On Fractional φ- and biφ-calculi

WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi On Fractional 𝝋- and bi𝝋-calculi AMER H. DARWEESH 1 , ABDELAZIZ M.D. MAGHRABI 2 1 Department of Mathemtics, Jordan University of Science and Technology, Irbid 22110, JORDAN 2 Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, TURKEY Abstract: - In this paper we introduce fractional 𝜑- and bi𝜑-calculi using Riemann-Liouville approach and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since they are not as common as classical calculus. This was also done for tanh-, bi-tanh- multiplicative, and bigeometric calculi and in the general case as well. Generalizations are also investigated where the homeomorphisms 𝜑, 𝜂 are arbitrary. Key-Words: - Derivative, integral, non-Newtonian calculus, fractional derivative, homeomorphism. Received: April 19, 2022. Revised: January 14, 2023. Accepted: February 9, 2023. Published: March 7, 2023. on 𝕀 = (0, ∞), the exponential-operations give rise to two pairs of calculi on functions 𝑓 : 𝕀 → 𝕀. This will be further explained later on in the second section. This paper is organized in the following way. In Section 2, we explain briefly the principles of 𝜑- and bi𝜑- calculi, and give examples regarding multiplicative and bigeometric calculi. Moreover, we mention the Newtonian versions of Caputo and RiemannLiouville approaches to this subject. Then, we introduce some theorems for 𝜑- and bi𝜑-calculi and we also mention theorems from [6] as well which are the stepping stones used in this paper. In Section 3, we define 𝜑- and bi𝜑- fractional calculi, and based on them we also define it with respect to non-Newtonian calculi, which are the bigeometric, tanh-, and bi-tanhfractional calculi. The multiplicative case is discussed in [6]. Moreover, we mention some results which are the relations between 𝜑- and bi𝜑- fractional calculi and the Newtonian fractional calculi considering the mentioned approaches. The notation is rather different than the one that was introduced in [2], which denotes the bijection as 𝛼 instead of 𝜑. This was done because the letter 𝜑 is more convenient when discussing fractional calculi since 𝛼 is commonly used for denoting the order. 1 Introduction In the seventeenth century, Isaac Newton and Gottfried Leibniz laid the foundations for the classical -or sometimes called- Newtonian calculus. That particular calculus has proved its mathematical strength. Indeed, it is the most applicable theory used in sciences. Fractional calculus, even though it is usually thought that it is a relatively new subject, it has dated back to 1695 when L’Hoptial wrote to Leibnitz asking about the interpretation of 1 𝑑𝑛 𝑓 𝑑𝑥 𝑛 when 𝑛 = , see [1]. In the previous century, many mathematicians have given different perspectives and approaches in an attempt to answer this question. The same question arises 2 𝑑∗(𝑛) 𝑓 𝑑𝜋(𝑛) when one considers or 𝑑𝑥 𝑛 . These are the 𝑑𝑥 𝑛 multiplicative and bigeometric derivatives respectively. In the period 1967 to 1970, Michael Grossman and Robert Katz initiated many calculi considering different operations and viewing classical calculus as an additive type that depend on addition and subtraction as their foundation [2]. Using that view, they came up with what we call multiplicative and bigeometric calculi [1-6], that which depends on multiplication and division. More precisely, defining 𝜑-𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 to represent the main algebraic operations performed on ℝ. The function 𝜑 is a bijection from ℝ onto an interval 𝕀 that induced the field and metric structures from ℝ onto 𝕀. Letting 𝜑(𝑥) = 𝑒 𝑥 , we see that E-ISSN: 2224-2678 87 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi 2 Elements of 𝝋- and bi𝝋Differentiation With this metric we can define, as usual, the limit of a function that is defined on a φ-nonNewtonian interval 𝕀. Consider an increasing homeomorphism 𝜑: ℝ → 𝕀. For 𝑥, 𝑦 ∈ 𝕀, we define the following operations: Definition 2 Let 𝕀 be 𝜑-interval, and 𝑓: 𝕀 → 𝕀. For 𝑥0 ∈ 𝕀, we define 1. 𝑥 ⊕𝜑 𝑦 = 𝜑(𝜑 −1 (𝑥) + 𝜑 −1 (𝑦)), 2. 𝑥 ⊖𝜑 𝑦 = 𝜑(𝜑 3. 𝑥 ⊗𝜑 𝑦 = 𝜑(𝜑 −1 −1 (𝑥) − 𝜑 (𝑥) × 𝜑 −1 −1 bi𝜑 − lim 𝑓(𝑥) = 𝐿 ∈ 𝕀 𝑥→𝑥0 (𝑦)), to be the limit from the metric (𝕀, 𝑑𝜑 ) to itself. That is, if 𝑑𝜑 (𝑓(𝑥), 𝐿) → 0 as 𝑑𝜑 (𝑥, 𝑥0 ) → 0. (𝑦)), 4. 𝑥 ⊘𝜑 𝑦 = 𝜑(𝜑 −1 (𝑥)/𝜑 −1 (𝑦)), In the next proposition, we see the relation between the usual limit and the bi𝜑-limit. 5. 𝑥 ≤ 𝑦 if and only if 𝜑 −1 (𝑥) ≤ 𝜑 −1 (𝑦). Proposition 3 Let 𝕀 be a 𝜑-interval, and 𝑓: 𝕀 → 𝕀. Then, It is easy to check that 𝕀 under the above operations becomes an ordered field. We call this field the 𝜑-non-Newtonian interval. Moreover, the following real-valued function 𝜑 defines a metric on 𝕀: bi𝜑 − lim 𝑓(𝑥) 𝑥→𝑥0 6. 𝑑𝜑 (𝑥, 𝑦) = |𝜑 −1 (𝑥) − 𝜑 −1 (𝑦)|. definition of bi𝜑-limit, we have |𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝐿)| → 0 as |𝜑 −1 (𝑥) − 𝜑 −1 (𝑥0 )| → 0. It is easy to check that the following properties are true: 1. For 𝛼, 𝛽 ∈ ℝ, we have 𝑥 𝑥 ⊗𝛽 ⊗ 𝑥 ⊗𝛼 = 𝑥 ⊗(𝛼+𝛽) . ⊗𝑥 In other words, = |𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝜑 −1 (𝑥)) − 𝜑 −1 (𝐿)| → 0 2. For 𝛼, 𝛽 ∈ ℝ, we have 𝑥 ⊗𝛼 ⊘ 𝑥 ⊗𝛽 = 𝑥 ⊗(𝛼−𝛽) . 3. For (𝑥 ⊗𝛽 ) ⊗𝛼 𝛼, 𝛽 ∈ ℝ, = 𝑥 ⊗(𝛼𝛽) . we have (𝑥 ⊗𝛼 ) ⊗𝛽 Hence, as |𝜑 −1 (𝑥) − 𝜑 −1 (𝑥0 )| → 0. |𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡) − 𝜑 −1 (𝐿)| → 0 = Remark 1 This metric is compatible with the operations on the field 𝕀, that is, the above operations are continuous. as |𝑡 − 𝜑 −1 (𝑥0 )| → 0. That is, lim 𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡) = 𝜑 −1 (𝐿). 𝑡→𝜑 −1 (𝑥0 ) Therefore, E-ISSN: 2224-2678 𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡)). 𝑥→𝑥0 𝑥 ⊗𝛼 = 𝜑([𝜑 −1 (𝑥)]𝛼 ). ⊗𝛽 lim 𝑡→𝜑 −1 (𝑥0 ) Proof. Let bi𝜑 − lim 𝑓(𝑥) = 𝐿. By the Moreover, for any 𝛼 ∈ ℝ, we define the 𝜑𝑎𝑙𝑝ℎ𝑎 power of 𝑥 ∈ 𝕀 by ⊗𝛼 = 𝜑( 88 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 𝜑( 𝑓 bi𝜑 (𝑥) = 𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡)) = 𝐿. lim −1 𝑡→𝜑 Amer H. Darweesh, Abdelaziz M. D. Maghrabi (𝑥0 ) bi𝜑 − lim [𝑓(𝑦) ⊖ 𝑓(𝑥)] ⊘ [𝑦 ⊖ 𝑥]. Definition 4 Let 𝕀 be a 𝜑-interval, and 𝑓: ℝ → 𝕀. For 𝑥0 ∈ 𝕀, we define Consider Equation 1. Using Proposition 3, one has 𝜑 − lim 𝑓(𝑥) = 𝐿 ∈ 𝕀 𝑥→𝑥0 𝑓 bi𝜑 (𝑥) = to be the limit from the usual metric on ℝ to the metric (𝕀, 𝑑𝜑 ). That is, 𝑑𝜑 (𝑓(𝑥), 𝐿) → 0 as |𝑥 − 𝑥0 | → 0. Proposition 5 𝑓: ℝ → 𝕀. Then, 𝜑 − lim 𝑓(𝑥) = 𝜑 ( lim 𝜑 𝑥→𝑥0 Proof. 𝑥→𝑥0 𝑥→𝑥0 By 𝑦→𝑥 = 𝜑 ( lim 𝜑 −1 ([𝑓(𝜑(𝑡)) ⊖ 𝑓(𝑥)] ⊘ −1 𝑡→𝜑 [𝜑(𝑡) ⊖ 𝑥])) ∘ 𝑓(𝑥)). 𝜑 − lim 𝑓(𝑥) = 𝐿. Let bi𝜑 − lim [𝑓(𝑦) ⊖ 𝑓(𝑥)] ⊘ [𝑦 ⊖ 𝑥] Let 𝕀 be a 𝜑-interval, and −1 𝑦→𝑥 (𝑥) 𝜑 −1 (𝑓(𝜑(𝑡)) − 𝜑 −1 (𝑓(𝑥)) ) = 𝜑 ( lim 𝑡→𝜑 −1 (𝑥) 𝑡 − 𝜑 −1 (𝑥) 𝑑 = 𝜑 ( (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝑡)|𝑡=𝜑−1 (𝑥) ) 𝑑𝑡 the definition of 𝜑-limit, we have = 𝜑 ∘ (𝜑 −1 ∘ 𝑓 ∘ 𝜑)′ ∘ 𝜑 −1 (𝑥). |𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝐿)| → 0 as |𝑥 − 𝑥0 | → 0. Therefore, lim 𝜑 −1 ∘ 𝑓(𝑥) = 𝜑 −1 (𝐿). This yields the following results. 𝜑 ( lim 𝜑 −1 ∘ 𝑓(𝑥)) = 𝐿. 𝑓 bi𝜑 (𝑥) = 𝑥→𝑥0 Proposition 8 The bi𝜑-derivative of a function 𝑓: 𝕀 → 𝕀 , where 𝕀 is a 𝜑-interval,is given by Or equivalently, 𝑥→𝑥0 Based on these types of limits, we can develop 𝜑- and bi𝜑-calculi, that is to define a derivative and an integral with respect to 𝜑operations. 𝑡→𝜑 = 𝜑 ∘ (𝜑 −1 ∘ 𝑓 ∘ 𝜑)′ ∘ 𝜑 −1 (𝑥), Remark 6 From now on, for the sake of convenience and brevity, we will use the operations ⊕, ⊖, ⊗, and ⊘ instead of ⊕𝜑 , ⊖𝜑 , ⊗𝜑 , and ⊘𝜑 . or in the other notations, 𝑑 𝑑 bi𝜑 𝑓(𝑥) = 𝜑 ( [𝜑 −1 ∘ 𝑓 ∘ 𝜑](𝑡)|𝑡=𝜑−1 (𝑥) ). 𝑑𝑥 𝑑𝑡 Definition 7 The bi𝜑-derivative of a function 𝑓: 𝕀 → 𝕀, where 𝕀 ⊆ ℝ with 𝜑: ℝ → 𝕀 is denoted and given by E-ISSN: 2224-2678 𝜑 −1 (𝑓(𝜑(𝑡)) − 𝜑 −1 (𝑓(𝑥)) ) (𝑥) 𝑡 − 𝜑 −1 (𝑥) 𝜑 ( lim −1 89 Volume 22, 2023 (1) WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi If we denote the 𝑛𝑡ℎ bi𝜑-derivative of 𝑓(𝑥) by 𝑑bi𝜑(𝑛) 𝑑𝑥 𝑛 𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡) − 𝜑 −1 (𝑓(𝑥)) ) 𝜑 ( lim 𝑡→𝜑 −1 (𝑥) 𝑡 − 𝜑 −1 (𝑥) 𝑓(𝑥) = 𝑓 bi𝜑(𝑛) (𝑥), we can easily obtain the following result. ln ∘ 𝑓 ∘ exp(𝑡) − ln(𝑓(𝑥)) ) 𝑡→ln𝑥 𝑡 − ln𝑥 ln(𝑓(𝑦)) − ln(𝑓(𝑥)) = 𝜑 (lim ) 𝑦→𝑥 ln𝑦 − ln𝑥 = 𝜑 ( lim Proposition 9 Let 𝕀 be a 𝜑-interval, and 𝑓: 𝕀 → 𝕀. Then, 𝑓 𝑏𝑖𝜑(𝑛) (𝑥) = 𝜑 ∘ (𝜑 −1 ∘ 𝑓 ∘ 𝜑) or in the other notations, (𝑛) ∘𝜑 −1 (𝑥), (here, 𝑥 ∈ 𝕀 asasubsetof ℝ) (2) = 𝑒 𝑥(ln𝑓(𝑥))′ , as it is expected in the bigeometric calculus. 𝑑 bi𝜑(𝑛) 𝑑𝑥 𝑛 𝑓(𝑥) = 𝜑( If we define 𝑓 from the Newtonian field ℝ into the 𝜑-non-Newtonian interval 𝕀, we can introduce a weaker version of differentiablity and integrability. 𝑑𝑛 −1 [𝜑 ∘ 𝑓 ∘ 𝜑](𝑡)|𝑡=𝜑−1 (𝑥) ). 𝑑𝑡 𝑛 Definition 12 The 𝜑-derivative of a function 𝑓: ℝ → 𝕀 , where 𝕀 ⊆ ℝ with 𝜑: ℝ → 𝕀 is denoted and given by Conversely, we can write the ordinary derivative in terms of bi𝜑-derivative as follows. 𝑓 𝜑 (𝑥) = 𝜑 − lim [𝑓(𝑦) ⊖𝜑 𝑓(𝑥)] ⊘𝜑 Proposition 10 Let 𝑔: ℝ → ℝ, 𝕀 be a 𝜑interval. Then, 𝜑 ∘ 𝑔 ∘ 𝜑 −1 : 𝕀 → 𝕀 and 𝑔(𝑛) (𝑡) = 𝜑 ∘ 𝑦→𝑥 [𝜑(𝑦) ⊖𝜑 𝜑(𝑥)]. 𝑑 bi𝜑(𝑛) [𝜑 ∘ 𝑔 ∘ 𝜑 −1 ] ∘ 𝜑(𝑡), 𝑑𝑥 𝑛 Proposition 13 The 𝜑-derivative of a function 𝑓: ℝ → 𝕀 , where 𝕀 is a 𝜑-interval, is given by or in the other notations, 𝑑𝑛 𝑔(𝑡) = 𝑑𝑡 𝑛 𝑑 𝜑( bi𝜑(𝑛) 𝑑𝑥 𝑛 𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝑓(𝑦)) 𝑓 (𝑥) = 𝜑 (lim ) 𝑦→𝑥 𝑥−𝑦 𝜑 = 𝜑( [𝜑 ∘ 𝑔 ∘ 𝜑 −1 ](𝑥)|𝑥=𝜑(𝑡) ). Example 11 Take 𝜑(𝑥) = 𝑒 𝑥 , with 𝕀 = (0, ∞), then for functions 𝑓 : 𝕀 → 𝕀, we can use Proposition 8 to define the bigeometric derivative as follows 𝑑𝑥 [𝜑 −1 ∘ 𝑓](𝑥)). Proposition 14 Let 𝑓: ℝ → 𝕀. Then, 𝑓 is 𝜑differentiable at 𝑥 ∈ ℝ if and only if 𝑓 ∘ 𝜑 −1 is bi𝜑-differentiable at 𝜑(𝑥). In this case, 𝑓 𝜑 (𝑥) = (𝑓 ∘ 𝜑 −1 )𝑏𝑖𝜑 (𝜑(𝑥)). 𝑑𝜋 𝑓 (𝑥) = 𝑑𝑥 E-ISSN: 2224-2678 𝑑 Proof. The proof follows from Propositions 8 and 13. 90 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi Theorem 15 Let 𝜑: ℝ → 𝕀 be differentiable with 𝜑′(𝑥) > 0 on ℝ, and let 𝑓: ℝ → 𝕀. Then 𝑓 is differentiable if and only if 𝑓 is 𝜑differentiable. 𝑓 𝑏𝑖𝜑 𝑑 (𝜑 −1 ∘ 𝑓)(𝑥) 𝑑𝑥 𝑑 𝑑𝑥 𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝑓(𝑦)) 𝑦→𝑥 𝑥−𝑦 =𝐿 = lim 𝜑 (lim 𝑦→𝑥 𝜑 (𝑓(𝑥)) − 𝜑 𝑥−𝑦 (𝑓(𝑦)) Therefore, 𝜑 (lim 𝑦→𝑥 and hence 𝜑 (𝑓(𝑥)) − 𝜑 𝑥−𝑦 −1 𝑑 𝑑 [𝜑 −1 ∘ 𝑓](𝜑(𝑡0 )) 𝜑(𝑡0 ). 𝑑𝑥 𝑑𝑥 𝑑𝑥 By Proposition 8, 𝑓 𝑏𝑖𝜑 (𝑥0 ) exists and 𝑓 ) = 𝐿, 𝑏𝑖𝜑 𝑑 𝑑𝑥 (𝑥) = 𝜑 ( (𝜑 −1 ∘ 𝑓)(𝑥) 𝑑 𝜑 −1 (𝑥) 𝑑𝑥 ). Now, suppose that 𝑓 is bi𝜑-differentiable. Then, 𝑓𝑏𝑖𝜑 (𝑥) = 𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝑓(𝑦)) lim = 𝜑 −1 (𝐿). 𝑦→𝑥 𝑥−𝑦 𝜑 −1 (𝑓(𝜑(𝑡)) − 𝜑 −1 (𝑓(𝑥)) ) (𝑥) 𝑡 − 𝜑 −1 (𝑥) 𝜑 ( lim −1 It follows that the function 𝜑 −1 ∘ 𝑓(𝑥) is differentiable at 𝑥. By the chain rule and the fact that 𝜑(𝑥) is differentiable everywhere, we conclude that 𝑓(𝑥) = 𝜑 ∘ 𝜑 −1 ∘ 𝑓(𝑥) is differentiable at 𝑥. 𝑡→𝜑 Therefore, Theorem 16 Let 𝜑: ℝ → 𝕀 be differentiable with 𝜑′(𝑥) > 0 on ℝ, and let 𝑓: 𝕀 → 𝕀. Then 𝑓 is differentiable if and only if 𝑓 is bi𝜑differentiable. Moreover, E-ISSN: 2224-2678 ). 𝑑 ) = 𝜑(𝐿). (𝑓(𝑦)) 𝜑 −1 (𝑥) 𝑑𝑥 [𝜑 −1 ∘ 𝑓](𝑥0 ) 𝑑 𝑑𝑥 −1 −1 [𝜑 ∘ 𝑓 ∘ 𝜑](𝜑 (𝑥0 )) = . 𝑑 𝑑𝑥 𝜑 −1 (𝑥0 ) By Proposition 13, we have 𝑓 𝜑 (𝑥) = 𝜑(𝐿). Hence, 𝑓 is 𝜑-differentiable at 𝑥. Now, suppose that 𝑓 is 𝜑-differentiable. Then, −1 𝑑 [𝜑 −1 ∘ 𝑓 ∘ 𝜑](𝑡0 ) = exists. Therefore, −1 (𝑥) = 𝜑 ( (𝜑 −1 ∘ 𝑓)(𝑥) Proof. Let 𝑓 be differentiable at 𝑥0 . By the inverse function theorem, the function 𝜑 −1 (𝑥) is differentiable on 𝕀. Hence, the function 𝜑 −1 ∘ 𝑓 is differentiable at 𝑥0 . Let 𝑡0 = 𝜑 −1 (𝑥0 ), then 𝜑(𝑡) is differentiable at 𝑡0 . By the chain rule, 𝜑 −1 ∘ 𝑓 ∘ 𝜑(𝑡) is differentiable at 𝑡0 , and Proof. Let 𝑓 be differentiable at 𝑥. By the inverse function theorem, the function 𝜑 −1 (𝑥) is differentiable on 𝕀. Hence, the function 𝜑 −1 ∘ 𝑓 is differentiable at 𝑥. That is, −1 𝑑 𝑑𝑥 = 𝜑 (lim 𝑦→𝑥 𝜑 −1 (𝑓(𝑥))−𝜑−1 (𝑓(𝑦)) 𝜑 −1 (𝑥)−𝜑−1 (𝑦) ) 𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝑓(𝑦)) = 𝜑 −1 (𝑓 𝑏𝑖𝜑 (𝑥)). lim 𝑦→𝑥 𝜑 −1 (𝑥) − 𝜑 −1 (𝑦) Since 91 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi 𝜑 −1 (𝑥) − 𝜑 −1 (𝑦) 𝑑 −1 lim = 𝜑 (𝑥) ≠ 0, 𝑦→𝑥 𝑥−𝑦 𝑑𝑥 𝑓 (𝑛) (𝑥) = 𝑛−1 ∑ ( we have 𝜑 −1 (𝑓(𝑥)) − 𝜑 −1 (𝑓(𝑦)) 𝑦→𝑥 𝑥−𝑦 lim = 𝜑 −1 (𝑓 𝑏𝑖𝜑 (𝑥)) 𝑘=0 𝑑 −1 𝜑 (𝑥). 𝑑𝑥 𝑓 ∗(𝑛) = exp( 𝑘≔𝑘1 +⋯+𝑘𝑛 𝑓 (𝑖) (𝑥) 𝑘 ) 𝑖) 𝑖! 𝑖=1,...,𝑛 ( (7) For a simpler variant of Faà di Bruno formula, refer to [7]. This gives a brief overview of multiplicative and bigeometric calculi. 𝑑𝑛 𝜑 (𝑑𝑥 𝑛 (𝜑 −1 ∘ 𝑓)(𝑥)). Example 19 The tanh-derivative for 𝑥 ∈ 𝕀 = (−1,1), is denoted and given by Proof. The proof will be done using mathematical induction and Proposition 13. Example 18 For 𝕀 = ℝ+ and 𝜑(𝑥) = 𝑒𝑥𝑝(𝑥), we obtain the geometric derivative (which is also called ∗derivative or multiplicative derivative) of 𝑓(𝑥). (𝑥) = lim ( ℎ→0 𝑓(𝑥+ℎ) 1 𝑓(𝑥) 𝑓(′𝑥) )ℎ = 𝑒 𝑓(𝑥) = 𝑒 (ln∘𝑓)′(𝑥) (𝑛) (𝑥) 𝑑𝑥 (𝑥) = 𝑒 (ln∘𝑓) 𝜋 ∗ one immediately = 𝑑𝑥 2𝑓(′𝑥) 2 𝑒 1−𝑓(𝑥) −1 2𝑓(′𝑥) 2 𝑒 1−𝑓(𝑥) +1 = 𝑑tanh−1 𝑓(𝑥) 𝑑𝑥 −1 𝑑tanh−1 𝑓(𝑥) 2 𝑑𝑥 𝑒 +1 𝑒 2 . Moreover, the 𝑛𝑡ℎ order tanh-derivative is given by (3) realizes ⋆(𝑛) (𝑥) = 𝑒 𝑒 (ln 1+𝑓(𝑥) (𝑛) ) 1−𝑓(𝑥) −1 (ln 1+𝑓(𝑥) (𝑛) ) 1−𝑓(𝑥) +1 Theorem 20 Let 𝑏𝑖𝜑 differentiable, then the (5) , 𝑛 = 0,1,2, . .. (9) 𝑑𝜑(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 = 𝑑bi𝜑(𝑛) 𝑓(𝜑 −1 (𝑡)) 𝑑𝑡 𝑛 𝑓(𝑥) be , 𝑥 = 𝜑 −1 (𝑡), 𝑛-times (10) Thus, the bi𝜑- derivative is a Gauss vector of the 𝜑-derivative. Equivalently, The multiplicative derivative and the additive derivative can be used to express each other. Indeed, we have the following equation E-ISSN: 2224-2678 𝑑⋆ 𝑓(𝑥) (4) 𝑥 𝑓 (𝑥) = (𝑓 (𝑥)) . 𝑓 ⋆ (𝑥) = 𝑓 By Theorem 17, we have Moreover, relation 𝑘1 +2𝑘2 +⋯+𝑛𝑘𝑛 =𝑛 −(𝑘 − 1)! 𝑛! 𝑘1 ! ⋅ … ⋅ 𝑘𝑛 ! Let 𝑓: ℝ → 𝕀, then 𝑓 𝜑(𝑛) (𝑥) = Theorem 17 𝑑 ∗(𝑛) 𝑓 ∑ (−𝑓(𝑥))−𝑘 ∏ We denote the 𝑛𝑡ℎ 𝜑-derivative by 𝑓 𝜑(𝑛) (𝑥). With this notation, one can obtain the following result. 𝑑𝑥 (6) Using Faà di Bruno formula on equation (4), one also arrives at the following It follows that the function 𝜑 −1 ∘ 𝑓(𝑥) is differentiable at 𝑥. By the chain rule, we conclude that 𝑓(𝑥) = 𝜑 ∘ 𝜑 −1 ∘ 𝑓(𝑥) is differentiable at 𝑥. 𝑑∗𝑓 𝑛 − 1 (𝑘) ) 𝑓 (𝑥)(ln ∘ 𝑓 ∗(𝑛−𝑘) )(𝑥) 𝑘 92 Volume 22, 2023 (8) WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 𝑑𝜑(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 𝑑𝜑(𝑛) 𝑓(𝑥) 𝑑𝜑 −1 (𝑥)𝑛 = = Amer H. Darweesh, Abdelaziz M. D. Maghrabi 𝑑bi𝜑(𝑛) 𝑓(𝑥) 𝑑 ∗(𝑛) 𝑓(𝑥) (11) 𝑑𝜑(𝑥)𝑛 𝑑bi𝜑(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 𝑑 ∗(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 Proof. This proof will be done using mathematical induction. Let 𝑥 = 𝜑 −1 (𝑡), then we have 𝑑𝑥 = 𝜑 −1 (𝑡)′𝑑𝑡. This implies, 𝑑𝜑 −1 (𝑓(𝑥)) 𝑑𝜑 𝑓(𝑥) = 𝜑( ) 𝑑𝑥 𝑑𝑥 = 𝜑( = 1 𝜑 −1 (𝑡)′ 𝑑𝜑 −1 (𝑓(𝜑 −1 (𝑡))) 𝑑bi𝜑 𝑓(𝜑 −1 (𝑡)) 𝑑𝑡 𝑑𝑡 𝑑 ∗(𝑛) 𝑓(𝑥) 𝑑ln𝑥 𝑛 ) 𝑑 ⋆(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 𝑑 ⋆(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 = 𝜑( 𝑑 𝑑𝑥 ) 𝑑𝜋(𝑛) 𝑓(𝑥) (14) 𝑑𝜋(𝑛) 𝑓(𝑥) (15) 𝑑exp𝑥 𝑛 𝑑𝑥 𝑛 = 𝑑bi⋆(𝑛) 𝑓(tanh−1 𝑥) 𝑑𝑡 𝑛 be 𝑛-times , 𝑥 = tanh−1 𝑡. (16) = 𝑑bi⋆(𝑛) 𝑓(𝑥) = (17) 𝑑tanh𝑥 𝑛 𝑑bi⋆(𝑛) 𝑓(𝑥) 𝑑𝑥 𝑛 . (18) 3 Elements of 𝝋- and bi𝝋-Riemann integration 𝑑bi𝜑(𝑘) 𝑓(𝜑 −1 (𝑡)) . = 𝑑𝑡𝑘 Using the structure of the metric 𝕀, one can define the boundedness of 𝑓: 𝐴 → 𝕀. Precisely, 𝑓 is 𝜑-bounded if 𝑑𝜑 (𝑓(𝑥), 𝜑(0)) ≤ 𝑀, for all 𝑥 ∈ 𝐴. That is, if |𝜑 −1 (𝑓(𝑥))| ≤ 𝑀, for all 𝑥 ∈ 𝐴. This concludes the proof. The other forms are obtained by manipulating the substitution 𝑥 = 𝜑 −1 (𝑡). Remark 21 The first form which includes the variables 𝑥 and 𝑡 were introduced to obtain a simple proof. E-ISSN: 2224-2678 (13) Remark 24 By Theorem (20), we can comprehend the relation between 𝜑- and bi𝜑calculi. Indeed, 𝜑-calculus is not only a weakened version of bi𝜑-calculus, rather bi𝜑calculus is the change in 𝜑-calculus with respect to 𝜑 −1 (𝑥), which is equivalent to stating that 𝜑-calculus is the change in bi𝜑calculus with respect to 𝜑(𝑥). 𝑑 bi𝜑(𝑘−1) 𝑓(𝜑 −1 (𝑡)) 1 𝑑 = 𝜑 ( −1 ′ 𝜑 −1 ( )) 𝜑 (𝑡) 𝑑𝑡 𝑑𝑡𝑘−1 𝜋 = 𝑑tanh−1 𝑥 𝑛 𝜑 −1 (𝑓 𝜑(𝑘−1) (𝑥))) Example 22 Let 𝑓 be 𝑛-times then = 𝑑 ⋆(𝑛) 𝑓(𝑥) 𝑑𝑘 𝜑 −1 (𝑓(𝑥)) 𝑑𝜑(𝑘) 𝑓(𝑥) = 𝜑( ) 𝑑𝑥 𝑘 𝑑𝑥 𝑛 𝑑𝑥 𝑘−1 , 𝑥 = ln𝑡. Which are equivalent to the forms, Hence, the theorem is true at 𝑛 = 1. Assume that it is true for 𝑛 = 𝑘 − 1, by the induction hypothesis we have, = 𝜑 (𝑑𝑥 𝑑𝑡 𝑛 Example 23 Let 𝑓(𝑥) 𝑏𝑖⋆ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒. Then, . 𝑑 𝑑𝑘−1 𝜑 −1 (𝑓(𝑥)) 𝑑𝜋(𝑛) 𝑓(ln𝑡) Which has equivalent forms, (12) 𝑑𝑥 𝑛 = Definition 25 Let 𝑓: 𝕀 → 𝕀 be 𝜑-bounded. Let 𝑎 < 𝑏 in 𝕀, and 𝑃 = {𝑥0 , 𝑥1 , … , 𝑥𝑛 } be a partition on [𝑎, 𝑏]. The function 𝑓 is called bi𝜑Riemann integrable if there is 𝐿 ∈ 𝕀 such that differentiable, 93 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi for any 𝜖 > 0 and any choice of 𝑥𝑖−1 ≤ 𝑐𝑖 ≤ 𝑥𝑖 , there is a 𝛿 > 0 satisfies: 𝜑(𝑥𝑖−1 )), 𝐿) < 𝜖 whenever 𝑠𝑢𝑝𝑖 |𝑥𝑖 − 𝑥𝑖−1 | < 𝛿. In this case we write whenever 𝑠𝑢𝑝𝑖 𝑑𝜑 (𝑥𝑖 , 𝑥𝑖−1 ) < 𝛿. In this case we write, lim ⊕𝜑 (𝑓(𝑐𝑖 ) ⊗𝜑 (𝜑(𝑥𝑖 ) ⊖𝜑 𝜑(𝑥𝑖−1 ))) 𝑑𝜑 (⊕𝑛𝑖=1 𝑓(𝑐𝑖 ) ⊗ (𝑥𝑖 ⊖ 𝑥𝑖−1 ), 𝐿) < 𝜖 𝑏 𝑛 𝑛→∞ 𝑖=1 ∫ 𝑓(𝑥)𝑑bi𝜑 𝑥 = lim ⊕ (𝑓(𝑐𝑖 ) ⊗ (𝑥𝑖 𝑛→∞ 𝑛→∞ 𝑖=1 ⊖ 𝑥𝑖−1 )) = 𝐿. 𝑏 𝑎 𝑏 ∫𝑎 𝑓 (𝑥)𝑑𝑥 is given by ∫ 𝑓(𝑥)𝑑 bi𝜑 𝑥 = 𝑏 𝑎 ∫ 𝑓(𝑥)𝑑𝑥 = (19) 𝑏 is given by 𝑏 𝑛 𝑎 𝑛→∞ 𝑖=1 𝑛 𝑖=1 = exp (∫ 𝑏 𝑎 ln𝑓(𝑥) 𝑑𝑥). 𝑥 𝑏 𝑏 ∫𝑎 𝑓(𝑥)𝑑 𝜑(𝑛) 𝑥 = 𝜑 (∫𝑎 𝜑 −1 (𝑓(𝑥))𝑑(𝑛) 𝑥) , 𝑛 ∈ 𝑁 ∪ {0}. (20) Proof. The proof will be done using mathematical induction. For 𝑛 = 0,1, it is clear. Assume that it holds true for 𝑛 = 𝑘 − 1, then we get Definition 28 Let 𝑓: ℝ → 𝕀 be 𝜑-bounded. Let 𝑎 < 𝑏 in 𝕀, and 𝑃 = {𝑥0 , 𝑥1 , … , 𝑥𝑛 } be partition on [𝑎, 𝑏]. The function 𝑓 is called 𝜑-Riemann integrable if there is 𝐿 ∈ 𝕀 such that for any 𝜖 > 0 and any choice of 𝑥𝑖−1 ≤ 𝑐𝑖 ≤ 𝑥𝑖 , there is a 𝛿 > 0 satisfies: 𝑑𝜑 (⊕𝑛𝑖=1 𝑓(𝑐𝑖 ) ⊗ (𝜑(𝑥𝑖 ) ⊖ E-ISSN: 2224-2678 𝑏 Theorem 30 Let ∫𝑎 𝑓(𝑥)𝑑 𝜑 𝑥 be the 𝜑-integral of 𝑓(𝑥). Then, = lim ∏ 𝑓(𝑐𝑖 )ln(𝑥𝑖 /𝑥𝑖−1 ) 𝑛→∞ 𝑖=1 = exp (∫ ln𝑓(𝑥)𝑑𝑥). ∫ 𝑓(𝑥)𝐝𝑥 = lim ⊕ (𝑓(𝑐𝑖 ) ⊗ (𝑥𝑖 ⊖ 𝑥𝑖−1 )) 𝑎 𝑛 𝑎 lim exp [∑ ln(𝑓(𝑐𝑖 ))(𝑥𝑖 − 𝑥𝑖−1 )] 𝑛→∞ Example 27 The bigeometric integral, denoted (𝑥) 𝑎 Example 29 The geometric integral, denoted 𝑏 𝒅𝑥 𝑏 ∫ 𝑓(𝑥)𝑑𝜑 𝑥 = 𝜑(∫ 𝜑 −1 (𝑓(𝑥))𝑑𝑥). It is worth mentioning that if 𝑓(𝑥) is Riemann integrable, and 𝜑 −1 (𝑥) is piecewise continuously differentiable on [𝑎, 𝑏], then 𝑏 ∫𝑎 𝑓 𝑖=1 It is clear that from the definition above if 𝑓 is Riemann integrable, and hence 𝜑 −1 ∘ 𝑓 is Riemann integrable, then 𝑓 is 𝜑-Riemann integrable and Remark 26 The definition above is independent of the choice of 𝑐𝑖 ∈ [𝑥𝑖−1 , 𝑥𝑖 ]. That is, if the above limit exists, then for any choice of {𝑐𝑖 }, the limit is the same. 𝑏 𝑑 𝜑 (∫𝑎 𝜑 −1 (𝑓(𝑥)) 𝜑 −1 (𝑥)𝑑𝑥). 𝑑𝑥 𝑛 = lim 𝜑 (∑ 𝜑 −1 ∘ 𝑓(𝑐𝑖 )(𝑥𝑖 − 𝑥𝑖−1 )) = 𝐿. 𝑛 𝑎 𝑏 ∫𝑎 𝑓(𝑥)𝑑 𝜑 𝑥 = 𝑏 𝑏 𝑏 ∫ 𝑓(𝑥)𝑑𝜑(𝑘) 𝑥 = ∫ (∫ 𝑓(𝑥)𝑑𝜑(𝑘−1) 𝑥) 𝑑 𝜑 𝑥 𝑎 94 𝑎 𝑎 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 𝑏 = 𝜑 (∫ 𝜑 𝑎 𝑏 −1 𝑏 (∫ 𝑓(𝑥)𝑑 𝑏 𝑎 Amer H. Darweesh, Abdelaziz M. D. Maghrabi 𝜑(𝑘−1) 𝐶 [ 𝐷 𝑥) 𝑑𝑥) = 𝜑 (∫ 𝜑 −1 (𝜑 (∫ 𝜑 −1 (𝑓(𝑥))𝑑(𝑘−1) 𝑥)) 𝑑𝑥) 𝑎 𝑏 𝑏 = 𝜑 (∫ [∫ 𝜑 𝑎 𝑎 𝑏 𝑎 −1 (𝑓(𝑥))𝑑 (𝑘−1) Γ𝜑 (𝛼) = ∫ = 𝜑 (∫ 𝜑 −1 (𝑓(𝑥))𝑑 (𝑘) 𝑥). (21) [𝐼 𝑓](𝑥) = 𝑒 𝑏 1+𝑓(𝑥) (𝑛) 2 ∫𝑎 ln 𝑑 𝑥 1−𝑓(𝑥) −1 𝑏 1+𝑓(𝑥) (𝑛) 2 ∫𝑎 ln 𝑑 𝑥 1−𝑓(𝑥) +1 , 𝑛 ∈ ℕ. [𝐼 (𝛼) 𝑓](𝑥) = the 𝑥 [𝐼 𝜑(𝛼) 𝑓](𝑥) = ∫ [𝜑(𝑥) ⊖ 𝜑(𝑡)]⊗(𝛼−1) Riemann-Liouville 𝑎 ⊗ 𝑓(𝑡) ⊘ Γ𝜑 (𝛼)𝑑 𝜑 𝑡. 𝑥 1 ∫ (𝑥 − 𝑡)𝛼−1 𝑓(𝑡)𝑑𝑡. Γ(𝛼) 𝑎 It is easy to see that [𝐼 Moreover, for 𝑛 − 1 ≤ ℜ(𝛼) < 𝑛, 𝑛: = ⌈𝛼⌉, by analytic continuation of the RL-integral to ℜ(𝛼) ≤ 0, the Riemann-Liouville fractional derivative is given by 𝜑(𝛼) 𝑥 1 𝑓](𝑥) = 𝜑 [ ∫ (𝑥 − 𝑡)𝛼−1 𝜑 −1 Γ(𝛼) 𝑎 ∘ 𝑓(𝑡)𝑑𝑡] = 𝜑[𝐼 (𝛼) (𝜑 −1 ∘ 𝑓)(𝑥)]. 𝑑𝑛 𝑥 1 (𝛼) ∫ (𝑥 [𝐷 𝑓](𝑥) = Γ(𝑛 − 𝛼) 𝑑𝑥 𝑛 𝑎 − 𝑡)𝑛−𝛼−1 𝑓(𝑡)𝑑𝑡. As an example, the fractional multiplicative Riemann-Liouville integral of order ℜ(𝛼) > 0 is defined as follows: Whereas the Caputo derivative is given by, E-ISSN: 2224-2678 = 𝜑 (∫ 𝑥 𝛼−1 𝑒 −𝑥 𝑑𝑥). Definition 33 Let 𝑓: ℝ → 𝕀, where 𝕀 is a 𝜑interval. The 𝜑- fractional Riemann-Liouville integral of order ℜ(𝛼) > 0 is denoted and given by (22) 4 Definitions of fractional 𝝋- and bi𝝋calculi For ℜ(𝛼) > 0, integral is given by, ∞ The definitions that will be discussed in this section are dealt with in a similar fashion to that logic used in the definitions of RiemannLiouville and Caputo. Indeed, we have the following definitions: Additionally, we have the tanh-integral 𝑒 𝜑(𝑥 𝛼−1 𝑒 −𝑥 )𝑑𝜑 𝑥 0 Example 31 It is clear from the definition of the multiplicative integral that ⋆(𝑛) 𝑙𝑖𝑚𝑡→∞ 𝜑(𝑡) 0 𝑎 [𝐼 ∗(𝑛) 𝑓](𝑥) = 𝑒 𝑥 1 ∫ (𝑥 𝑓](𝑥) = Γ(𝑛 − 𝛼) 𝑎 − 𝑡)𝑛−𝛼−1 𝑓 (𝑛) (𝑡)𝑑𝑡. Definition 32 For ℜ(𝛼) > 0, define the 𝜑Gamma function by 𝑥] 𝑑𝑥) [𝐼 (𝑛) (ln∘𝑓)](𝑥) (𝛼) 95 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 [𝐼 ∗(𝛼) Amer H. Darweesh, Abdelaziz M. D. Maghrabi 𝑥 1 𝑓](𝑥) = exp [ ∫ (𝑥 − 𝑡)𝛼−1 (ln Γ(𝛼) 𝑎 𝑏𝑖𝜑(𝛼) [𝐼𝑎 𝑓](𝑥) ∘ 𝑓)(𝑡)𝑑𝑡]. [𝐼 ⋆(𝛼) 𝑓](𝑥) = 𝑒 𝑥 2 ∫𝑎 ln 1+𝑓(𝑥) (𝛼) 𝑑 𝑥 1−𝑓(𝑥) 𝑥 1+𝑓(𝑥) (𝛼) 2 ∫𝑎 ln 𝑑 𝑥 1−𝑓(𝑥) −1 +1 𝑏𝑖𝜑(𝛼) [𝐼𝑎 𝑓](𝑥) . 𝑎 ⊘ Γ𝜑 (𝛼)𝑑 𝑏𝑖𝜑 𝑡. −1 𝜑 (𝑥) 1 = 𝜑[ ∫ (𝜑 −1 (𝑥) Γ(𝛼) 𝜑−1 (𝑎) − 𝑠)𝛼−1 (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝑠)𝑑𝑠] 𝑥 1 ∫ (𝜑 −1 (𝑥) − Γ(𝛼) 𝑎 𝑑 𝜑 −1 (𝑡))𝛼−1 (𝜑 −1 ∘ 𝑓)(𝑡) 𝜑 −1 (𝑡)𝑑𝑡] 𝑑𝑡 = 𝜑[ Proposition 34 The 𝜑- fractional RiemannLiouville integral operator satisfies the property (𝛼) = 𝜑 [𝐼𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝜑−1 (𝑥))]. 𝐼 𝜑(𝛼) ⊗ 𝐼 𝜑(𝛽) 𝑓 = 𝐼 𝜑(𝛼+𝛽) 𝑓 = 𝐼 𝜑(𝛽) ⊗ 𝐼 𝜑(𝛼) 𝑓. Proof. For ℜ(𝛼), ℜ(𝛽) > 0 Proposition 36 The bi𝜑- fractional RiemannLiouville integral operator satisfies the property 𝐼 𝜑(𝛼) ⊗ 𝐼 𝜑(𝛽) 𝑓(𝑥) = 𝜑[𝐼 (𝛼) (𝜑 −1 ∘ 𝑓)(𝑥)] ⊗ 𝜑[𝐼 (𝛽) (𝜑 −1 ∘ 𝑓)(𝑥)] 𝑏𝑖𝜑(𝛼) 𝐼𝑎 = 𝜑[𝐼 (𝛼) (𝜑 −1 ∘ 𝑓)(𝑥)𝐼 (𝛽) (𝜑 −1 ∘ 𝑓)(𝑥)] = 𝜑[𝐼 (𝛼) 𝐼 (𝛽) (𝜑 −1 ∘ 𝑓)(𝑥)] = 𝜑[𝐼 (𝛼+𝛽) (𝜑 −1 ∘ 𝑓)(𝑥)] 𝑏𝑖𝜑(𝛽) ⊗ 𝐼𝑎 𝑏𝑖𝜑(𝛼+𝛽) 𝑓 = 𝐼𝑎 𝑏𝑖𝜑(𝛽) = 𝐼𝑎 𝑓 𝜑(𝛼) ⊗ 𝐼𝑎 𝑓. Proof. For ℜ(𝛼), ℜ(𝛽) > 0, one has = 𝐼 𝜑(𝛼+𝛽) 𝑓(𝑥). 𝑏𝑖𝜑(𝛼) 𝐼𝑎 Similarly, 𝐼 𝜑(𝛼) ⊗ 𝐼 𝜑(𝛽) 𝑓(𝑥) = 𝐼 𝜑(𝛽+𝛼) 𝑓(𝑥) = 𝐼 𝜑(𝛼+𝛽) 𝑓(𝑥). This proves the assertion. 𝑏𝑖𝜑(𝛽) ⊗ 𝐼𝑎 (𝛼) 𝑓(𝑥) = 𝜑 [𝐼𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝜑 −1 (𝑥))] (𝛼) (𝛽) ⊗ 𝜑 [𝐼𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝜑 −1 (𝑥))] (𝛽) = 𝜑 [𝐼𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝜑 −1 (𝑥))𝐼𝜑−1 (𝑎) (𝜑 −1 Definition 35 Let 𝑓: 𝕀 → 𝕀, where 𝕀 is a 𝜑interval. The bi𝜑- fractional Riemann-Liouville integral of order ℜ(𝛼) > 0 is denoted and given by E-ISSN: 2224-2678 = ∫ [𝑥 ⊖ 𝑡]⊗(𝛼−1) ⊗ 𝑓(𝑡) It is easy to see that (Check) Moreover, the tanh- fractional integral is denoted and given by 𝑒 𝑥 (𝛼+𝛽) ∘ 𝑓 ∘ 𝜑)(𝜑 −1 (𝑥))] = 𝜑 [𝐼𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓 ∘ 𝜑)(𝜑 −1 (𝑥))] 96 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 𝑏𝑖𝜑(𝛼+𝛽) = 𝐼𝑎 The other part is similar. Amer H. Darweesh, Abdelaziz M. D. Maghrabi Definition 38 Let 𝑛 − 1 < 𝛼 < 𝑛 and 𝑓: 𝕀 → 𝕀, where 𝕀 is a 𝜑-interval. The bi𝜑- fractional Riemann-Liouville derivative of order 𝛼 is denoted and given by 𝑓(𝑥). Definition 37 Let 𝑛 − 1 < 𝛼 < 𝑛 and 𝑓: ℝ → 𝕀, where 𝕀 is a 𝜑-interval. The 𝜑- fractional Riemann-Liouville derivative of order 𝛼 is denoted and given by [𝐷𝜑(𝛼) 𝑓](𝑥) = 𝑏𝑖𝜑(𝛼) [𝐷𝑎 𝑑𝜑(𝑛) 𝑥 ∫ [𝜑(𝑥) 𝑑𝑥 𝑛 𝑎 𝑏𝑖𝜑(𝛼) [𝐷𝑎 − 𝑡)𝑛−𝛼−1 𝜑 −1 ∘ 𝑓(𝑡)𝑑𝑡) (𝛼) = 𝜑 [𝐷𝜑−1 (𝑎) (𝜑 −1 ∘ 𝑓)(𝜑 −1 (𝑥))]. (28) ∘ 𝑓(𝑡)𝑑𝑡) (25) As examples we have the multiplicative and tanh- versions, respectively, which are denoted and given by 𝑑 ⋆(𝑛) 𝑑𝑥 𝑛 (𝑥−𝑡)𝑛−𝛼−1 Γ(𝑛−𝛼) )𝑑𝑡 , ℜ(𝑛 − 𝛼) > 0, [𝐷⋆(𝛼) 𝑓](𝑥) = 𝑏 1+𝑓(𝑥) (𝑛−𝛼) 𝑑 𝑥 2 ∫ ln −1 𝑒 𝑎 1−𝑓(𝑥) 𝑏 1+𝑓(𝑥) (𝑛−𝛼) 𝑥 2 ∫ ln 𝑑 𝑒 𝑎 1−𝑓(𝑥) E-ISSN: 2224-2678 +1 , ℜ(𝑛 − 𝛼) > 0. −1 𝜑 (𝑥) 1 𝑑𝑏𝑖𝜑(𝑛) 𝜑( ∫ (𝜑 −1 (𝑥) = Γ(𝑛 − 𝛼) 𝜑−1 (𝑎) 𝑑𝑥 𝑛 − 𝜑 −1 (𝑡))𝑛−𝛼−1 𝜑 −1 ∘ 𝑓(𝑡)𝑑𝑡) 𝑥 1 𝑑 ∫ (𝑥 − 𝑡)𝑛−𝛼−1 𝜑 −1 𝑑𝑥 𝑛 Γ(𝑛 − 𝛼) 𝑎 𝑥 ∫ (𝑓(𝑡) 𝑑𝑥 𝑛 𝑎 𝑓](𝑥) 𝑥 𝑑𝑛 1 = 𝜑( 𝑛 ∫ (𝜑 −1 (𝑥) 𝑑𝑥 Γ(𝑛 − 𝛼) 𝑎 = 𝜑[𝐷 (𝛼) (𝜑 −1 ∘ 𝑓)(𝑥)]. 𝑑 ∗(𝑛) 𝑥 ∫𝑎 [𝑥 ⊖ − 𝜑 −1 (𝑡))𝑛−𝛼−1 𝜑 −1 ∘ 𝑓(𝑡)𝑑𝑡) 𝑥 𝑑𝜑(𝑛) 1 𝜑(𝛼) [𝐷 𝑓](𝑥) = 𝜑( ∫ (𝑥 𝑑𝑥 𝑛 Γ(𝑛 − 𝛼) 𝑎 [𝐷 ∗(𝛼) 𝑓](𝑥) = 𝑑𝑥 𝑛 It is easy to see that It is easy to see that = 𝜑( 𝑑𝑏𝑖𝜑(𝑛) 𝑡]⊗(𝑛−1−𝛼) ⊗ 𝑓(𝑡) ⊘ Γ𝜑 (𝛼)𝑑 𝑏𝑖𝜑 𝑡. 𝑓(𝑡) . ⊖ 𝜑(𝑡)]⊗(𝑛−1−𝛼) ⊗ Γ𝜑 (𝛼)𝑑 𝜑 𝑡 𝑛 𝑓](𝑥) = Definition 39 The 𝜑- fractional Caputo derivative of order 𝛼 is denoted and given by (26) [ 𝐶 𝐷𝜑(𝛼) 𝑓](𝑥) = 𝑥 1 ∫ (𝑥 − 𝑡)𝛼−1 𝜑 −1 (𝑓 𝜑(𝑛) (𝑡))𝑑𝑡), 𝜑( Γ(𝑛 − 𝛼) 𝑎 ℜ(𝑛 − 𝛼) > 0. (27) 97 = 𝜑([𝐶 𝐷(𝛼) 𝜑 −1 ∘ 𝑓](𝑥)) (29) As examples, Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 [ 𝐶 𝐷∗(𝛼) 𝑓](𝑥) = 𝑥 ∫𝑎 (𝑓 ∗(𝑛) (𝑡) (𝑥−𝑡)𝑛−𝛼−1 Γ(𝑛−𝛼) 𝑒 [𝐶 𝐷⋆(𝛼) 𝑓](𝑥) = 𝑒 Amer H. Darweesh, Abdelaziz M. D. Maghrabi 𝑥 ∫𝑎 𝑓(tanh−1 𝑡)𝑑 ⋆(𝛼) tanh−1 𝑡, ℜ(𝛼) > 0. 𝑑𝑡 ) , ℜ(𝑛 − 𝛼) > 0. (30) 𝑥 1+(𝑓⋆(𝑛) (𝑥)) 𝑥 1+(𝑓⋆(𝑛) (𝑥)) 2 ∫𝑎 ln 2 ∫𝑎 ln 1−(𝑓⋆(𝑛) (𝑥)) 𝑑(𝑛−𝛼) 𝑥 1−(𝑓⋆(𝑛) (𝑥)) 𝑑(𝑛−𝛼) 𝑥 ℜ(𝑛 − 𝛼) > 0. [𝐼 bi⋆(𝛼) 𝑓](𝑥) = (34) This notation in (34) is just an abbreviation, since equation (32) is rather tedious. It means that all the arguments would change from 𝑡, 𝑥 → tanh−1 (𝑡), tanh−1 (𝑥) everywhere except at the boundaries of integration. −1 , +1 Definition 41 The bi𝜑- fractional RiemannLiouville derivative of order 𝛼 is denoted and given by (31) [𝐷bi𝜑(𝛼) 𝑓](𝑥) 𝑥 1 𝑑bi𝜑(𝑛) 𝜑( ∫ (𝜑 −1 (𝑥) = 𝑑𝜑 −1 (𝑥)𝑛 Γ(𝑛 − 𝛼) 𝑎 − 𝜑 −1 (𝑡))𝑛−𝛼−1 𝜑 −1 (𝑓(𝜑 −1 (𝑡)))𝑑𝜑 −1 (𝑡)). Theorem (20) paves the way for the bi𝜑fractional calculi without the use of heavy machinery. It is an immediate out-growth of it in some sense. In the following definitions, the subscript 𝜑 −1 (𝑡) is to clarify that the operations are carried out with respect to 𝜑 −1 (𝑡). (𝛼) = 𝜑([𝐷𝜑−1 (𝑡) 𝑓](𝑥)), ℜ(𝑛 − 𝛼) > 0. As examples, Definition 40 The bi𝜑- fractional integral of order 𝛼 is denoted and given by 𝑥 𝑛−𝛼−1 (ln ) 𝑡 𝑑𝜋(𝑛) 𝑥 𝜋(𝛼) Γ(𝑛−𝛼) )𝑑ln𝑡 , ∫ (𝑓(ln𝑡) [𝐷 𝑓](𝑥) = 𝑑ln𝑥 𝑛 𝑎 [𝐼 bi𝜑(𝛼) 𝑓](𝑥) ℜ(𝑛 − 𝛼) > 0 𝑥 1 ∫ (𝜑 −1 (𝑥) = 𝜑( Γ(𝛼) 𝑎 −1 − 𝜑 (𝑡))𝛼−1 𝜑 −1 (𝑓(𝜑 −1 (𝑡)))𝑑𝜑 −1 (𝑡)), [𝐷bi⋆(𝛼) 𝑓](𝑥) = 𝑑bi⋆(𝑛) 𝑑tanh−1 𝑥 𝑛 𝑅𝑒(𝛼) > 0. (𝛼) = 𝜑([𝐼𝜑−1 (𝑡) 𝑓](𝑥)) 𝑥 (ln )𝛼−1 𝑡 𝑥 ∫𝑎 (𝑓(ln𝑡) Γ(𝛼) )𝑑ln𝑡 , ℜ(𝛼) > 0. E-ISSN: 2224-2678 𝑥 ∫𝑎 𝑓(tanh−1 𝑡)𝑑 ⋆(𝛼) tanh−1 𝑡 . (37) Definition 42 The bi𝜑- fractional (32) Caputo derivative of order 𝛼 is denoted and given by [𝐶 𝐷bi𝜑(𝛼) 𝑓](𝑥) = 𝜑( As examples, [𝐼 𝜋(𝛼) 𝑓](𝑥) = (36) 𝜑 −1 (𝑡))𝛼−1 𝜑 −1 (𝑓 (33) 1 𝑥 ∫ (𝜑 −1 (𝑥) − Γ(𝑛−𝛼) 𝑎 bi𝜑(𝑛) −1 (𝛼) (𝜑 (𝑡)))𝑑𝜑 −1 (𝑡)) = 𝜑([𝐶 𝐷𝜑−1 (𝑡) 𝑓](𝑥)), ℜ(𝑛 − 𝛼) > 0 Remark 43 One can see that the Hadamard fractional derivative is the logarithm of the 98 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi bigeometric RL derivative of 𝑒 𝑓 . That is, it is a RL derivative on the manifold 𝕀 under the diffeomorphism 𝜑. This hints that many of the fractional derivatives that are defined may indeed be a RL derivative on a given manifold, from a differential geometric point of view. [ 𝐶 𝐷𝜋(𝛼) 𝑓](𝑥) = 𝐶 [ 𝐷 2 2 (0, ∞) defined by 𝜑(𝑥) = 𝑎𝑟𝑐𝑡𝑎𝑛𝑥, 𝜂(𝑥) = 𝑒 𝑥 . Then we have, and, 𝑥 ∫𝑎 (𝑓 𝜋(𝑛) (ln𝑡) bi⋆(𝛼) 𝜋 𝜋 Example 45 Consider 𝜑: ℝ → (− , ), 𝜂: ℝ → 𝑥 (ln )𝑛−𝛼−1 𝑡 Γ(𝑛−𝛼) 𝑓](𝑥) = )𝑑ln𝑡 , 𝑥 ∫𝑎 𝑓 ⋆(𝑛) (tanh−1 𝑡)𝑑 ⋆(𝛼) tanh−1 𝑡 𝑏 ∫ 𝑓(𝑥)𝑑 bi(𝜑,𝜂) 𝑥 (40) 𝑎 𝑎 Where the tangent function is on domain 𝜋 𝜋 (− , ). 2 2 Remark 46 Many other fractional calculi may be defined under the influence of a diffeomorphism defined as a composition of finitely many diffeomorphisms. 5 Conclusion In this paper, the very basic definitions of fractional calculus are established in the relatively new 𝜑-calculi and bi𝜑-calculi, which are promising to be of great use. Indeed, they give an interpretation of the so-called 𝜓 −fractional calculus under the scope of the discussed subject, as Remark 43 mentions. This paper also reveals a new form of the discussed calculi as seen in the first section which is useful in proofs. We have also arrived at an important link which in future papers will make establish relations between 𝜑-calculi and bi𝜑calculi in a smooth and practical way as well as a relation to the Newtonian versions, where various analogs such as the 𝜑 −gamma function. 𝜂 −1 ∘ 𝑓(𝑥) − 𝜂 −1 ∘ 𝑓(𝑦) 𝑓 bi(𝜑,𝜂) (𝑥) = lim 𝜂 ( ) 𝑦→𝑥 𝜑 −1 (𝑥) − 𝜑 −1 (𝑦) ′ (𝜂 −1 ∘𝑓) (𝑥) 𝜑 −1 (𝑥)′ ) And the bi(𝜑, 𝜂)-integral, 𝑏 (42) 𝑏 ∫𝑎 𝑓(𝑥)𝑑 bi(𝜑,𝜂) 𝑥 = 𝜂 (∫𝑎 𝜑 −1 (𝑥)′ 𝜂−1 ∘ 𝑓(𝑥)𝑑𝑥) 𝑏 = exp (∫ 𝑠𝑒𝑐 2 (𝑥)ln𝑓(𝑥)𝑑𝑥) (41) These definitions also allows one to calculate various 𝜑- and bi𝜑-fractional derivatives, and integrals. Hence, all the results that holds true for Caputo and Riemann-Liouville differintegrals are also true under the influence of the homeomorphism 𝜑. For a more general case, consider a function 𝑓: 𝕀1 → 𝕀2 , where 𝕀1 , 𝕀2 ⊆ ℝ are ordered fields equipped with the usual metric and their field structure are based on the algebraic operations similar to those in the beginning of the second section under the influence of the homeomorphisms 𝜑: ℝ → 𝕀1 , 𝜂: ℝ → 𝕀2 . Then, we can define the bi(𝜑, 𝜂)derivative in a similar fashion, where = 𝜂( 𝑓(′𝑥) 𝑓 bi(𝜑,𝜂) (𝑥) = exp ( ) 𝑓(𝑥)𝑠𝑒𝑐 2 (𝑥) (43) Remark 44 The domains of the homeomorphisms may be a subset of the real numbers. E-ISSN: 2224-2678 99 Volume 22, 2023 WSEAS TRANSACTIONS on SYSTEMS DOI: 10.37394/23202.2023.22.10 Amer H. Darweesh, Abdelaziz M. D. Maghrabi References: [1] Kochubei A, Luchko Y, Handbook of Fractional Calculus with Applications, de Gruyter, Berlin, 2019. [2] Agamirza Bashirov, Emine Mısırlı, Y ̈ucel Tando ̆gdu, and Ali ̈Ozyapıcı. On modeling with multiplicative differential equations. Applied Mathematics-A Journal of Chinese Universities, 26(4):425– 438, 2011. 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Conflict of Interest The authors have no conflicts of interest to declare that are relevant to the content of this article. Creative Commons Attribution License 4.0 (Attribution 4.0 International, CC BY 4.0) This article is published under the terms of the Creative Commons Attribution License 4.0 https://creativecommons.org/licenses/by/4.0/deed.en _US E-ISSN: 2224-2678 100 Volume 22, 2023