Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani a,b, Douglas Anderson c a Deparmen of Mahemaics, The Universiy of Jordan, Amman, Jordan. b Deparmen of Mahemaics, Faculy of Science, Universiy of Hail, Saudi Arabia. c Deparmen of Mahemaics, Concordia College, Moorhead, MN, USA. Absrac Le G= (V, σ, µ) be a fuzzy graph. Le H be he graph consruced from G as follows V(H) =V(G), wo poins u and v are adjacen in H if and only if u and v are adjacen and degree fuzzy equiable in G. H is called he adjacency inheren fuzzy equiable graph of G or fuzzy equiable associae graph of G and is denoed by e ef (G). In his paper we inroduced he concep of fuzzy equiable associae graph and obain some ineresing resuls for his new parameer in fuzzy equiable associae graph. c 2016 All righs reserved. Keywords: Fuzzy equiable dominaing se, fuzzy equiable associae graph, pre-e-fuzzy equiable graph, degree equiable fuzzy graph. 2010 MSC: 26A33. 1. Inroducion Fracional differenial equaions have been of grea ineres for he las hiry years because of heir applicaions in applied sciences, see [6, 8] and [12]. The main definiions which are of wide use are he Riemann-Liouville definiion and he Capuo definiion, see [10, 11]. Corresponding auhor Email addresses: roshdi@ju.edu.jo (Roshdi Khalil), horani@ju.edu.jo, m.alhorani@uoh.edu.sa (Mohammed Al Horani), andersod@cord.edu (Douglas Anderson) Received 2016-01-11
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 141 (i) Riemann - Liouville Definiion. For α [n 1, n), he α derivaive of f is D α a (f)() = 1 d n Γ(n α) d n a f(x) dx. ( x) α n+1 (ii) Capuo Definiion. For α [n 1, n), he α derivaive of f is D α a (f)() = 1 Γ(n α) a f (n) (x) dx. ( x) α n+1 However, he following are some of he sebacks of one definiion or he oher: (i) The Riemann-Liouville derivaive does no saisfy D α a (1) = 0 (D α a (1) = 0 for he Capuo derivaive), if α is no a naural number. (ii) All fracional derivaives do no saisfy he known formula of he derivaive of he produc of wo funcions: D α a (fg) = fd α a (g) + gd α a (f). (iii) All fracional derivaives do no saisfy he known formula of he derivaive of he quoien of wo funcions: D α a (f/g) = gdα a (f) fd α a (g) g 2. (iv) All fracional derivaives do no saisfy he chain rule: D α a (f g)() = f (α)( g() ) g (α) (). (v) All fracional derivaives do no saisfy: D α D β f = D α+β f, in general. (vi) All fracional derivaives, especially he Capuo definiion, assume ha he funcion f differeniable. is In [9], he auhors gave a new definiion of a (local) fracional derivaive which is a naural exension o he usual firs derivaive as follows: Le f : [0, ) R be a given funcion. Then for all > 0 and α (0, 1), define T α (f) via f( + ε 1 α ) f() T α (f)() = lim. ε 0 ε T α is called he conformable fracional derivaive of f of order α. Le f (α) () sand for T α (f)(). I hen follows ha f (α) f( + ε 1 α ) f() () = lim. ε 0 ε
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 142 If f is α differeniable in some (0, b), b > 0, and lim 0 +f (α) () exiss, hen define f (α) (0) = lim 0 +f (α) (). According o his definiion, he following saemens are rue, see [9], see also [1, 2, 3, 4, 5, 7], 1. T α ( p ) = p p α for all p R. 2. T α (sin 1 α α ) = cos 1 α α. 3. T α (cos 1 α α ) = sin 1 α α. 4. T α (e 1 α α ) = e 1 α α. Furher, all he classical properies of he derivaive hold. This suggess ha one may ry o solve (local) fracional differenial equaions using he same echniques for solving ordinary differenial equaions. I is he purpose of his paper o sudy he mehod of undeermined coefficiens o find paricular soluions for linear fracional differenial equaions. The concep of fracional polynomials, fracional exponenial, and fracional rigonomeric funcions are inroduced, and hen applied for he undeermined coefficiens mehod. 2. Fracional Polynomial Funcions Le X be a vecor space, and T : X X, be a linear operaor on X. A subspace M X, is called invarian under T, if T (M) M. Such a concep was used in he mehod of undeermined coefficiens for ordinary differenial equaions. For such a siuaion, T was he differenial operaor, and M was aken o be he space generaed by polynomials, exponenial funcions, and rigonomeric funcions, added or muliplied. Definiion 2.1. Le n N, he se of naural numbers, and α (0, 1). We call α a facor of n if here exiss k N such ha kα = n. For example, 1 is a facor of 2 wih k = 4, and 1 is a facor of 1 wih k = 3. Bu 3 is no a facor 2 3 5 of 1. Definiion 2.2. A fracional polynomial of degree n and facor α is a funcion of he form P (x) = a n x n + a n 1 x n α +... + a n k 1 x α + a n k, where a j R, he se of real numbers. We wrie P (x) is an (n, α) fracional polynomial. If a n = 0, we ake n o be he smalles n for which α is a facor. For example, x+x 1 2 4 is a (1, 1 2 ) fracional polynomial, and 2x 3 2 +5x x 1 2 +7 is a (2, 1 2 ) fracional polynomial. In addiion, x + x 2 3 1 is a (1, 1 3 ) fracional polynomial. Here he coefficien of x 1 3 is 0. Le J(α) be he se of all (n, α) polynomials for all n N and fixed α; clearly, J(α) is a subspace of he space of all coninuous funcions on [0, ). Le G(α) be he space of all funcions of he form c 1 sin ( ) α α + c2 cos ( ) α α, where c1, c 2 R; E(α) be he space of all funcions of he form ce α α (, where c R; and le M(α) be he space of all funcions of he form e α α P 1 () sin ( ) α α + P2 () cos ( ) ) α, α where P 1, P 2 J(α).
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 143 Theorem 2.3. The subspaces J(α), G(α), E(α), and M(α) are invarian under he linear map T α. Proof. Using Theorem 2.2 in [9], and he facs, see also [9], he resul follows. T α (e 1 α α ) = e 1 α α, T α (sin 1 α α ) = cos( 1 α α ), T α (cos 1 α α ) = sin( 1 α α ), T α ( p ) = p p α, We should remark ha he spaces J(1), G(1), E(1), and M(1) are invarian under T 1, which is he firs derivaive. 3. Mehod of Undeermined Coefficiens Le us sar wih he following definiion. Definiion 3.1. A differenial equaion is called an (n, α) fracional differenial equaion if i is of he form a n y (n) + a n 1 y (n α) +... + a n k 1 y (α) + a n k y = f(), (3.1) where α is a facor of he naural number n. If a n = 0, we ake n o be he smalles n for which α is a facor. The differenial equaion (3.1) is called a fracional differenial equaion of order n and facor α. Since T α is linear, see [9], Theorem 2.2, hen one can easily see ha equaion (3.1) is linear. In his secion we consider equaions of he form y (α) + ay = f(), (3.2) where f() is an elemen of one of he spaces J(α), G(α), E(α), and M(α). Le us wrie y h for he soluion of he homogenous equaion y (α) + ay = 0, and y p for any paricular soluion of y (α) + ay = f(). Then as in he case of ordinary differenial equaions, he general soluion is y g = y h + y p. The equaion y (α) + ay = 0 can be wrien as (T α + ai)y = 0, where I is he ideniy operaor on he space of coninuous funcions on [0, ). Hence, y h is an elemen of he kernel of he operaor T α + ai. Now, since T α (e 1 α α ) = e 1 α α, i follows ha T α (e a α α ) = ae a α α. I follows ha he kernel of T α + ai consiss of he funcions be a α α, for b R. Thus y h for y (α) + ay = 0 is y h = be a α α. Remark 3.2. I is ineresing o observe ha he general soluion of he homogenous par of equaion (3.2) is y h = e r 1 α α and o noice ha one can form he auxiliary equaion r + a = 0, where y (α) is replaced by r. So r = a, and y h = e a 1 α α. For example, he equaion y ( 1 2 ) y = 0, has auxiliary equaion r 1 = 0, so r = 1, and hence y h = e 2. Now, o find y p using he mehod of undeermined coefficiens, he funcion f() mus be in one of he spaces J(α), G(α), E(α), and M(α). This is because such spaces are invarian under T α, and hence under T α + ai.
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 144 Consequenly, if f() = ce α, hen y p mus be be α, where b is o be deermined by subsiuing be α in y (α) + ay = ce α. If f() = c 1 sin α + c 2 cos α, hen y p mus be b 1 sin α + b 2 cos α, where b 1 and b 2 are o be deermined by subsiuing in he equaion y (α) + ay = c 1 sin α + c 2 cos α. Similarly if f() is an (n, α) fracional polynomial. Remark 3.3. The above discussion works for he form of y p as long as here is no similariy beween y h and f(). In such a case, we have o modify he form of y p as we will show laer in his paper. Example 3.4. Le us consider he general soluion of he following fracional differenial equaions: (1) y ( 1 2 ) + 2y = sin. Soluion. The auxiliary equaion of y ( 1 2 ) +2y = 0 is r+2 = 0, so r = 2. Hence y h = be 2(2) = be 4. Now, y p = A sin + B cos, noing here is no similariy beween y h and any of he erms of y p. Subsiuing y p in he equaion y ( 1 2 ) + 2y = sin, we ge A = 8 and B = 2. Hence 17 17 (2) y ( 1 3 ) y =. y g = be 4 + 8 17 sin 2 17 cos. Soluion. The auxiliary equaion of y ( 1 3 ) y = 0 is r 1 = 0, so r = 1. Hence y h = be 3 3. Again, y p = a + b 2 3 + c 1 3 + d, since here is no similariy beween any of he erms of y p and y h. Subsiuing y p in he equaion y ( 1 3 ) y =, we ge a = 1, b = 1, c = 2 3, d = 2 9. (3) y ( 1 2 ) 3y = e cos. Soluion. One can easily see ha y h = be 6. As for y p, he form is y p = e (c 1 + c 2 + c3 )(A sin + B cos ), noing ha here is no similariy beween y h and any of he erms of y p. So jus subsiuing y p in our equaion and deermine he coefficiens. 4. The case of similariy Wha if here is a similariy beween y h and any of he erms of y p? Here is such an example y ( 1 2 ) y = 5e 2. (4.1)
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 145 Here y h = e 2, and he form of y p is y p = be 2. Bu if we subsiue y p in equaion (4.1), we will no be able o deermine b. Hence we ry y p = b e 2. Subsiue such y p in equaion (4.1) o ge Hence b = 10. So y g = c e 2 + 10 e 2. b e 2 + 1 2 be2 b e 2 = 5e 2. Thus in case of similariy beween any of he erms of y p and y h we muliply he assumed form of y p by α, whenever he equaion is y (α) + by = f(). Open Problem 1. Is here an auxiliary equaion for equaions a n y (n) + a n 1 y (n α) +... + a n k 1 y (α) + a n k y = f(), wih a leas one coefficien of some derivaive(no fracional) no equal o zero? As an example y + y ( 3 2 ) y = 0? Open Problem 2. If f() does no belong o any of he spaces J(α), G(α), E(α), and M(α), is here a mehod of undeermined coefficiens o find y p? For example, in he case of (3.2), we have he following affirmaive resul. Lemma 4.1. Le f, a : [ 0, ) [0, ) be coninuous, and le y 0 R. Then he unique soluion of he iniial value problem y (α) () + a()y() = f(), y( 0 ) = y 0, is given by y() = y 0 e a(τ)τ α 1dτ 0 + e s a(τ)τ α 1dτ f(s)s α 1 ds, [ 0, ). (4.2) 0 Proof. Le y be given by (4.2). Using he conformable fracional derivaive rules, D α y() = y 0 a()e a(τ)τ α 1dτ 0 + f()e a(τ)τ α 1dτ a()e s a(τ)τ α 1dτ f(s)s α 1 ds 0 = a()y() + f(), which complees he proof of he lemma. References [1] T. Abdeljawad, On conformable fracional calculus, J. Compu. Appl. Mah., 279 (2015), 57 66. 1 [2] T. Abdeljawad, M. Al Horani, R. Khalil, Conformable Fracional Semigroups of Operaors, J. Semigroup Theory Appl., 2015 (2015), 11 pages. 1 [3] M. Abu Hammad, R. Khalil, Abel s formula and Wronskian for conformable fracional differenial equaions, In. J. Differenial Equaions Appl., 13 (2014), 177 183. 1 [4] B. Bayour, D. F. M. Torres, Exisence of soluion o a local fracional nonlinear differenial equaion, J. Compu. Appl. Mah., (2016), (in press). 1 [5] N. Benkheou, S. Hassani, D. F. M. Torres, A conformable fracional calculus on arbirary ime scales, J. King Saud Univ., 28 (2016), 93 98.1 [6] T. Caraballoa, M. Abdoul Diopb, A. A. Ndiayeb, Asympoic behavior of neural sochasic parial funcional inegro-differenial equaions driven by a fracional Brownian moion, J. Nonlinear Sci. Appl., 7 (2014), 407 421. 1
R. Khalil, M. Al Horani, D. Anderson, J. Mah. Compuer Sci. 16 (2016), 140 146 146 [7] A. Gokdogan, E. Unal, E. Celik, Exisence and Uniqueness Theorems for Sequenial Linear Conformable Fracional Differenial Equaions, o appear in Miskolc Mahemaical Noes. 1 [8] M. Hao, C. Zhai, Applicaion of Schauder fixed poin heorem o a coupled sysem of differenial equaions of fracional order, J. Nonlinear Sci. Appl., 7 (2014), 131 137. 1 [9] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new Definiion of Fracional Derivaive, J. Compu. Appl. Mah., 264 (2014), 65 70.1, 2, 3 [10] A. Kilbas, H. Srivasava, J. Trujillo, Theory and applicaions of fracional differenial equaions, Mah. Sudies. Norhholland, NewYork, (2006). 1 [11] K. S. Miller, An inroducion o fracional calculus and fracional differenial equaions, J.Wiley and Sons, New York, (1993). 1 [12] J. A. Nanware, D. B. Dhaigude, Exisence and uniqueness of soluions of differenial equaions of fracional order wih inegral boundary condiions, J. Nonlinear Sci. Appl., 7 (2014), 246 254. 1