Type of Leibniz Rule on Riemann-Liouville Variable-Order Fractional Integral and Derivative Operator
aa r X i v : . [ m a t h . G M ] J a n Type of Leibniz Rule on Riemann-Liouville Variable-OrderFractional Integral and Derivative Operator
Dagnachew Jenber a,⋆ , Mollalign Haille b a Department of Mathematics, Addis Ababa Science and Technology University, Addis Ababa, EthiopiaDepartment of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia, P.O.Box 79Email: djdm [email protected] b Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia, P.O.Box 79Email: [email protected]
Abstract
In this paper, types of Leibniz Rule for Riemann-Liouville Variable-Order fractional inte-gral and derivative Operator is developed. The product rule, quotient rule, and chain ruleformulas for both integral and differential operators are established. In particular, there arefour types of product rule formulas: Product rule type-I, Product rule type-II, Product ruletype-III and Product rule type-Iv. Quotient rule type-I, quotient rule type-II, quotient ruletype-III, and quotient rule type-Iv formulas developed from product rule types. There arefour types of chain rule formulas: chain rule type-I, chain rule type-II, chain rule type-III,and chain rule type-Iv.
Keywords:
Fractional integral inequalities, Riemann-Liouville variable-order fractionalintegral, Leibniz RuleMSC 2010: 26D10, 26A33, 26A24
1. Introduction
Fractional calculus, that is fractional derivative and integral of an arbitrary real order, hasa history of more than three hundred years (see [1],[2] and the references therein). In 1993,Samko and Ross [3] firstly proposed the notion of variable-order integral and differentialoperators and some basic properties. Lorenzo and Hartley [4] summarized the researchresults of the variable-order fractional operators and then investigated the definitions ofvariable-order fractional operators in different forms. After that, some new extensions andvaluable application potentials of the variable-order fractional differential equation modelshave been further explored [5]. It has become a research hotspot and has aroused wideconcern in the last ten years. Different kind of definitions of fractional derivatives andintegrals are available in the literature. Forexample, Riemann-Liouville, Riesz, Caputo,Coimbra, Hadamard, Gr¨unwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov, conformable ⋆ corresponding authorDagnachew Jenber Preprint submitted to arXiv January 20, 2021 nd others (see [6],[7], [15] and the references therein). Excepting conformable fractionalderivative (see [9]) the other definition violates basic properties of Leibniz rule that holds forinteger order calculus, like product rule and chain rule. V.E. Tarasov proved that fractionalderivatives of non-integer orders can not satisfy the Leibniz rule (see [13],[14]). There aresome attempts to define new type of fractional derivative such that the Leibniz rule holds(see [10],[11],[12]). This paper established a Leibnize rule type formula like product rule,quotient rule and chain rule for Riemann-Liouville variable-order fractional derivative andintegral operator. We will leave linearity property for the reader to check, since it is obviousand straightforward.
2. Preliminaries
Throughout this paper, we will use the following definitions.
Definition 1.
Given ℜ ( z ) > , we define the gamma function, Γ( z ) , as Γ( z ) = Z ∞ t z − e − t dt Γ( z ) is a holomorphic function in ℜ ( z ) > . In the following definition of Riemann-Liouville variable-order fractional integral, we usedthe abbreviation RL stands for Riemann-Liouville.
Definition 2. (see[8]) Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) . Then the left Riemann-Liouvillefractional integral of order α ( ., . ) for function f ( t ) is defined by RLa I α ( .,. ) t f ( t ) = Z ta ( t − s ) α ( t,s ) − Γ( α ( t, s )) f ( s ) ds, t > a (1) Definition 3. (see[8]) Let α : [ a, b ] × [ a, b ] −→ (0 , . Then the left Riemann-Liouvillefractional derivative of order α ( ., . ) for function f ( t ) is defined by RLa D α ( .,. ) t f ( t ) = ddt (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19) = ddt Z ta ( t − s ) − α ( t,s ) Γ(1 − α ( t, s )) f ( s ) ds, t > a (2)
3. Main Result
For the Reimann-Liouville variable-order fractional integral operator, from Theorem (1),we get, product rule formulas and from the consequence of this Theorem, product rule type-I,product rule type-II, product rule type-III and product rule type-IV are obtained.2 heorem 1.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a, s > c . Then for functions f and g the following equality holds (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19)(cid:18) RLc I β ( .,. ) s (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s ( f g )( s ) (cid:19) = (cid:18) RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19)(cid:19) + (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19) × (cid:18) RLc I β ( .,. ) s f ( s ) (cid:19) + (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s g ( s ) (cid:19) (3) Proof.
Since f ( x ) g ( x ) = ( f ( x ) − f ( y ))( g ( x ) − g ( y )) + f ( y ) g ( x ) + f ( x ) g ( y ) − f ( y ) g ( y ) . (4)Now, multiplying equation (4) by ( t − x ) α ( t,x ) − / Γ( α ( t, x )) and integrate from a to t withrespect to x , we have Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) f ( x ) g ( x ) dx = Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) ( f ( x ) − f ( y ))( g ( x ) − g ( y )) dx + Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) f ( y ) g ( x ) dx + Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) f ( x ) g ( y ) dx − Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) f ( y ) g ( y ) dx which means RLa I α ( .,. ) t ( f g )( t ) = RLa I α ( .,. ) t (cid:18) ( f ( t ) − f ( y ))( g ( t ) − g ( y )) (cid:19) + f ( y ) (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19) + g ( y ) (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) − (cid:18) f ( y ) g ( y ) (cid:19)(cid:18) RLa I α ( .,. ) t (1) (cid:19) (5)Now, multiplying equation (5) by ( s − y ) β ( s,y ) − / Γ( β ( s, y )) and integrate from c to s with3espect to y , we get, Z sc ( s − y ) β ( s,y ) − Γ( β ( s, y )) RLa I α ( .,. ) t ( f g )( t ) dy = Z sc ( s − y ) β ( s,y ) − Γ( β ( s, y )) (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( y ))( g ( t ) − g ( y )) (cid:19) dy + Z sc ( s − y ) β ( s,y ) − Γ( β ( s, y )) f ( y ) (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19) dy + Z sc ( s − y ) β ( s,y ) − Γ( β ( s, y )) g ( y ) (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) dy − Z sc ( s − y ) β ( s,y ) − Γ( β ( s, y )) (cid:18) f ( y ) g ( y ) (cid:19)(cid:18) RLa I α ( .,. ) t (1) (cid:19) dy which means (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19)(cid:18) RLc I β ( .,. ) s (1) (cid:19) = (cid:18) RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19)(cid:19) + (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s f ( s ) (cid:19) + (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s g ( s ) (cid:19) − (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s ( f g )( s ) (cid:19) which means (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19)(cid:18) RLc I β ( .,. ) s (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s ( f g )( s ) (cid:19) = (cid:18) RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19)(cid:19) + (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19) × (cid:18) RLc I β ( .,. ) s f ( s ) (cid:19) + (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s g ( s ) (cid:19) From Theorem (1), we established the following corollary (1), corollary (2), corollary (3),and corollary (4).
Corollary 1 ( Product rule type-I ).
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a ,and t > c . Then (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19)(cid:18) RLc I β ( .,. ) t (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) t ( f g )( t ) (cid:19) (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19)(cid:18) RLc I β ( .,. ) t f ( t ) (cid:19) + (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) t g ( t ) (cid:19) (6) Proof.
From Theorem (1), equation (3). Letting s = t completes the proof. Corollary 2 ( Product rule type-II).
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a .Then (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19)(cid:18) RLa I β ( .,. ) t (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLa I β ( .,. ) t ( f g )( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa I β ( .,. ) t f ( t ) (cid:19) + (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I β ( .,. ) t g ( t ) (cid:19) (7) Proof.
From Theorem (1), equation (3). Letting s = t and a = c completes the proof. Corollary 3 ( Product rule type-III).
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a .Then (cid:18) RLa I α ( .,. ) t ( f g )( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) (8) Proof.
From Theorem (1), equation (3). Letting s = t , a = c , and α ( ., . ) = β ( ., . ) completesthe proof. Corollary 4 ( Product rule type-IV).
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a .Then (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) (9) Proof.
From Theorem (1), equation (3). Letting s = t , a = c , and α ( ., . ) = β ( ., . ) and f = g completes the proof. Remark 1.
Quotient rule type-I, quotient rule type-II, quotient rule type-III, and quotientrule type-IV formulas is the same as product rule types, that is, from equation (6), equa-tion (7), equation (8), and equation (9) respectively by letting g = 1 /h such that h is nonzero. Theorem 2.
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a , n ∈ N . Then for function f n the following equality holds (cid:18) RLa I α ( .,. ) t f n ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − ( n − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) n (10) Proof.
Use mathematical induction. For n = 2, equation (10) becomes product rule type-Iv.Now, assume that equation (10) is true for n = k . Let us show that equation (10) also holdsfor n = k + 1, we have, (cid:18) RLa I α ( .,. ) t f k +1 ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t f k ( t ) f ( t ) (cid:19) (11)5ow, use product rule type-III for the right-hand side of equation (11). Then we have, (cid:18) RLa I α ( .,. ) t f k +1 ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t f k ( t ) f ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t f k ( t ) (cid:19) (12)now, using our assumption for n = k is true, equation (12) becomes, (cid:18) RLa I α ( .,. ) t f k +1 ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t f k ( t ) f ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t f k ( t ) (cid:19) = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t (1) (cid:19) − ( k − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) k = (cid:18) RLa I α ( .,. ) t (1) (cid:19) − k (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) k +1 . This completes the proof.For the Reimann-Liouville variable-order fractional integral operator, the following Theo-rem (3) established chain rule type-I and from the consequence of this Theorem we canobtain Chain rule type-II, chain rule type-III and chain rule type-IV.
Theorem 3.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a , g ( f ) = ( g ◦ f )( x ) , where f := f ( x ) and for f ( t ) > c . Then we have (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19) = (cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) (13) Proof.
This Theorem can be proved in two different approachs.6 ethod-I:
Using Riemann-Liouville variable-order fractional integral definition, we’ve (cid:18)
RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t f ( t ) g ( s ) (cid:19)(cid:19) = (cid:18) RLc I β ( .,. ) s (cid:18) g ( s ) (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s g ( s ) (cid:19) which implies (cid:18) RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t f ( t ) g ( s ) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s g ( s ) (cid:19) (14)now suppose s = f ( t ), then equation (14) becomes (cid:18) RLc I β ( .,. ) f ( t ) (cid:18) RLa I α ( .,. ) t f ( t )( g ◦ f )( t ) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) (15)use product rule type-III for the left-hand side of equation (15), that is, RLc I β ( .,. ) f ( t ) (cid:20)(cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19)(cid:21) = (cid:18) RLc I β ( .,. ) f ( t ) (cid:18) RLa I α ( .,. ) t f ( t )( g ◦ f )( t ) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) which means (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) = RLc I β ( .,. ) f ( t ) (cid:20)(cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19)(cid:21) = (cid:18) RLc I β ( .,. ) f ( t ) (cid:18) RLa I α ( .,. ) t f ( t )( g ◦ f )( t ) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) this implies 7 RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) this implies (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19) = (cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) Method-II:
Let g ( f ) = ( g ◦ f )( x ), where f := f ( x ) and then multiplying this equation by( t − x ) α ( t,x ) − / Γ( α ( t, x )) and integrate with respect to x from a to t , we get, Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) g ( f ) dx = Z ta ( t − x ) α ( t,x ) − Γ( α ( t, x )) ( g ◦ f )( x ) dx which means g ( f ) (cid:18) RLa I α ( .,. ) t (1) (cid:19) = RLa I α ( .,. ) t ( g ◦ f )( t ) (16)multiply equation (16) by ( f ( t ) − f ( x )) β ( f ( t ) ,f ( x )) − / Γ( β ( f ( t ) , f ( x ))) and integrate withrespect to f ( x ) from c to f ( t ), that is, Z f ( t ) c ( f ( t ) − f ( x )) β ( f ( t ) ,f ( x )) − Γ( β ( f ( t ) , f ( x ))) (cid:18) RLa I α ( .,. ) t (1) (cid:19) g ( f ( x )) df ( x )= Z f ( t ) c ( f ( t ) − f ( x )) β ( f ( t ) ,f ( x )) − Γ( β ( f ( t ) , f ( x ))) RLa I α ( .,. ) t ( g ◦ f )( t ) df ( x )which means (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) = (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) this implies (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19) = (cid:18) RLc I β ( .,. ) f ( t ) g ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) f ( t ) (1) (cid:19) Corollary 5 ( Chain rule type-II).
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a , g ( f ) = ( g ◦ f )( x ) , where f := f ( x ) and for f ( t ) > c . Then we have, (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19) = (cid:18) RLc I α ( .,. ) f ( t ) g ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I α ( .,. ) f ( t ) (1) (cid:19) (17) Proof.
From Theorem (3), equation (13). Letting α = β completes the proof. Corollary 6 ( Chain rule type-III).
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a , g ( f ) = ( g ◦ f )( x ) , where f := f ( x ) and for f ( t ) > c . Then we have, (cid:18) RLa I α ( .,. ) t ( g ◦ f )( t ) (cid:19) = (cid:18) RLa I α ( .,. ) f ( t ) g ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLa I α ( .,. ) f ( t ) (1) (cid:19) (18) Proof.
From Theorem (3), equation (13). Letting α = β and a = c completes the proof. Corollary 7 ( Chain rule type-IV).
Let α : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a ∈ R , t > a , g ( f ) = ( g ◦ f )( x ) , where f := f ( x ) and for f ( t ) > c . Then we have, (cid:18) RLa I α ( .,. ) t ( f ◦ f )( t ) (cid:19) = (cid:18) RLa I α ( .,. ) f ( t ) f ( f ( t )) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLa I α ( .,. ) f ( t ) (1) (cid:19) (19) Proof.
From Theorem (3), equation (13). Letting α = β , a = c and f = g completes theproof.In the following Theorem (4), equation (20) mentions the relationship between variable-order Riemann-Liouville integrals of addition, subtraction and product of two functions withrespect to two different variables beautifully. The consequences of this theorem becomes morebeautifull. Theorem 4.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a, s > c . Then for functions f and g : Lc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19) = (cid:18) RLa I α ( .,. ) t ( f ( t ) g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s f ( s ) g ( s ) (cid:19) + 12 (cid:20)(cid:18) RLa I α ( .,. ) t ( f ( t ) − g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s ( f ( s ) − g ( s )) (cid:19) − (cid:18) RLa I α ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s ( f ( s ) + g ( s )) (cid:19)(cid:21) (20) Proof.
Since, ( f ( t ) − f ( s ))( g ( t ) − g ( s ))= f ( t ) g ( t ) + f ( s ) g ( s )+ 12 (cid:20)(cid:18) f ( t ) − g ( t ) (cid:19)(cid:18) f ( s ) − g ( s ) (cid:19) − (cid:18) f ( t ) + g ( t ) (cid:19)(cid:18) f ( s ) + g ( s ) (cid:19)(cid:21) (21)applying the operator RLa I α ( .,. ) t on equation (21) and use linearity property, we have, RLa I α ( .,. ) t (cid:18) ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19) = RLa I α ( .,. ) t (cid:18) f ( t ) g ( t ) (cid:19) + RLa I α ( .,. ) t (cid:18) f ( s ) g ( s ) (cid:19) + 12 (cid:20) RLa I α ( .,. ) t (cid:18) f ( t ) − g ( t ) (cid:19)(cid:18) f ( s ) − g ( s ) (cid:19) − RLa I α ( .,. ) t (cid:18) f ( t ) + g ( t ) (cid:19)(cid:18) f ( s ) + g ( s ) (cid:19)(cid:21) (22)which means using linearity property, RLa I α ( .,. ) t (cid:18) ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19) = RLa I α ( .,. ) t ( f ( t ) g ( t )) + f ( s ) g ( s ) (cid:18) RLa I α ( .,. ) t (1) (cid:19) + 12 (cid:20) ( f ( s ) − g ( s )) (cid:18) RLa I α ( .,. ) t ( f ( t ) − g ( t )) (cid:19) − ( f ( s ) + g ( s )) (cid:18) RLa I α ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:21) (23)10pplying the operator RLc I β ( .,. ) s on equation (23) and use linearity property, we get, RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19) = RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) g ( t )) (cid:19) + (cid:18) RLc I β ( .,. ) s f ( s ) g ( s ) (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:19) + 12 (cid:20)(cid:18) RLc I β ( .,. ) s ( f ( s ) − g ( s )) (cid:18) RLa I α ( .,. ) t ( f ( t ) − g ( t )) (cid:19)(cid:19) − (cid:18) RLc I β ( .,. ) s ( f ( s ) + g ( s )) (cid:18) RLa I α ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:19)(cid:21) which means RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s ))( g ( t ) − g ( s )) (cid:19) = (cid:18) RLa I α ( .,. ) t ( f ( t ) g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s f ( s ) g ( s ) (cid:19) + 12 (cid:20)(cid:18) RLa I α ( .,. ) t ( f ( t ) − g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s ( f ( s ) − g ( s )) (cid:19) − (cid:18) RLa I α ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) s ( f ( s ) + g ( s )) (cid:19)(cid:21) Corollary 8.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a, s > c . Then for functions f and g the following equality holds (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLc I β ( .,. ) s (cid:18) RLa I α ( .,. ) t ( f ( t ) − f ( s )) (cid:19)(cid:19) = (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19) (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLc I β ( .,. ) s (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) s f ( s ) (cid:19) (cid:18) RLc I β ( .,. ) s (1) (cid:19) − − (cid:18) RLa I α ( .,. ) t f ( t ) (cid:19)(cid:18) RLc I β ( .,. ) s f ( s ) (cid:19) (24) Proof.
From equation (20), let f = g and use product rule type-IV. Corollary 9.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a, s > c . Then for functions f and g the following equality holds RLa I α ( .,. ) t ( f ( t ) g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) t (1) (cid:19) + (cid:18) RLa I α ( .,. ) t (1) (cid:19)(cid:18) RLc I β ( .,. ) t f ( t ) g ( t ) (cid:19) + 12 (cid:20)(cid:18) RLa I α ( .,. ) t ( f ( t ) − g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) t ( f ( t ) − g ( t )) (cid:19) − (cid:18) RLa I α ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:18) RLc I β ( .,. ) t ( f ( t ) + g ( t )) (cid:19)(cid:21) = 0 (25) Proof.
From equation (20), letting s = t completes the proof.The next Theorem (5) will show us how to operate with Riemann-Liouville variable-orderfractional integral operator of the product of two functions with two-variable. Theorem 5.
Let α, β : [ a, b ] × [ a, b ] −→ (0 , ∞ ) , a, c ∈ R , t > a, s > c . Then for functions F and G the following equality holds RLa I α ( .,. ) t RLc I β ( .,. ) s F ( t, s ) G ( t, s )= (cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s F ( t, s ) (cid:19)(cid:19) × (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s G ( t, s ) (cid:19)(cid:19) (26) Proof.
Applying product rule type-III repeatedly, that is,
RLa I α ( .,. ) t RLc I β ( .,. ) s F ( t, s ) G ( t, s )= RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s F ( t, s ) G ( t, s ) (cid:19) = RLa I α ( .,. ) t (cid:18)(cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLc I β ( .,. ) s F ( t, s ) (cid:19)(cid:18) RLc I β ( .,. ) s G ( t, s ) (cid:19)(cid:19) = (cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (cid:18)(cid:18) RLc I β ( .,. ) s F ( t, s ) (cid:19)(cid:18) RLc I β ( .,. ) s G ( t, s ) (cid:19)(cid:19) = (cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s F ( t, s ) (cid:19)(cid:19) × (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s G ( t, s ) (cid:19)(cid:19) this implies RLa I α ( .,. ) t RLc I β ( .,. ) s F ( t, s ) G ( t, s )= (cid:18) RLc I β ( .,. ) s (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (1) (cid:19) − (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s F ( t, s ) (cid:19)(cid:19) × (cid:18) RLa I α ( .,. ) t (cid:18) RLc I β ( .,. ) s G ( t, s ) (cid:19)(cid:19) emark 2. To find the product rule, quotient rule, and chain rule formulas for Riemann-Liouville variable-order fractional derivative operator, use definition (2), that is,
RLa D α ( .,. ) t f ( t ) = ddt (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19) (27) where α : [ a, b ] × [ a, b ] −→ (0 , , a ∈ R and t > a . For example, let’s see the next Theorem (6)which is product rule type-III. Theorem 6.
Let α : [ a, b ] × [ a, b ] −→ (0 , , a ∈ R , t > a . Then (cid:18) RLa D α ( .,. ) t ( f g )( t ) (cid:19) = − (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t (1) (cid:19) + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t f ( t ) (cid:19) + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t f ( t ) (cid:19) (28) Proof.
From definition (2), we have,
RLa D α ( .,. ) t f ( t ) g ( t ) = ddt (cid:18) RLa I − α ( .,. ) t f ( t ) g ( t ) (cid:19) (29)now use product rule type-III for the right-hand side of equation (29), we have, RLa D α ( .,. ) t f ( t ) g ( t ) = ddt (cid:18) RLa I − α ( .,. ) t f ( t ) g ( t ) (cid:19) = ddt (cid:18)(cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:19) (30)13ow use Leibniz product Rule for the right-hand side of equation (30), we have, RLa D α ( .,. ) t f ( t ) g ( t ) = ddt (cid:18) RLa I − α ( .,. ) t f ( t ) g ( t ) (cid:19) = ddt (cid:18)(cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:19) = (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19) ddt (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19) ddt (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19) + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19) ddt (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19) = − (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa I − α ( .,. ) t f ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t (1) (cid:19) + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t f ( t ) (cid:19) + (cid:18) RLa I − α ( .,. ) t (1) (cid:19) − (cid:18) RLa I − α ( .,. ) t g ( t ) (cid:19)(cid:18) RLa D α ( .,. ) t f ( t ) (cid:19) Authors’ contributions:
All authors worked jointly and all the authors read and approvedthe final manuscript.
Funding:
This research received no external funding.
Conflicts of Interest:
The authors declare no conflict of interest.