2m 4 (cos θ) 16(2i +3) 0 m=0 G (4) ( 1 2( + b +(b ) cos θ ) ) sin θ dθ, i 4. (43) Tking into considertion the ssumption G (4) (x) M,thenwehve c i M(b )4 (i +1) 4 16(2i +3) 6 m=0 π 0 B m L i+2m 4 (cos θ) sin θ dθ. (44) From Bernstein type inequlity (see [34]), it is esy to see tht Li+2m 4 (cos θ) 2 sin θ <, 0 m 6, (i 4)π nd hence (44) together with the lst inequlity leds to the estimtion c i < M(b )4 (i +1) 4 16(2i +3) = M(b )4 (i +1) 4 16(2i +3) 2π i 4 6 B m m=0 Finlly it is esy show tht for ll i 6, 2π i 4 4(2i +5) (2i 5)(2i 1)(2i + 7)(2i + 11)(2i +15). (45) c i < M(b )4 2π 8i. i 4 This completes the proof of the theorem. 6 Numericl results nd discussions In this section, the two proposed lgorithms in Section 4 re pplied to solve liner nd nonliner fourth-order two point boundry vlue problems. The numericl results ensure tht the two lgorithms re very efficient nd ccurte. Exmple 1 Consider the fourth-order liner boundry vlue problem (see [35]): y (4) (x) y (2) (x) y(x)=(x 4)e x, 0 x 2, y(0) = 2, y (0) = 1, y(2) = 0, y (2) = e 2. (46) The exct solution of (46)is y(x)=(2 x)e x.
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 12 of 16 Tble 1 Mximum bsolute error of y y N for Exmple 1 N E 4 1.283929 10 5 6 2.28788 10 8 8 3.45852 10 11 10 4.17094 10 14 12 2.03407 10 15 14 1.96477 10 15 Tble 2 Comprison between the reltive errors for Exmple 1 x 1OM [35] 2OMs[35] GJGOMM(N =8) GJGOMM(N =12) 0.25 5.39791 10 3 8.11851 10 3 1.09077 10 10 2.75173 10 15 0.50 1.57835 10 2 2.09205 10 2 1.14685 10 10 1.33747 10 15 0.75 2.54797 10 2 2.93281 10 2 8.24739 10 11 4.50799 10 16 1.00 3.14713 10 2 3.09264 10 2 3.3689 10 12 5.37406 10 16 1.25 3.22814 10 2 2.65317 10 2 7.75226 10 11 2.27451 10 16 1.50 2.73173 10 2 1.82938 10 2 9.08649 10 11 2.54606 10 15 1.75 1.64910 10 2 8.68533 10 3 7.53861 10 11 9.75198 10 16 Tble 3 Mximum bsolute error of y y N for Exmple 2 for N =2,4,6,8,10,12 N E 1 E 2 E 3 E 4 2 1.36037 10 6 2.11456 10 6 8.70645 10 7 8.46647 10 8 4 1.23719 10 8 2.69013 10 8 5.15533 10 9 5.83994 10 10 6 4.34874 10 11 1.19637 10 10 1.93656 10 11 4.17821 10 12 8 2.65565 10 13 4.8217 10 13 1.57874 10 13 3.44169 10 14 10 9.10383 10 15 9.76996 10 15 1.19904 10 14 8.88178 10 15 12 8.65974 10 15 8.65974 10 15 1.06581 10 14 8.43769 10 15 Tble 1 lists the mximum bsolute errors E, which resulted from the ppliction of GJ- GOMM, for vrious vlues of N,whileinTble2 we disply comprison between the reltive errors obtined by the ppliction of the two methods nmely, the first-order method (1OM) nd the second-order methods (2OMs) developed in [35] with the reltive errors resulting from the ppliction of GJGOMM. Exmple 2 Consider the following fourth-order nonliner boundry vlue problem (see [36, 37]): y (4) (x)= 6e 4y(x), 0<x <4 e, y(0) = 1, y (0) = 1 e, y(4 e)=ln(4), y (4 e)= 1 4. (47) The exct solution of the bove problem is y(x)=ln(e + x). In Tble 3, we list the mximum bsolute errors using GJCOMM for vrious vlues of N. Let E 1, E 2, E 3,ndE 4 denote the mximum bsolute errors if the selected colloction points rerespectively,thezerosoftheshiftedlegendrepolynomill N+1 (x), the shifted Chebyshev polynomils of the first nd second kinds TN+1 (x), U N+1 (x), nd the shifted symmetric Jcobi polynomil R (2,2) N+1 (x), while Figures 1 nd 2 disply comprison between the mxi-
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 13 of 16 Figure 1 Comprison between the errors obtined by the ppliction of GJCOMM for N = 4,for different choices to the selected colloction points. Figure 2 Comprison between the errors obtined by the ppliction of GJCOMM for N = 6,for different choices to the selected colloction points. mum bsolute errors resulting from the ppliction of GJCOMM for N =4nd6,respectively. Tble 3 nd Figures 1 nd 2 show tht the best choice mong the previous choices for the colloction points re obtined if the selected colloction points re the zeros of the polynomil R (2,2) N+1 (x). Tble 4 displys comprison between the errors obtined by the ppliction of GJCOMM for N = 4, with the errors resulting from the ppliction of the three methods developed in [36, 37].This comprison scertins tht our resultsre more ccurte thn those obtined in [36, 37]. Exmple 3 Consider the following fourth-order nonliner boundry vlue problem (see [37]): y (4) (x)=y 2 (x)+g(x), 0 < x <1, y(0) = 0, y (0) = 0, y(1) = 1, y (1) = 1, (48)
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 14 of 16 Tble 4 Comprison between the bsolute errors for Exmple 2 x Method in [36] Methodin[37]( φ 2 y ) Methodin[37]( ψ 2 y ) GJCOMM(N =4) 0.0 0.0 0.0 0.0 2.55351 10 15 0.2 1.79 10 4 4.1883 10 4 4.1012 10 6 6.94449 10 8 0.4 3.25 10 4 1.2786 10 3 1.2574 10 5 6.30457 10 8 0.6 4.08 10 4 1.9971 10 3 1.9762 10 5 4.48952 10 8 0.8 4.02 10 4 2.0753 10 3 2.0714 10 5 6.49632 10 8 1.0 2.94 10 4 1.3038 10 3 1.3163 10 5 1.47025 10 8 1.2-1.8581 10 4 1.9026 10 6 2.10697 10 8 4-e 2 10 9 2.4400 10 15 0 7.77156 10 16 Tble 5 Mximum bsolute error of y y N for Exmple 3 for N =2,4,6 N E 2 5.489 10 17 4 2.265 10 17 6 2.10662 10 17 Tble 6 Comprison between the bsolute errors for Exmple 3 x Method in [37]( ψ 2 y ) Methodin[37]( φ 2 y ) GJCOMM(N =2) 0.0 0.0 0.0 0.0 0.2 8.1093 10 10 3.5906 10 5 2.03045 10 18 0.4 2.0542 10 9 1.0188 10 4 9.04773 10 18 0.6 2.2272 10 9 1.3579 10 4 2.10718 10 17 0.8 1.0115 10 9 8.5908 10 5 3.74696 10 17 1.0 0.0 5.5799 10 13 5.489 10 17 where g(x)= x 10 +4x 9 4x 8 4x 7 +8x 6 4x 4 +120x 48. The exct solution of the bove problem is y(x)=x 5 2x 4 +2x 2. In Tble 5, we list the mximum bsolute errors using GJCOMM for vrious vlues of N. Let E denote the mximum pointwise errors if the selected colloction points re the zeros of the polynomil R (2,2) N+1 (x). Moreover, Tble 6 displys comprison between the errors obtined by the ppliction of GJCOMM with the method developed in [37]forthecse N = 2. The comprison scertins tht our results re more ccurte thn those obtined by [37]. Exmple 4 Consider the following nonliner fourth-order boundry vlue problem (see [38]): y (4) (x) e x y (x)+y(x)+sin ( y(x) ) =1 ( 2 + e x) sinh(x)+sin ( sinh(x)+1 ), 0 x 1, (49) y(0) = y (0) = 1, y(1) = 1 + sinh(1), y (1) = cosh(1), with the exct solution y(x)=sinh(x)+1. In Tble 7, the bsolute errors re listed for vrious vlues of N.Inordertocomprethe bsolute errors obtined by pplying GJCOMM with those obtined by pplying RHKSM in [38], we list the bsolute errors obtined by the ppliction of RHKSM in the lst column
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 15 of 16 Tble 7 Comprison between the bsolute errors for Exmple 4 x GJCOMM (N =4) GJCOMM(N =6) GJCOMM(N =8) RKHSM(u 101 1 )[38] 0.0 0.0 0.0 0.0 0.0 0.1 1.74774 10 11 2.42005 10 14 9.34812 10 17 2.78 10 8 0.2 8.7269 10 11 1.5889 10 14 8.29415 10 18 8.09 10 8 0.3 2.27345 10 11 3.59562 10 14 2.17857 10 18 1.20 10 7 0.4 1.213 10 10 6.36858 10 14 6.03901 10 17 1.25 10 7 0.5 6.32272 10 12 1.87133 10 15 2.06595 10 16 9.56 10 8 0.6 1.27768 10 10 6.44606 10 14 1.21431 10 17 4.82 10 8 0.7 1.44017 10 11 4.09586 10 14 1.10589 10 16 7.38 10 9 0.8 9.92086 10 11 1.56333 10 14 8.74301 10 16 1.07 10 8 0.9 2.37061 10 11 2.57225 10 14 3.38271 10 17 7.08 10 9 1.0 4.44089 10 16 2.70617 10 16 2.27249 10 16 0.0 of this tble. This tble shows tht the pproximte solution of problem (49)obtinedby using GJCOMM is of high efficiency nd more ccurte thn the pproximte solution obtined by RHKSM [38]. 7 Concluding remrks In this rticle, novel opertionl mtrix of derivtives of certin generlized Jcobi polynomils is derived nd used for introducing spectrl solutions of liner nd nonliner fourth-order two point boundry vlue problems. The two spectrl methods, nmely the Glerkin nd colloction methods re employed for this purpose. The min dvntges of the introduced lgorithms re their simplicity in ppliction, nd lso their high ccurcy, since highly ccurte pproximte solutions cn be chieved by using smll number of terms of the suggested expnsion. The numericl results re convincing nd the resulting pproximte solutions re very close to the exct ones. Competing interests The uthors declre tht they hve no competing interests. Authors contributions The uthors declre tht they crried out ll the work in this mnuscript nd red nd pproved the finl mnuscript. Author detils 1 Deprtment of Mthemtics, Fculty of Science, University of Jeddh, Jeddh, Sudi Arbi. 2 Deprtment of Mthemtics, Fculty of Science, Ciro University, Giz, Egypt. 3 Deprtment of Mthemtics, Fculty of Industril Eduction, Helwn University, Ciro, Egypt. 4 Deprtment of Mthemtics, Fculty of Science, Shqr University, Shqr, Sudi Arbi. Acknowledgements The uthors re grteful to the referees for their vluble comments nd suggestions which hve improved the mnuscript in its present form. Received: 22 My 2015 Accepted: 11 Jnury 2016 References 1. Rshidini, J, Ghsemi, M: B-spline colloction for solution of two-point boundry vlue problems. J. Comput. Appl. Mth. 235(8), 2325-2342 (2011) 2. Elbrbry, EME: Efficient Chebyshev-Petrov-Glerkin method for solving second-order equtions. J. Sci. Comput. 34(2), 113-126 (2008) 3. Julien, K, Wtson, M: Efficient multi-dimensionl solution of PDEs using Chebyshev spectrl methods. J. Comput. Phys. 228, 1480-1503 (2009) 4. Cnuto, C, Hussini, MY, Qurteroni, A, Zng, TA: Spectrl Methods in Fluid Dynmics. Springer, Berlin (1988) 5. Doh, EH, Abd-Elhmeed, WM: Efficient spectrl-glerkin lgorithms for direct solution of second-order equtions using ultrsphericl polynomils. SIAM J. Sci. Comput. 24, 548-571 (2002) 6. Doh, EH, Abd-Elhmeed, WM, Youssri, YH: Second kind Chebyshev opertionl mtrix lgorithm for solving differentil equtions of Lne-Emden type. New Astron. 23/24, 113-117 (2013) 7. Bhrwy, AH, Hfez, RM, Alzidy, JF: A new exponentil Jcobi pseudospectrl method for solving high-order ordinry differentil equtions. Adv. Differ. Equ. 2015, 152 (2015)
Abd-Elhmeed et l. Advnces in Difference Equtions (2016) 2016:22 Pge 16 of 16 8. Doh, EH, Bhrwy, AH, Abd-Elhmeed, WM: Jcobi spectrl Glerkin method for elliptic Neumnn problems. Numer. Algorithms 50(1), 67-91 (2009) 9. Doh, EH, Abd-Elhmeed, WM, Bhrwy, AH: New spectrl-glerkin lgorithms for direct solution of high even-order differentil equtions using symmetric generlized Jcobi polynomils. Collect. Mth. 64(3), 373-394 (2013) 10. Doh, EH, Abd-Elhmeed, WM, Bssuony, MA: New lgorithms for solving high even-order differentil equtions using third nd fourth Chebyshev- Glerkin methods. J. Comput. Phys. 236, 563-579 (2013) 11. Doh, EH, Abd-Elhmeed, WM, Bhrwy, AH: Efficient spectrl ultrsphericl-glerkin lgorithms for the direct solution of 2nth-order liner differentil equtions. Appl. Mth. Model. 33, 1982-1996 (2009) 12. Bernrdi, C, Giuseppe, C, Mdy, Y: Some spectrl pproximtions of two-dimensionl fourth-order problems. Mth. Comput. 59(199),63-76 (1992) 13. Shen, J: Efficient spectrl-glerkin method I. Direct solvers of second-nd fourth-order equtions using Legendre polynomils. SIAM J. Sci. Comput. 15(6), 1489-1505 (1994) 14. Shen, J: Efficient spectrl-glerkin method II. Direct solvers of second-nd fourth-order equtions using Chebyshev polynomils. SIAM J. Sci. Comput. 16(1), 74-87 (1995) 15. Noor, MA, Mohyud-Din, ST: An efficient method for fourth-order boundry vlue problems. Comput. Mth. Appl. 54(7), 1101-1111 (2007) 16. Khn, A, Khndelwl, P: Non-polynomil sextic spline pproch for the solution of fourth-order boundry vlue problems. Appl. Mth. Comput. 218(7), 3320-3329 (2011) 17. Lshien, IF, Rmdn, MA, Zhr, WK: Quintic nonpolynomil spline solutions for fourth order two-point boundry vlue problem. Commun. Nonliner Sci. Numer. Simul. 14(4), 1105-1114 (2009) 18. Doh, EH, Bhrwy, AH: Efficient spectrl-glerkin lgorithms for direct solution of fourth-order differentil equtions using Jcobi polynomils. Appl. Numer. Mth. 58(8), 1224-1244 (2008) 19. Doh, EH, Bhrwy, AH: A Jcobi spectrl Glerkin method for the integrted forms of fourth-order elliptic differentil equtions. Numer. Methods Prtil Differ. Equ. 25(3),712-739 (2009) 20. Agrwl, RP: Boundry Vlue Problems for Higher-Order Differentil Equtions. World Scientific, Singpore (1986) 21. Öztürk, Y, Gülsu, M: An opertionl mtrix method for solving Lne-Emden equtions rising in strophysics. Mth. Methods Appl. Sci. 37(15), 2227-2235 (2014) 22. Bhrdwj, A, Pndey, RK, Kumr, N, Dutt, G: Solution of Lne-Emden type equtions using Legendre opertionl mtrix of differentition. Appl. Mth. Comput. 218(14), 7629-7637 (2012) 23. Bhrwy, AH, Zky, MA: Numericl simultion for two-dimensionl vrible-order frctionl nonliner cble eqution. Nonliner Dyn. 80(1-2),101-116 (2015) 24. Bhrwy, AH, Th, TM, Alzhrni, EO, Blenu, D, Alzhrni, AA: New opertionl mtrices for solving frctionl differentil equtions on the hlf-line. PLoS ONE 10(5),e0126620 (2015).doi:10.1371/journl.pone.0126620 25. Sdtmndi, A, Dehghn, M: A new opertionl mtrix for solving frctionl-order differentil equtions. Comput. Mth. Appl. 59(3), 1326-1336 (2010) 26. Mleknejd, K, Bsirt, B, Hshemizdeh, E: A Bernstein opertionl mtrix pproch for solving system of high order liner Volterr- Fredholm integro-differentil equtions. Mth. Comput. Model. 55(3), 1363-1372 (2012) 27. Zhu, L, Fn, Q: Solving frctionl nonliner Fredholm integro-differentil equtions by the second kind Chebyshev wvelet. Commun. Nonliner Sci. Numer. Simul. 17(6), 2333-2341 (2012) 28. Abd-Elhmeed, WM: On solving liner nd nonliner sixth-order two point boundry vlue problems vi n elegnt hrmonic numbers opertionl mtrix of derivtives. Comput. Model. Eng. Sci. 101(3), 159-185 (2014) 29. Abd-Elhmeed, WM: New Glerkin opertionl mtrix of derivtives for solving Lne-Emden singulr-type equtions. Eur.Phys.J.Plus130, 52 (2015) 30. Abrmowitz, M, Stegun, IA: Hndbook of Mthemticl Functions: With Formuls, Grphs, nd Mthemticl Tbles. Dover, New York (2012) 31. Andrews, GE, Askey, R, Roy, R: Specil Functions. Cmbridge University Press, Cmbridge (1999) 32. Doh, EH, Abd-Elhmeed, WM, Ahmed, HM: The coefficients of differentited expnsions of double nd triple Jcobi polynomils. Bull. Irn. Mth. Soc. 38(3),739-766 (2012) 33. Guo, B-Y, Shen, J, Wng, L-L: Optiml spectrl-glerkin methods using generlized Jcobi polynomils. J. Sci. Comput. 27(1-3), 305-322 (2006) 34. Chow, Y, Gtteschi, L, Wong, R: A Bernstein-type inequlity for the Jcobi polynomil. Proc. Am. Mth. Soc. 121(3), 703-709 (1994) 35. Xu, L: The vritionl itertion method for fourth order boundry vlue problems. Chos Solitons Frctls 39(3), 1386-1394 (2009) 36. Wzwz, AM: The numericl solution of specil fourth-order boundry vlue problems by the modified decomposition method. Int. J. Comput. Mth. 79(3),345-356 (2002) 37. Singh, R, Kumr, J, Nelknti, G: Approximte series solution of fourth-order boundry vlue problems using decomposition method with Green s function. J. Mth. Chem. 52(4), 1099-1118 (2014) 38. Geng, F: A new reproducing kernel Hilbert spce method for solving nonliner fourth-order boundry vlue problems. Appl. Mth. Comput. 213, 163-169 (2009)