Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 4356371, 11 pages http://dx.doi.org/10.1155/2016/4356371 Research Article Fractional Mathematical Operators and Their Computational Approximation José Crespo and Francisco Javier Montáns Escuela Tecnica´ Superior de Ingenier´ıa Aeronautica´ y del Espacio, Universidad Politecnica´ de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Correspondence should be addressed to Francisco Javier Montans;´ [email protected] Received 27 January 2016; Revised 30 July 2016; Accepted 9 August 2016 Academic Editor: Manuel Doblare´ Copyright © 2016 J. Crespo and F. J. Montans.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Usual applied mathematics employs three fundamental arithmetical operators: addition, multiplication, and exponentiation. However, for example, transcendental numbers are said not to be attainable via algebraic combination with these fundamental operators. At the same time, simulation and modelling frequently have to rely on expensive numerical approximations of the exact solution. The main purpose of this article is to analyze new fractional arithmetical operators, explore some of their properties, and devise ways of computing them. These new operators may bring new possibilities, for example, in approximation theory and in obtaining closed forms of those approximations and solutions. We show some simple demonstrative examples. 1. Introduction parameters [2, 3] rendering linear equations. We will also see possible applications of the presented operators in constitu- Science mostly hinges on mathematics which has been built tive modelling. on three fundamental operations: the addition, the multipli- Consider the classification described in Table 1. cation, and the exponentiation. We have become extremely As observed in Table 1, in the last column, exponentiation used to them. However, the rigidity of these preset tools may is expressed in terms of a higher operation. This operation is attimesmakeitdifficulttofindwaysandsolutionsfornature called tetration and is commonly considered as the first hyper- problems that are not easily expressed in terms of traditional operation because it has a higher rank (=4) than exponenti- operators. For example, transcendental numbers are said not ation (=3). Preliminary and fundamental works in this line to be attainable via algebraic combination with these funda- have been performed by Peano [4] with his Peano Arithmetic mental operators, which are a long standing human inven- axioms, Ackermann [5] with the nonprimitive recursive func- tion. At the same time, simulation and modelling have to deal tions, and Goodstein [6] with the hyperoperational sequence, with some situations in which the only solution is to compute among others. an expensive numerical approximation of the exact solution, However, intermediate-rank operations are also possible which frequently cannot be written in closed form in terms and may be useful. The problem we address is how to compute of traditional operators. a hyperoperation of rank =3/2(from now on, 3/2). To The purpose of this work is to present and apply some this respect, this was already an open question formulated operations beyond the rigidity of the known fundamental byRutsovandRomerio[7]orWilliams[8].Wewillfirstset operations, such as an intermediate operation between addi- the proper environment, namely, the arithmetical continuous tion and multiplication. The ideas have some parallelism spectrum. Some definitions are useful to understand some with fractional derivatives [1] which, for example, allow the properties that are going to be necessary to compute 3/2,as accurate characterization of viscous materials with just two well as to establish a comfortable notation for our purposes. 2 Mathematical Problems in Engineering Table 1: Hierarchical classification of fundamental operations. Op. rank, Operational name Conventional notation Rank-wise notation (()) = ( + 1) +1⋅ =⋅2 0 Successor 0 1 +=⋅2 ⋅=⋅2 1 Addition 1 2 ⋅=2 ⋅=⋅2 2 Multiplication 2 3 = 2⋅=⋅2 3 Exponentiation # 3 4 2. The Arithmetical Continuous Spectrum Table 2: Proposed hyperoperational notation. The arithmetical continuous spectrum focuses on the idea of Operationalnature Notation Operation Example(=3) ⋅ ⋅=3=2 2⋅ 3 =8 having a continuous spectrum of arithmetical operations as it Direct 3 3 occurs with real numbers. Here we note that there are many R-inverse / /=3= 8/ √8=2 possibilities to define hyperoperations of intermediate ranks. 3 L-inverse \ \=2= 8\ log2(8) = 3 The reader is referred to the explanation and discussion in [9]. 3 The fractional operators that are herein defined have the pur- pose of seeking improved and more compact interpolations in engineering fields in general and in constitutive modelling / = = / in particular. 2 In order to connect operations of different ranks, the H \ = = / environment set of hyperoperations must be previously 2 defined. For this purpose, a set of definitions and lemmas will facilitate the task of establishing fractional hyperoperators so (⋅) / (∗) ≡ (⋅) \ (∗) ≡ (⋅) / (∗) . 2 2 and their computational approximations. (1) Definition 1. Let H be the set that contains every binary hyperoperation of rank , ∈ R. H ⊂ H is the space of For instance, in the case of rank =3, the relations between integer rank hyperoperations, and H ⊂ H is the space of the base , the exponent ,andtheresult are as follows: fractional rank hyperoperations; that is, H ⊂ H ⊂ H. =2 The suggested notation described in Table 2 will be employed in order to extend a universal notation for hyper- ⇓ operations of rank ∈R. Consider , , ∈ R,whichresem- =⋅= bles the notion of base, exponent, and result, respectively. 3 The names of R- and L-inverses remind those of root-kind inverse and log-kind inverse, but note that the notation is also √ =/=(⋅)= / (2) such that the bar is slanted right for the R-inverse operator 3 3 3 and left for the L-inverse operator. Some trivial particular cases are those for =1and =2 log=\=(⋅) \ = 3 3 3 =1 so (⋅) / (∗) ≠ (⋅) \ (∗) . 3 3 ⇓ Definition 2 (hyperoperational mean). All rank arithmeti- +=⋅= H 1 cal operation in has a corresponding hyperoperational mean, which is defined in the following way: /==− 1 (, ) = (⋅ ) / 2, >. (3) \==− +1 1 This definition is the natural extension of ranks =1 so (⋅) / (⋅) ≡ (⋅) \ (⋅) ≡ (⋅) − (⋅) =2 1 1 (arithmetical mean) and (geometrical mean), =2. + 1 (, ) = =(⋅)2, / 2 1 ⇓ 2 (4) 2 ⋅=⋅= 2 (, ) = √⋅=(⋅) /2. 2 2 3 Mathematical Problems in Engineering 3 Definition 3 (hyperoperational sequence, HS). Hyperopera- According to Definition 5, an example of rank =1is tional sequence of base is a sequence HS(, ) so that the shown. RS1(3, ) satisfy arithmetical mean, 1, th term is 1 +−1 −1 :1 (1,−1)=0 = , = ⋅ = (, ) ; 2 +1 HS (5) −1 =(0 ⋅2)−1 =(0 ⋅ 2) \ 1 =0⋅2−3=−3, that is, 2 1 (, ) ≡ { } HS 0 =1 =0, ={...,(),(⋅) , ( ⋅ ⋅),...} 1 ==3, (6) + : ( , )= = 0 2 , ={..., ⋅ 1, ⋅ 2, ⋅ 3,...}. 2 1 2 0 1 2 +1 +1 +1 2 =(1 ⋅2)−0 =(1 ⋅ 2) /0 =6−0=6, Definition 4 (neutral element). The neutral element for 2 1 hyperoperations of rank is the number such that (14) 3 =(2 ⋅ 2) /1 =12⋅2−3=9, ⋅ = 2 1 (7) andthenobviously 4 =(3 ⋅ 2) /2 =18⋅2−6=12, 2 1 / =, 5 =(4 ⋅ 2) /3 =24⋅2−9=15, (8) 2 1 \=. . Definition 5 (recursive sequence, RS). Let a recursive (, ) sequence RS ,ofrank ,base , and neutral element , =(−1 ⋅ 2) /−2 =3⋅=3⋅=⋅. 2 1 2 2 be the sequence built by its corresponding mean, ,andthe following pair of starting values: As another example, consider this time rank =3. RS3(3, ) must satisfy the corresponding hyperoperational {0,1} = {,} . (9) mean, 3, The sequence can be run either forwards (direct values) = =1, in this way, 0 3 1 ==3, 2 =( ⋅ 2) /, (10) +1 2 :3 (3,1)=2, or backwards (inverse values), in this way √0 1 3 3 2 =(1 ⋅ 2) /0 = 1 =3 =27=3⋅2=3, 4 3 4 =( ⋅ 2) \ . (15) −1 (11) 3 3 +1 √1 2 √ 27 3 3 =(2 ⋅ 2) /1 = 2 = 27 =3⋅3=3 , 4 3 4 Generally, RS is based on the relation that adjacent terms . satisfy. This relation is the hyperoperational mean . (−1,+1)=, =(−1 ⋅ 2) /−2 =⋅. (12) 4 3 4 ( ⋅ )/2=. −1 +1 +1 It can be checked that this makes recursive sequences, RS, coincide up to rank =3,∈N,withthecorresponding If solved for +1 the forwards direction of RS is obtained. hyperoperational sequences, HS. In turn, they also coincide On the contrary, if solved for −1 it yields the expression of with the Goodstein function if the neutral elements are the backwards direction same for the corresponding rank of the Goodstein function; that is, +1 =( ⋅ 2) /−1 forwads, +1 (13) 1 =0, (16) −1 =( ⋅ 2) \ −1 backwards. +1 =1, >1,∈N. 4 Mathematical Problems in Engineering Lemma 9 (RS starting set). Let RS(, ) be a recursive m2 (S2 (2,2, 1) ,S, S2 (2,2, 9)) =m= m2 (S2 (2,2, 2) ,S, S2 (2,2, 8)) =m= m2 (S2 (2,2, 4) ,S, S2 (2, 6)) sequence of base .Then,thefirstthreetermsofRS(, ) are √21 ·2· 29 = √22 ·2· 28 = √24 ·2· 26 =32= 32 {⋅0, ⋅ 1, ⋅ 2} = {,, ⋅ 2} +1 +1 +1 +1 (21) m1 (S1 (2,2, 1) ,S, S1 (2,2, 9)) =m= m1 (S1 (22,, 2) ,S, S1 (2,2, 8)) =m= m1 (S1 (2,2, 4) ,S, S1 (2,2, 6)) 2·1+2·92 · 1 + 2 · 9 22·2+2·8· 2 + 2 · 8 22·4+2·6· 4 + 2 · 6 the term being the right neutral element of +1. = = =10= 10 2 2 2 Proof. The right neutrality of with ≥2is imposed to the 022 4 8 1616 32 64 7 8 9 10 n 2 2 2 2 2 =1 ⋅1= value of due to multiplicative recursion, +1 . According to Definition 5, 0 2 4 6 8 10 12 141616 18 20 2·n (2,0)=1, 1 2 3 4 5 6 7 8 9 10 n 0 0 =, (22) Inch 6 5 4 3 2 1 1 =.