Copyright 2018 Tech Science Press CMES, vol.115, no.1, pp.67-84, 2018

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1 Copyight 208 ech Science ess CMES vol.5 no. pp hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving hee-dimensional Multi-em Factional- Ode DEs with Vaiable Coefficients Jiaquan Xie 3 * Fuqiang Zhao 3 Zhibin Yao 3 and Jun Zhang 2 Abstact: In this pape the thee-vaiable shifted Jacobi opeational matix of factional deivatives is used togethe with the collocation method fo numeical solution of theedimensional multi-tem factional-ode DEs with vaiable coefficients. he main chaacteistic behind this appoach is that it educes such poblems to those of solving a system of algebaic equations which geatly simplifying the poblem. he appoximate solutions of nonlinea factional DEs with vaiable coefficients thus obtained by theevaiable shifted Jacobi polynomials ae compaed with the exact solutions. Futhemoe some theoems and lemmas ae intoduced to veify the convegence esults of ou algoithm. Lastly seveal numeical examples ae pesented to test the supeioity and efficiency of the poposed method. Keywods: hee-vaiable shifted Jacobi polynomials multi-tem factional-ode DEs vaiable coefficients numeical solution convegence analysis. Intoduction he elliptic patial diffeential equations have been applied in vaious fields of engineeing and science. Many impotant phenomena in electomagnetics viscoelasticity fluid mechanics electochemisty biological population models signals pocessing [Apaci (984); Apaci and Roache (972); Myint- and Debnath (2007); Spot and Caey (996); Wang Zhong and Zhang (2006); Cebeci (2002)] can be well descibed by elliptic factional diffeential equations. Fo that eason we need a eliable and efficient technique fo the solution of factional diffeential equations. he eseach of numeical solution is still an impotant subject. Vaious numeical methods have been poposed to solve such poblems. hese methods include meshless methods [Dehghan and Shiadi (205); Hu Li and Cheng (2005)] spline collocation methods [Faiweathe Kaageoghis and Maack (20); Abushama and Bialecki (2008)] finitediffeence methods [Bitt synkov and ukel (200); Boisvet (98); Singe and ukel (2006)] finite element method [Cialet (2002)] Chebyshev polynomials method [Ghimie College of Mechanical Engineeing aiyuan nivesity of Science and echnology aiyuan China. 2 aiyuan Institute of China Coal echnology Engineeing Goup aiyuan China. 3 Collaboative Innovation Cente of aiyuan Heavy Machiney Equipment aiyuan China. * Coesponding autho: Jiaquan Xie. xjq @63.com. CMES. doi:0.3970/cmes

2 68 Copyight 208 ech Science ess CMES vol.5 no. pp ian and Lamichhane (206)] new wavelet based full-appoximation [Shialashetti Kantli and Deshi (206)] and Domain decomposition method [Zhang Zhang and Yin (2008)]. In Ai et al. [Ai and Asif (207)] the authos utilied Haa collocation method fo theedimensional elliptic patial diffeential equations. R. C. Mittal and S. Dahiya used cubic B- spline diffeential quadatue method to obtain the numeical solution of thee-dimensional telegaphic equation in Mittal et al. [Mittal and Dahiya (207)]. In Sivastava et al. [Sivastava Awasthi and Chauasia (207)] they poposed the educed diffeential tansfom method to solve two and thee-dimensional second ode hypebolic telegaph equations. In this pape we conside the thee-dimensional multi-tem factional-ode DEs with vaiable coefficients of the following fom using thee-vaiable shifted Jacobi polynomials: x y L L L 2 3 u x y u x y u x y u x y a x y b x y cx y d x y x y x u x y u x y e x y k x y l x y u x y f x y y x y x y known function and u x y Diichlet bounday conditions: u x y0 g x y0 u x y L g x y L u x0 g x0 u x L g x L u 0 y g 0 y u L y g L y. () denotes the Caputo deivative f x y is a is the solution function to be detemined. Subject to the he cuent pape is oganied as follows: In next Section the definitions of factional calculus and shifted Jacobi polynomials and function appoximation ae intoduced. he diffeential opeational matix of one-vaiable shifted Jacobi polynomials is given in Section 3. In Section 4 the eo bound and convegence analysis is investigated though some theoems and lemmas. In Section 5 we utilie the thee-vaiable shifted Jacobi polynomials to solve thee-dimensional DEs with vaiable coefficients. In Section 6 seveal numeical examples ae illustated to test the poposed method. Finally a conclusion is dawn in Section 7. (2) 2 eliminaies and notations 2. he factional deivative in the Caputo sense Definition. he Riemann-Liouville factional integal opeato of ode vv 0 defined as [Zhao Huang Xie et al. (207)] is

3 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 69 x v v J f x x f d v 0 x 0 0 v 0 J f x f x. Definition 2. he Caputo factional deivatives of ode v is defined as Zhao et al. [Zhao Huang Xie et al. (207)] m x v mv m mv d D f x J D f x x f d m v m x 0 0 m m v d m D is the classical diffeential opeato of ode m. Fo the Caputo deivative we have v Dx 0 fo v v x fo v. v Recall that fo v N the Caputo diffeential opeato coincides with the usual diffeential opeato of an intege ode. Simila to the intege-ode diffeentiation the Caputo s factional diffeentiation is a linea opeation i.e. D v f x g x D v f x D v g x (6) and ae constants. 2.2 Jacobi polynomials he well-known Jacobi polynomials ae defined on the inteval [- ] and can be geneated with the aid of the following ecuence fomula [Bhawy and Zaky (205)]: i 2 2 2i t 2i 2i 2 2i i 2i 2 i i 2i i2 t i 23 i i 2i 2 i t t t 0 and t t In ode to use these polynomials on the inteval x0 L Jacobi polynomials by intoducing the change of vaiable (3) (4) (5) we define the so-called shifted 2x t. Let the shifted Jacobi L

4 70 Copyight 208 ech Science ess CMES vol.5 no. pp x polynomials i be denoted by L geneated fom: x Li. hen L i L i 2i i 2i 2 i i 2i Li 2 x i i i 2i 2 Li x 2 2 2x 2i 2i 2i 2 L x x x L x and L x. 2 L 2 he analytical fom of the shifted Jacobi polynomials Li x can be (7) of degeei is given by i ik k Li x x (8) k k i i k! k! L i i k k0 i i 0 i Li L i! i! Li he othogonality condition of shifted Jacobi polynomials is L j x L k x w L x dx hk (9) 0 L L k k w L x x L x i j and hk 2k k! k. 0 i j Definition 3. Suppose that Li x n0. is the sequence of one-vaiable shifted Jacobi polynomials on the inteval0 L. hee-vaiable Jacobi polynomials ae defined on the domain 0 L 0 L 0 L as follows: L i L2 j L3 k 2 3 x i j k 0 x y x y i j k 0 2 x y (0) x y in the domain 0 L 0 L 0 L heoem. he polynomials W x y wl i x wl 2 j y wl 3 k On the hand the following popety is held: ae othogonal with espect to the weight function x y i jk x y W x y dxdyd hl i x hl 2 j y hl 3 k ii jj kk.

5 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 7 Lemma. If the poduct of j j x and k x x and x j ae jth and kth ae witten as shifted Jacobi polynomials jk jk jk 0 Q x x coefficient If j k If j k; jk 0 j k if j then k jk j k l l lj else end min k jk j k ll l0 0 j k if j then min j jk j k l l l0 else end 2 min k 2 jk j k ll lj ae detemined as follows: oof. See Bohanifa et al. [Bohanifa and Sadi (205)]. Lemma 2. If then i x j x and x k ae i j and kth shifted Jacobi polynomial

6 72 Copyight 208 ech Science ess CMES vol.5 no. pp i j k jk i n0 l0 il q x x x w x dx jk n i i l n l l i n l 2 i l! l! jk wee intoduced in Lemma. n () 2.3 Function appoximation A thee-vaiable continuous function u x y in the domain 0 L 0 L 0 L can be expanded in tems of thee-vaiable shifted Jacobi polynomials as i0 j0 k 0 u x y u x y 2 3 u u x y x y W x y dxdyd i j k hl i hl 2 j hl 3 k In pactice the N tuncated seies with espect to all thee vaiables xyand can be used an appoximation fo the given function u x y N N N u x y u x y u x y x y x y (2) N i0 j0 k 0 and x y ae the unknown coefficients and thee-vaiable Jacobi polynomials vectos ae defined as = u u u u u u u u u N N NN 0 NN NNN x y 000 x y 00 N x y 00 x y 0 N x y NN 0 x y NNN x y. he following popety of the poduct of two vectos x y and x y intoduced and applied in solving the thee-dimensional DEs with vaiable coefficients. x y x y V V x y (4) V and V ae espectively N 3 vecto and N N opeational matix of poduct. heoem 2. he enties of the matixv in Eq. (4) ae computed as: (3) is 3 3

7 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 73 N N N V 2 2 V q q q mn nn l mn nn l h h h m n l m n l 0 N q ae intoduced by Lemma 2. mim njn lkl m n l i0 j0 k 0 oof. See Sadi et al. [Sadi Amini and Cheng (207)]. 3 he diffeential opeational matix of one-vaiable shifted Jacobi polynomials vecto Lemma 3. he fist-ode deivative of the vecto d x dx x x can be expessed by D (5) D is the N N C i j i j D dij 0 othewise C i j opeational matix of deivative given by 2 2 j ij i j! 2 j L i i j j i j i j 2 j F ;. j 2 2 j Fo the poof see Doha et al. [Doha Bhawy and E-Eldien (202)]. heoem 3. Let x be one-vaiable shifted Jacobi polynomials vecto and let also v 0 then v D x v D x (6) v D is the N N Caputo sense and is defined by: opeational matix of deivative of ode v in the

8 74 Copyight 208 ech Science ess CMES vol.5 no. pp D v v v v v v v v v N v and v v v v i j l0 i0 i i2 i N v v v v N0 N N2 N N i k v is given by ik v L j i i k h j k i k v i k! j j Note that in jl j l l k v l l k v 2 j l! l!. v D the fist v ows ae all eos. oof. See Doha et al. [Doha Bhawy and E-Eldien (202)]. (7) 4 Eo bound and convegence analysis In this section we show that the given method in the pevious sections is convegent. Fo ou pupose we will need the following definitions and theoems to obtain an eo bound fo the poposed method in the Jacobi-weighted Sobolev Space. Definition 4. We define F span x y 0 m n l N N mnl as the finite-dimensional polynomials space. heoem 4. Suppose that i u x y C i i2 i3 i i 0 N. i i2 i3 x y Jacobi appoximate solution to u x y fom seies of the F and If u N x y is the N un x y is the aylo u x y of ode N espect to each vaiables xy and then an eo bound can be pesented as follows:

9 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 75 N 3 3 M u x y 2 un x y B 2 W N! N u x y M max M m n M m n max 0 m nn x y N m mn n x y and B s is the well-known Beta function. oof. See Sadi et al. [Sadi Amini and Cheng (207)]. Definition 5. o deive appoximation esults we intoduce the Jacobi-weighted Space: W F v v is measuable and v W equipped with the following nom and semi-nom: k 2 v W X v k k X x y W 0 k 0 v v W X W0 l 3 l v Xv li l l l2 l 3 x y i 2 3 l l l l l2 l2 l3 l3 W 0 x y w x y w x y w x y li l i heoem 5. Fo any u F N and 0 W holds:. 2 3 the following estimate u un N N (8) W W is a positive constant independent of any function N and. oof. See Sadi et al. [Sadi Amini and Cheng (207)]. Remak. Let u F and N N W u F the following estimates holds fo all uf u u u u N 2 N F L W. W be the Jacobi appoximation to u. hen

10 76 Copyight 208 ech Science ess CMES vol.5 no. pp Numeical implementation In the section we use the thee-vaiable shifted Jacobi polynomials to solve theedimensional factional-ode DEs with vaiable coefficients. Similaity the functions ax y bx y cx y d x y ex y k x y and l x y ae also appoximated by the thee-vaiable shifted Jacobi polynomials as: a x y x y A b x y x y B c x y x y C d x y x y D e x y x y E k x y x y K l x y x y L. A B C D E K and L can be obtained by Eq. (3). sing Eqs. (2) (4) (5) and (6) we have Sadi et al. [Sadi Amini and Cheng (207)] u x y x y x y a x y x y A x y A x x x x (20) = y x y A x y x y A = D I x D I A x y x y y I 2D I2 B x y x y u x y b x y x y B x y B y y y = x x y B 2 2 x y x y B = I D I y x y I D C x y x y u x y c x y x y C x y C = x y x y C x y x y C = I D x y D I D x y x y u x y d x y x y D x y D x x x d x = y x y D = D I x y x y D dx (9) (2) (22) (23)

11 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 77 x y y x y u x y e x y x y E x y E y y y d I D I 2 2 = x x y E = x y x y E dy I D I E x y 2 2 x y I D K x y x y u x y k x y x y K x y K d = x y x y K = I D x y x y K d l x y u x y x y x y L L x y. (26) I ae and 2 2 I and 2 N N N N identity matices espectively. Substituting Eqs. (20)-(26) into Eq. () we get D I A x y I2 D I2 B x y I D C x y D I I2 D I2 I D D x y E x y K x y L x y f x y. Fo the Diichlet bounday condition (2) we have x y g x y x y L3 g x y L3 x g x x L2 g x L2 y g y L y g L y Eq. (27) togethe with Eq. (28) constitutes a system of algebaic equations. hen dispesing the unknown vaiables xy and as the following way: L 2i L2 2 j L3 2l xi y j l i j l N. (29) 2N 2N 2N hen we have (24) (25) (27) (28)

12 78 Copyight 208 ech Science ess CMES vol.5 no. pp D I A xi y j l I 2 D I2 B xi y j l I D C xi y j l D I D xi y j l I2 D I2 E xi y j l I D K xi y j l L xi y j l f xi y j l xi y j0 g xi y j0 xi y j L3 g xi y j L3 xi 0 l g xi 0 l xi L2 l g xi L2 l 0 y j l g 0 y j l L y j l g L y j. Solving this system the unknown coefficient matix can be obtained. hen using Eq. u x y is found. (2) the unknown solution function 6 Numeical expeiments Example. Conside the following thee-dimensional multi-tem factional-ode DEs with vaiable coefficients u x y.5 u x y.25 u x y x y 2 x 2 y u x y u x y u x y x y u x y x y 2 x 2 y 2 x y with the Diichlet bounday conditions: u x y u x u y u x y2 4 x y u x2 4 x u 2 y 4 y. (30) (3) he analytical solution of this poblem is u x y x y. When N 2 4 and 6 the absolute eos at some values of xy and ae shown in ab.. ab. shows that the absolute eos decease as N inceases. able : he absolute eos at some values of x y with N 2 46 x y Anal. Sol. N 2 4 N N 6 (000) e e e-6 ( ) e e e-6 ( ) e e e-6 ( ) e e e-6 () e e e-7 ( ) e e e-6 (.5.5.5) e e e-6 ( ) e e e-6 (222) e e e-6

13 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 79 Example 2. Conside the following thee-dimensional factional-ode DEs with vaiable coefficients x y u x y u x y u x y u x y a x y b x y cx y d x y x y x u x y u x y e x y k x y l x y u x y f x y y a x y x y bx y x y cx y x y 2 2 d x y xy ex y x y k x y x y l x y xy and f x y 6 x y xy x y y x x y x y x y. 3 u x y0 u x0 u 0 y 0 Subject to the Diichlet bounday conditions: ux y xy x y ux x x u y y y he analytical solution of this poblem is ux y xy x 2 y 2 2. (32). Example 2 and Example 3 show that the numeical solutions appoximate to the exact solutions well as N becomes bigge. y 2 2 (i) When x 3 u x y y. When N 3 4 and 5 the gaphs of 3 9 the numeical solutions at and 0.9 ae shown in Fig.. Figue : he numeical solution u y 3 at and 0.9 when N 3 4 and 5

80 Copyight 208 ech Science ess CMES vol.5 no. pp.67-84 208 2y 4 2 2 (ii) When x 23 u x y y. When 3 4 3 9 N and 5 the gaphs of the numeical solutions at 0.30.6 and 0.9 ae shown in Fig. 2. Figue 2: he numeical solution 2 u y 3 at 0.

14 80 Copyight 208 ech Science ess CMES vol.5 no. pp y (ii) When x 23 u x y y. When N and 5 the gaphs of the numeical solutions at and 0.9 ae shown in Fig. 2. Figue 2: he numeical solution 2 u y 3 at and 0.9 when N 3 4 and 5 Example 3. Conside the following thee-dimensional second-ode DEs with vaiable coefficients x y u x y u x y u x y a x y b 2 x y c 2 x y f 2 x y x x a x y y b x y x c x y xy and sinh sinh sinh f x y x y xy x y. With the Diichlet bounday conditions: 0 sinh sinh sinh 0 sinh sinh sinh 0 sinh sinh sinh sinh 2sinh 2sinh 2 sinh 2sinh 2sinh 2 sinh 2sinh 2sinh 2. u x y x y u x x u y y u x y x y u x x u y y he analytical solution of this poblem is u x y sinh x sinh y sinh. when u x y x y (33) 0.5 sinh.5 sinh sinh. he gaphs of the numeical and analytical solutions when N 23and 4 ae shown in Figs. 3-6.

15 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 8 Figue 3: Analytical solution Figue 4: Numeical solution with N 2. Figue 5: Numeical solution with N 3.

82 Copyight 208 ech Science ess CMES vol.5 no. pp.67-84 208 Figue 6: Numeical solution with N 4. Example 4. Conside Eq.

N the 2-nom eo 2 2 x y with M 2 at y 0.3 0.6 ae shown in ab. 2. ab. 2 shows that the numeical pecision can achieve e5 e6 only small seies tems ae expanded.

16 82 Copyight 208 ech Science ess CMES vol.5 no. pp Figue 6: Numeical solution with N 4. Example 4. Conside Eq. (33) we define the 2-nom eo as M 2 2 N N i j l i j l 2 x y 2 0 u x y u x y dx u x y u x y M i j l 2 M. Whee u x y and u x y N When N 2 N 3 and 4 ae the appoximate and exact solutions espectively. N the 2-nom eo 2 2 x y with M 2 at y ae shown in ab. 2. ab. 2 shows that the numeical pecision can achieve e5 e6 only small seies tems ae expanded. able 2: he 2-nom eo 2 x y 2 x y with N 23and 4 N 2 N 3 N e e e-6 7 Conclusions In this aticle we have studied a numeical scheme to solve thee-dimensional multi-tem factional-ode DEs with vaiable coefficients. Ou appoach is based on the theevaiable shifted Jacobi polynomials and thei opeational matices of factional deivatives togethe with a set of suitable collocation nodes. he appoximation of the solution togethe with imposing the collocation nodes is utilied to educe the computation of this poblem to some algebaic equations. he numeical esults show that ou method is convegent as N inceases. Acknowledgement: his wok was suppoted by the Collaboative Innovation Cente of

17 hee-vaiable Shifted Jacobi olynomials Appoach fo Numeically Solving 83 aiyuan Heavy Machiney Equipment ostdoctoal Statup Fund of aiyuan nivesity of Science and echnology ( ) the Natual Science Foundation of Shanxi ovince (2070D2235) National College Students Innovation and Entepeneuship oject ( ) and ( ). Refeences Abushama A.; Bialecki B. (2008): Modified nodal cubic spline collocation fo oisson equation. Society fo Industial and Applied Mathematics vol. 46 pp Apaci V. (984): Convection Heat ansfe. entice Hall New Jesey. Ai I.; Asif M. (207): Haa wavelet collocation method fo thee-dimensional elliptic patial diffeential equations. Computes & Mathematics with Applications vol. 73 no. 9 pp Bhawy A. H.; Zaky M. A. (205): A method based on the Jacobi tau appoximation fo solving multi-tem time-space factional patial diffeential equations. Jounal of Computational hysics vol. 28 pp Boisvet R. (98): Families of high ode accuate discetiations of some elliptic poblems. Siam Jounal on Scientific & Statistical Computing vol. 2 pp Bohanifa A.; Sadi K. (205): A new opeational appoach fo numeical solution of genealied functional intego-diffeential equations. Jounal of Computational and Applied Mathematics vol. 279 pp Bitt S.; synkov S.; ukel E. (200): A compact fouth ode scheme fo the Helmholt equation in pola coodinates. Jounal of Scientific Computing vol. 45 pp Cebeci. (2002): Convection Heat ansfe. Hoions ublishing Inc. Spinge Long Beach Califonia Heidelbeg. Cialet. (2002): he finite element method fo elliptic poblems. Noth Holland. Dehghan M.; Shiadi M. (205): Numeical solution of stochastic elliptic patial diffeential equations using the meshless method of adial basis functions. Engineeing Analysis with Bounday Elements vol. 50 pp Doha E. H.; Bhawy A. H.; E-Eldien S. S. (202): A new Jacobi opeational matix: An application fo solving factional diffeential equations. Applied Mathematical Modelling vol. 36 no. 0 pp Faiweathe G.; Kaageoghis A.; Maack J. (20): Compact optimal quadatic spline collocation methods fo the Helmholt equation. Jounal of Computational hysics vol. 230 pp Ghimie B. K.; ian H. Y.; Lamichhane A. R. (206): Numeical solutions of elliptic patial diffeential equations using Chebyshev polynomials. Computes & Mathematics with Applications vol. 72 no. 4 pp Hu H.; Li Z.; Cheng D. (2005): Radial basis collocation methods fo elliptic bounday value poblems. Computes & Mathematics with Applications vol. 50 pp Mittal R. C.; Dahiya S. (207): Numeical simulation of thee-dimensional telegaphic equation using cubic B-spline diffeential quadatue method. Applied Mathematics &

18 84 Copyight 208 ech Science ess CMES vol.5 no. pp Computation vol. 33 pp Myint-.; Debnath L. (2007): Linea patial diffeential equations fo scientists and enginees. Bikhause. Roache. (972): Computational fluid dynamics. Hemosa ess Albuqueque New Mexico. Sadi K.; Amini A.; Cheng C. (207): Low cost numeical solution fo theedimensional linea and nonlinea integal equations via thee-dimensional Jacobi polynomials. Jounal of Computational & Applied Mathematics vol. 39 pp Shialashetti S. C.; Kantli M. H.; Deshi A. B. (206): New wavelet based fullappoximation scheme fo the numeical solution of nonlinea elliptic patial diffeential equations. Alexandia Engineeing Jounal vol. 55 no. 3 pp Singe I.; ukel E. (2006): Sixth ode accuate finite diffeence schemes fo the Helmholt equation. Jounal of Computational Acoustics vol. 4 pp Spot W.; Caey G. (996): A high-ode compact fomulation fo the 3D oisson equation. Nume. Methods atial Diffeential Equations vol. 83 pp Sivastava V. K.; Awasthi M. K.; Chauasia R. K. (207): Reduced diffeential tansfom method to solve two and thee dimensional second ode hypebolic telegaph equations. Jounal of King Saud nivesity Engineeing Sciences vol. 29 no. 2 pp Wang J.; Zhong W.; Zhang J. (2006): A geneal meshsie fouth ode compact diffeence discetiation scheme fo 3D oisson equation. Applied Mathematics & Computation vol. 83 pp Zhang K.; Zhang R.; Yin Y. (2008): Domain decomposition methods fo linea and semilinea elliptic stochastic patial diffeential equations. Applied Mathematics & Computation vol. 95 no. 2 pp Zhao F.; Huang Q.; Xie J.; Li Y.; Ma M. et al. (207): Chebyshev polynomials appoach fo numeically solving system of two-dimensional factional DEs and convegence analysis. Applied Mathematics & Computation vol. 33 pp

A matrix method based on the Fibonacci polynomials to the generalized pantograph equations with functional arguments

A matrix method based on the Fibonacci polynomials to the generalized pantograph equations with functional arguments A mati method based on the Fibonacci polynomials to the genealized pantogaph equations with functional aguments Ayşe Betül Koç*,a, Musa Çama b, Aydın Kunaz a * Coespondence: aysebetuloc @ selcu.edu.t a