Multi-variable Conformable Fractional Calculus

Transcription

1 Filomat 32: (208), Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Multi-variable Conformable Fractional Calculus Nazlı Yazıcı Gözütok a, Uğur Gözütok a a Karadeniz Technical University, Department of Mathematics, 6080, Trabzon, Turkey Abstract Conformable fractional derivative is introduced by the authors Khalil et al In this study we develop their concept and introduce multi-variable conformable derivative for a vector valued function with several variables Introduction For many years, many definitions of fractional derivative have been introduced by various researchers One of them is the Riemann-Liouville fractional derivative and the second one is the so-called Caputo derivative But these are not all purpose definitions Recently, a new fractional derivative has been introduced in [5] and one can see that the new derivative suggested in this paper satisfies all the properties of the standard one Applications of new defined fractional derivative called conformable fractional derivative are studied in papers [2 4] However, definitions given in the literature are only for the real valued functions In this paper, we introduce conformable fractional derivative definition for the vector valued functions with several real variables Our paper has the following order: In Section 2, some basic definitions and theorems appeared in the literature are given In Section 3, α derivative of a vector valued function, conformable Jacobian matrix are defined; relation between α derivative and usual derivative of a vector valued function is revealed; chain rule for multi-variable conformable derivative is given In Section 4, conformable partial derivatives of a real valued function with n variables is defined and relation between conformable Jacobian matrix and conformable partial derivatives is given 200 Mathematics Subject Classification Primary 26B2; Secondary 26A33 Keywords Conformable fractional derivative, conformable partial derivatives, conformable jacobian matrix Received: 22 January 207; Accepted: 2 April 207 Communicated by Dragan S Djordjević addresses: nazliyazici@ktuedutr (Nazlı Yazıcı Gözütok), ugurgozutok@ktuedutr (Uğur Gözütok)

2 2 Basic Definitions and Theorems NY Gözütok, U Gözütok / Filomat 32: (208), In this section we will give some definitions and properties introduced in [, 5] Definition 2 Given a function f : [0, ) R The conformable derivative of the function f of order α is defined by T α ( f )(x) = f (x + hx α ) f (x) h () for all x > 0, α (0, ) If the conformable derivative of a function f exists for an α (0, ), then the function f is called α differentiable Theorem 22 [5] If a function f : [0, ) R is α differentiable at t 0 > 0, α (0, ], then f is continuous at t 0 Theorem 23 [] Let α (0, ] and f, be α differentiable at a point t > 0 Then () T α (a f + b ) = at α ( f ) + bt α ( ), for all a, b R (2) T α (t p ) = pt p α for all p R (3) T α (λ) = 0, for all constant functions f (t) = λ (4) T α ( f ) = f T α ( ) + T α ( f ) (5) T α ( f ) = T α( f ) f T α ( ) 2 (6) If, in addition, f is differentiable, then T α ( f )(t) = t α d dt f (t) Theorem 24 Assume f, : (0, ) R be two α differentiable functions where α (0, ] α differentiable and for all t with t 0 and f (t) 0 we have Then f is T α ( f )(t) = T α ( )( f (t))t α ( f )(t) f (t) α (2) 3 α Derivative of a Vector Valued Function Definition 3 Let f be a vector valued function with n real variables such that f (x,, x n ) = ( f (x,, x n ),, f m (x,, x Then we say that f is α differentiable at a = (a,, a n ) R n where each a i > 0, if there is a linear transformation L : R n R m such that n ) f (a,, a n ) L(h) = 0 (3) where h = (h,, h n ) and α (0, ] The linear transformation is denoted by D α f (a) and called the conformable derivative of f of order α at a Remark 32 For m = n =, Definition 3 equivalent to Definition 2 Theorem 33 Let f be a vector valued function with n variables If f is α differentiable at a = (a,, a n ) R n, each a i > 0, then there is a unique linear transformation L : R n R m such that n ) f (a,, a n ) L(h) = 0

3 Proof Suppose M : R n R m satisfies Then, NY Gözütok, U Gözütok / Filomat 32: (208), α ) f (a,, a n ) M(h) = 0 L(h) M(h) L(h) ( f (a + h a α,, a n ) f (a)) ( f (a + h a α,, a n ) f (a)) M(h) + If x R n, then tx 0 as t 0 Hence for x 0 we have L(tx) M(tx) 0 = = tx Therefore L(x) = M(x) L(x) M(x) x Example 34 Let us consider the function f defined by f (x, y) = sinx and the point (a, b) R 2 such that a, b > 0, then D α f (a, b) = L satisfies L(x, y) = xa α cos a To prove this, note that f (a + h a α, b + h 2 b α ) f (a, b) L(h, h 2 ) (h,h 2 ) (0,0) (h, h 2 ) sin(a + h a α ) sin a h a α cos a = (h,h 2 ) (0,0) h 2 + h2 2 sin(a + h a α ) sin a h a α cos a h 0 h = h 0 sin(a + h a α ) sin a h a α cos a = a α cos a a α cos a = 0 Definition 35 Consider the matrix of the linear transformation D α f (a) : R n R m with respect to the usual bases of R n and R m This m n matrix is called the conformable Jacobian matrix of f at a, and denoted by f α (a) Example 36 If f (x, y) = sin x, then f α (a, b) = [ a α cos a 0 ] Theorem 37 Let f be α differentiable at a = (a,, a n ) R n, each a i > 0 If f is differentiable at a, then D α f (a) = D f (a) L α a where D f (a) is the usual derivative of f and L α a (a α x,, an α x n ) Proof It is suffices to show that n ) f (a,, a n ) D f (a) L α a (h) = 0 = 0 is the linear transformation from R n to R n defined by L α a (x,, x n ) =

4 NY Gözütok, U Gözütok / Filomat 32: (208), Let ɛ = (ɛ,, ɛ n ) = (h a α,, h n a α n ), then ɛ 0 as h 0 On the other hand, if we put M = max{(a α ) 2 i a i > 0, i =, 2,, n} > 0, then ɛ = h 2 (a α ) h 2 n(a α n ) 2 h 2 M + + h2 nm = nm Hence we have Finally, nm ɛ n ) f (a,, a n ) D f (a) L α a (h) n ) f (a) D f (a)(ɛ) = f (a + ɛ) f (a) D f (a)(ɛ) ɛ 0 ɛ nm = f (a + ɛ) f (a) D f (a)(ɛ) nm = nm0 = 0 ɛ 0 ɛ This completes the proof Theorem 37 is the generalized case of the part (6) of Theorem 23 Also matrix form of the Theorem 37 is given by the following: f α (a) = f (a) a α a α a α n where f (a) is the usual Jacobian of f and L α a, a α a α a α n is the matrix corresponding to linear transformation Theorem 38 If a vector valued function f with n variables is α differentiable at a = (a,, a n ) R n, each a i > 0, then f is continuous at a R n Proof Since We have f (a + h a α,, a n + h n an α ) f (a,, a n ) = n ) f (a) L(h) + L(h) n ) f (a) L(h) + L(h) f (a + h a α,, a n + h n an α ) f (a,, a n ) n ) f (a) L(h) + L(h)

5 NY Gözütok, U Gözütok / Filomat 32: (208), By taking its of the two sides of the inequality as h 0, we have f (a + h a α,, a n + h n an α ) f (a,, a n ) f (a + h a α + L(h) = 0,, a α ) f (a) L(h) Let (ɛ,, ɛ n ) = (h a α,, h n a α n ), then ɛ 0 as h 0 Since f (a + ɛ) f (a) 0 ɛ 0 we have f (a + ɛ) f (a) = 0 ɛ 0 Hence f is continuous at a R n Theorem 39 (Chain Rule) Let x R n, y R m If f (x) = ( f (x),, f m (x)) is α differentiable at a = (a,, a n ) R n, each a i > 0 such that α (0, ] and (y) = ( (y),, p (y)) is α differentiable at f (a) R m, all f i (a) > 0 such that α (0, ] Then the composition f is α differentiable at a and D α ( f )(a) = D α ( f (a)) f (a) α D α f (a) (4) where f (a) α is the linear transformation from R m to R m defined by f (a) α (x,, x m ) = (x f (a) α,, x m f m (a) α ) Proof Let L = D α f (a), M = D α ( f (a)) If we define (i) ϕ(a + h a α,, a n ) = f (a + h a α,, a n ) f (a) L(h), (ii) ψ( f (a) + k f (a) α,, f n (a) + k n f n (a) α ) = ( f (a) + k f (a) α,, f n (a) + k n f n (a) α ) ( f (a)) M(k), (iii) ρ(a + h a α,, a n ) = f (a + h a α,, a α ) f (a) M f (a) α L(h), then ϕ(a + h a α,, a n ) (iv) = 0, (v) k 0 ψ( f (a) + k f (a) α,, f n (a) + k n f n (a) α ) k and we must show that ρ(a + h a α,, a α ) = 0 = 0,

6 NY Gözütok, U Gözütok / Filomat 32: (208), Now, ρ(a + h a α,, a n + h n an α ) = ( f (a + h a α,, a n + h n an α )) ( f (a)) M f (a) α L(h) = ( f (a + h a α,, a n + h n a α n ),, f m (a + h a α,, a n + h n a α ( f (a)) M f (a) α ( f (a + h a α,, a n + h n a α n ) f (a) ϕ(a + h a α,, a n + h n a α by(i) = [ ( f (a + h a α,, a n + h n an α ),, f m (a + h a α,, a n + h n a α ( f (a)) M( f (a) α ( f (a + h a α,, a n + h n a α n ) f (a)))] + M f (a) α (ϕ(a + h a α,, a n + h n a α = [ ( f (a + h a α,, a n + h n an α ),, f m (a + h a α,, a n + h n a α ( f (a)) M( f (a) α ( f (a + h a α,, a n + h n a α n ) f (a),, f m (a + h a α,, a n + h n an α ) f m (a)))] + M f (a) α (ϕ(a + h a α,, a n + h n a α = [ ( f (a + h a α,, a n + h n an α ),, f m (a + h a α,, a n + h n a α ( f (a)) M([ f (a + h a α,, a n + h n a α n ) f (a)] f (a) α,, [ f m (a + h a α,, a n + h n a α n ) f m (a)] f m (a) α )] + M f (a) α (ϕ(a + h a α,, a n + h n a α If we put u i = [ f i (a +h a α,, a n +h n a α n ) f i (a)] f i (a) α, i =,, n, then we have f i (a +h a α,, a n +h n a α n ) = f i (a) + u i f i (a) α and u 0 as h 0 Therefore, ρ(a + h a α,, a n + h n a α n ) = [ ( f (a + h a α,, a n + h n a α n ),, f m (a + h a α,, a n + h n a α ( f (a)) M(u)] + M f (a) α (ϕ(a + h a α,, a n + h n a α = ψ( f (a) + u f (a) α,, f m (a) + u m f m (a) α ) + M f (a) α (ϕ(a + h a α,, a n + h n a α by(ii) Thus we must show (vi) u 0 ψ( f (a) + u f (a) α,, f m (a) + u m f m (a) α ) u = 0, M f (a) α (ϕ(a + h a α,, a (vii) = 0 For (vi), it is obvious from (v) For (vii), the linear transformation satisfies M f (a) α (ϕ(a + h a α,, a n + h n a α K ϕ(a + h a α,, a n + h n a α n ) such that K > 0 Hence, from (iv), (vii) holds Corollary 30 For, m = n = p =, Theorem 39 states that T α ( f )(a) = T α ( f (a))t α f (a) f (a) α

7 NY Gözütok, U Gözütok / Filomat 32: (208), Above corollary says that Theorem 39 is the generalized case of Theorem 24 Corollary 3 Let all conditions of Theorem 39 be satisfied Then where ( f ) α (a) = α ( f (a)) f (a) α f 2 (a) α f m (a) α f (a) α f 2 (a) α f m (a) α f α (a) is the matrix corresponding to the linear transformation f (a)α Theorem 32 Let f be a vector valued function with n variables such that f (x,, x n ) = ( f (x,, x n ),, f m (x,, x Then f is α differentiable at a = (a,, a n ) R n, each a i > 0 if and only if each f i is, and D α f (a) = (D α f (a),, D α f m (a)) Proof If each f i is α differentiable at a and L = (D α f (a),, D α f m (a)), then Therefore, f (a + h a α,, a n + h n an α ) f (a) L(h) = ( f (a + h a α,, a n + h n a α n ) f (a) D α f (a)(h),, f m (a + h a α,, a n + h n a α n ) f m (a) D α f m (a)(h)) n ) f (a) L(h) n f i (a + h a α,, a α ) f i (a) D α f i (a)(h) i= If f is α differentiable at a, then f i = π i f is α differentiable at a by Theorem 39 = 0 Theorem 33 Let α (0, ] and f, be α differentiable at a point a = (a,, a n ) R n, each a i > 0 Then (i) D α (λ f + µ )(a) = λd α f (a) + µd α (a) for all λ, µ R (ii) D α ( f )(a) = f (a)d α (a) + (a)d α f (a) Proof (i) follows from the definition, thus we omitted the proof of (i)

8 For (ii), let a + h a α,, a n = A, then NY Gözütok, U Gözütok / Filomat 32: (208), ( f )(A) ( f )(a) ( f (a)d α (a) + (a)d α f (a))(h) f (A) (A) f (a) (A) (A)D α f (a)(h) f (a) (A) f (a) (a) f (a)d α (a)(h) + (A)D α f (a)(h) (a)d α f (a)(h) + = (A) f (A) f (a) Dα f (a)(h) + f (a) (A) (a) Dα (a)(h) + D α (A) (a) f (a)(h) (A) (a) K = 0 This completes the proof 4 Conformable Partial Derivatives In this section we introduce the definition of conformable partial derivative of a real valued function with n variables by the following Definition 4 Let f be a real valued function with n variables and a = (a,, a n ) R n be a point whose ith component is positive Then the it f (a,, a i + ɛa α,, a i n ) f (a,, a n ), (5) ɛ 0 ɛ if it exists, is denoted by α f (a), and called the ith conformable partial derivative of f of order α (0, ] at a xα Theorem 42 Let f be a vector valued function with n variables If f is α differentiable at a = (a,, a n ) R n, each a i > 0, then α ( α f i (a) ) f i (a) exists for i m, j n and the conformable Jacobian of f at a is the m n matrix Proof Let f (x,, x n ) = ( f (x,, x n ),, f m (x,, x Suppose first that m =, so f (x,, x n ) R Define h : R R n by h(y) = (a,, y,, a n ) with y in the jth place Then α f i (a) = D α ( f h)(a j ) Hence, by

9 NY Gözütok, U Gözütok / Filomat 32: (208), Corollary 3, h (a j ) α 0 0 ( f h) α (a j ) = f α (h(a j )) = f α (a) 0 0 h j (a j ) α h n (a j ) α a α a α 0 j 0 0 a α n Since ( f h) α (a j ) has the single entry α 0 h α (a j ) 0 0 a α j = f α (a) 0 f i (a), this shows that α f i (a) exists and is the jth entry of the n matrix f α (a) The theorem now follows for arbitrary m since, by Theorem 32, each f i is α differentiable and the ith row of f α (a) is ( f i ) α (a) 5 Conclusion In our study, we have extended the idea given by Khalil at al in [5] and called it multi-variable conformable fractional calculus Multi-variable conformable fractional derivative has many interesting properties and is related to classical multi-variable calculus We have given important theorems and corollaries which reveal this relation and we have obtained useful results Acknowledgements The first author would like to thank TUBITAK(The Scientific and Technological Research Council of Turkey) for their financial supports during her doctorate studies References [] T Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (205) [2] DR Anderson, DJ Ulness, Newly defined conformable derivatives, Advances in Dynamical Systems and Applications 0(2) (205) [3] N Benkhettou, S Hassani, DFM Torres, A conformable fractional calculus on arbitrary time scales, Journal of King Saud University 28() (206) [4] H Batarfi, J Losada, JJ Nieto, W Shammakh, Three-point boundary value problems for conformable fractional differential equations, 205 (205) 7 [5] R Khalil, M Al Horani, A Yousef, M Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (204) 65 70