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 Depatment of Mathematics, Faculty of Science, Selcu Univesity, Konya, Tuey b Mustafa Kemal Univesity, Yayladagı Vocational School, Hatay, Tuey Abstact: In this study, a collocation method based on the Fibonacci opeational mati is poposed to solve genealized pantogaph equations with linea functional aguments. Some illustative eamples ae given to veify the efficiency and effectiveness of the poposed method. Keywods: Pantogaph equations, Mati method, Polynomial appoimation, Opeational matices, Fibonacci polynomials MSC: 65D25. 1. Intoduction Many phenomena in applied banches that fail to be modeled by the odinay diffeential equations can be descibed by the Delay diffeential equations. Many eseaches have studied diffeent applications of those equations in vaiety of applied sciences such as biology, physics, economy and electodynamics (see [1-4]). Pantogaph equations with popotional delays play an impotant ole in this contet. The eistence and uniqueness of the analytic solutions of the multi-pantogaph equation ae investigated in [5]. A numeical appoach to multi-pantogaph equations with vaiable coefficients is also studied in [6]. An etension of the multi-pantogaph equation is nown to be the genealized pantogaph equation with functional aguments defined as J m1 ( m.) y p y g I a b unde the mied conditions, [, ] (1.1) j0 0 whee,, c, m1 c y, 01 m 1, a b (1.2) 0 and ae eal and/o comple coefficients; analytical functions defined in inteval a b. p and g ae the
In ecent yeas, many eseaches have developed diffeent numeical appoaches to the genealized pantogaph equations as vaiational iteation method [7], diffeential tansfom appoach [8], Taylo method [9], collocation method based on Benoulli mati [10] and Bessel collocation method [11]. In this study, we investigate a collocation method based on the Fibonacci polynomial opeational mati fo the numeical solution of the genealized pantogaph equation (1.1). Even the Fibonacci numbes have been nown fo a long time, the Fibonacci polynomials ae vey ecently defined to be an impotant agent in the wold of polynomials [12-13]. Compaed to the methods of the othogonal polynomials, the Fibonacci appoach has poved to give moe pecise and eliable esults in the solution of diffeential equations [14]. This study is oganized as follows: In the second pat, a shot eview of the Fibonacci polynomials is pesented. A Fibonacci opeational mati fo the solution of the pantogaph equation is developed in Section 3. Some numeical eamples ae given in Section 4 to illustate efficiency and effectiveness of the method. 2. Opeational matices of the Fibonacci polynomials The Fibonacci polynomials F ae detemined by following geneal fomula [12-13] 1 1 with F and Fibonacci polynomials. 2 2.1 Fibonacci Seies Epansions F 1 F F 1, fo 1, (2.1) F. Now, we will mention some mati elations in tems of To obtain an epansion fom of the analytic solution of the pantogaph equation, we use the Fibonacci collocation method as follows: Suppose that the equation (1.1) has a continuous function solution that can be epessed in the Fibonacci polynomials y a F. (2.2) 1 Then, a tuncated epansion of N -Fibonacci polynomials can be witten in the vecto fom N F A 1 y a F (2.3)
whee the Fibonacci ow vecto A ae given, espectively, by and 2.2 Mati elations of the deivatives: F and the unnown Fibonacci coefficients column vecto F F F F (2.4) 1 2 N 1 N T. 1 2 1 The -th ode deivative of (2.3) can be witten as A a a a N (2.5) N (0) whee a ( ) N ( ) ( ) F A 1 y a F, 0,1,..., n (2.6) (0) a, y y and T ( ) ( ) ( ) ( ) A a1 a2 a N (2.7) is the coefficient vecto of the polynomial appoimation of -th ode deivative. Then, thee eists a elation between the Fibonacci coefficients as ( 1) A D A, 0,1,, n (2.8) whee D is N N opeational mati fo the deivative defined by [14] ( j i) i sin, j i D d i, j 2. (2.9) 0, j i Maing use of Eq.s (2.6) and (2.8)yields ( ) y, 0,1,, n F D A. (2.10) 3. Solution pocedue fo the pantogaph diffeential equations Let us ecall the m -th ode linea pantogaph-diffeential equation, J m1 ( m.) y p y g I a b, [, ]. (3.1) j0 0 The fist step in the solution pocedue is to define the collocation points in the domain I, so that, b a i a ( i 1), i 1,2,, N, a i b. (3.2) N 1
Then, collocating poblem (3.1)at the points in (3.2) yields J m1 ( m.) y i p i y i g i i N N j0 0, 1(1), m 1 (3.3) The system (3.3) can, altenatively, be ewitten in the mati fom whee J m1 m Y P Y G (3.4) j0 0 j j P j Pj ( 1 ) 0... 0 0 Pj ( 2)... 0 0 0... Pj ( N ) and G ( ) ( ) ( ) T. g 1 g 2 g N Theefoe, the -th ode deivative of the unnown function at the collocation points can be witten in the mati fom as o equivalently, F D A ( ) y i ( i ), i 1, 2,..., N Y y ( ) 1 ( ) ( ) y 2 y ( ) ( ) ( ) FD A. ( N ) (3.5) To epess the functional tems of Eq. (1.1) as in the fom (2.3), let i in the elation (3.5), then obtain o, i i, 11 () y i N Y () j y y y () () () F D A (3.6) 1 2 j N F D A (3.7)
whee F ae Fibonacci opeational matices coesponding to the coefficients. Then, eplacing (3.5) and (3.7) in equation (3.4) gives the fundamental mati equation fo the poblem (3.1) as J m1 m FD PFj D A G (3.8) j0 0 which coesponds to a system of N algebaic equations fo the unnown Fibonacci coefficients a, 1, 2,..., N. In othe wods, when we denote the epession in the sum by W w s, t, fo s 1, 2,..., N and t 1, 2,..., N, we get Thus, the augmented mati of Eq. (3.9) becomes WA = G. (3.9) W; G. (3.10) On the othe hand, with the help of Eq.(2.10), the conditions (1.2) can be conveted to following mati fom whee, m1 U j c y, (3.11) 0 and m1 U j u, cf D A, 0 F F F F 1 2 N. (3.12) Theefoe, the augmented mati of the specified conditions is U ; u u u :. (3.13) j j j1 j2 jn j Consequently, (3.10) togethe with (3.13) can be witten in the new augmented mati fom * * W : G. (3.14) This fom can also be achieved by eplacing some ows of the mati (3.10) by the ows of (3.13) o adding those ows to the mati (3.10) povided that vecto A (theeby vecto of the coefficients * det( W ) 0. Finally, the a ) is detemined by applying some numeical methods designed especially to solve the system of linea equations. On the othe hand, when the singula case * det( ) 0 W appeas, the least squae methods ae inevitably available to each the best possible appoimation. Theefoe, the appoimated solution can be obtained.
This would be the Fibonacci seies epansion of the solution to the poblem (3.1) with specified conditions. Accuacy of the Results: We can, now, poceed with a shot accuacy analysis of the poblem in a simila way to [15]. As the tuncated Fibonacci seies epansion is an appoimate solution of Eq. (1.1) with (1.2), it must be satisfy the following equality fo [ a, b] o When m1 ( m.) J E y p y g 0 j0 0 10 E ( is any positive intege). ma 10 10 ( is any intege) is pescibed, the tuncation limit N is inceased until the diffeence E at each of the collocation points becomes smalle than the desied value 10. 4. Numeical esults In this pat, thee illustative eamples ae given in ode to claify the findings of the pevious section. The eos of the poposed method ae compaed with those of the eos occued in the solutions by some othe methods in Table 1-3 fo two sample eamples. It is noted hee that the numbe of collocation points in the eamples is indicated by the capital lette N. Eample 1 [10-11] Conside the following linea pantogaph equation 3 2 y y y 2, 0 1 4 2 with the initial conditions y 0 y 0 0. y The eact solution of this poblem is nown 2. When the solution pocedue in Section 3 is applied to the poblem, the solution of the linea algebaic system gives the
numeical appoimation of the solution to the poblem. It is notewothy that the method y eaches the eact solution 2 even fo N 3. Eample 2 [9,16-17] Now, conside the following equation with vaiable coefficient given in 1 /2 1 y e y y, 0 1 2 2 2 y 0 1. The eact solution is also nown to be y ep. A compaison of the absolute eos of the poposed appoach, Taylo method [16] and the eponential appoach [17] is given in Table1 fo N 5. Anothe compaison of the pesent method with the methods of Taylo polynomials [9,16] is also given fo N 9 and N 8 in Table 2. These esults veify that the Fibonacci appoach is bette at least one decimal place in accuacy than the othes. Table 1 Compaison of the absolute eos of diffeent appoimation techniques to Eample 2 Pesent Method Eponential appoach [17] Taylo polynomial appoach [16] N=5 N=5 N=5 0.2 0.2553 E-05 0.32778 E-02 0.271 E-06 0.4 0.1965 E-05 0.32081 E-02 0.882 E-05 0.6 0.3874 E-05 0.44444 E-02 0.682 E-04 0.8 0.4833 E-05 0.46898 E-02 0.293 E-03 1.0 0.2690 E-04 0.12864 E-01 0.912 E-03 Table 2 Compaison of the absolute eos of diffeent appoimation techniques to Eample Pesent Method Taylo polynomial appoach [16] Taylo Method [9] N=9 N=9 N=8 0.2 0 0.271 E-06 1.44 E-12 0.4 0 0.882 E-05 7.524 E-10 0.6 0 0.682 E-04 2.953 E-08 0.8 0.1 E-08 0.293 E-03 4.018 E-07 1.0 0.1 E-08 0.912 E-03 3.059 E-06 Eample 3 [6,18] Finally, let us conside the pantogaph equation with vaiable coefficients 0.5 0.75 0 1 y y e sin(0.5 ) y 0.5 2e cos 0.5 sin 0.25 y 0.25, 0 1 y
which has the eact solution cos y e. Computed esults ae compaed with the esults of the Taylo [6] and Boubae [18] mati methods in Table 3. Table 3 Compaison of the absolute eos of diffeent appoimation techniques to Eample 3 Pesent Method Taylo Mati Method [6] Baubae Mati Method [18] N=5 N=9 N=12 N=5 N=9 N=9 N=12 0.2 0.18903 E-5 0.12102 E-10 0.18730 E-14 0.69082 E-6 0.1300 E-8 0.121 E-10 0 0.4 0.62395 E-6 0.96855 E-11 0.14897 E-14 0.42924 E-4 0.1434 E-6 0.968 E-11 0.310 E-14 0.6 0.13542 E-5 0.71954 E-11 0.11151 E-14 0.47443 E-3 0.2058 E-5 0.719 E-11 0.110 E-14 0.8 0.15097 E-5 0.68229 E-11 0.68964 E-15 0.25855 E-2 0.1212 E-4 0.682 E-11 0.200 E-14 1.0 0.47735 E-4 0.75830 E-9 0.13256 E-12 0.95631 E-2 0.4003 E-4 0.758 E-9 0.562 E-12 Acnowledgements This study was suppoted by Reseach Pojects Cente (BAP) of Selcu Univesity. Also, the authos would lie to than the Selcu Univesity and TUBITAK fo thei suppots. We have denoted hee that a mino pat of this study was pesented oally at the "2nd Intenational Euasian Confeence on Mathematical Sciences and Applications (IECMSA- 2013)", Saajevo, August, 2013. Refeences 1. Ajello, W. G., Feedman, H. I., Wu, J.: A model of stage stuctued population gowth with density dependent time delay, SIAM J. Appl. Math., 52, 855-869 (1992). 2. Kuang, Y.: Delay diffeential equations with applications in population dynamics, Academic Pess, Boston (1993). 3. Dehghan, M., Shaei, F.: The use of the decomposition pocedue of Adomian fo solving a delay diffeential equation aising in electodynamics, Phys. Scipt., 78(6), 065004 (2008). 4. Ocendon, J. R., Tayle, A. B.: The dynamics of a cuent collection system fo an electic locomotive, Poc. Roy. Soc. Lond. A, 322, 447-468 (1971). 5. Liu, M. Z., Dongsong, L.: Popeties of analytic solution and numeical solution of multipantogaph equation, Appl. Math. Comp., 155, 853-871 (2004). 6. Seze, M., Yalcinbas, S., Sahin, N.: Appoimate solution of multi pantogaph equation with vaiable coefficients, J. Comp. Appl. Math., 214, 406-416 (2008). 7. Saadatmandi, A., Dehghan, M.: Vaiational iteation method fo solving a genealized pantogaph equation, Comp. Math. Appl., 58, 2190-2196 (2009)
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