Applications o local ractional calculus to engineering in ractal time-space: Local ractional dierential equations with local ractional derivative Yang XiaoJun Department o Mathematics and Mechanics, China University o Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu, 2218, P. R.C dyangiaojun@163.com This paper presents a better approach to model an engineering problem in ractal-time space based on local ractional calculus. Some eamples are given to elucidate to establish governing equations with local ractional derivative. Key words: local ractional calculus, ractal time-space, local ractional derivative, local ractional dierential equation MSC21: 26A27, 28A8, 26A99 1 Introduction Local ractional calculus is a generalization o dierentiation and integration o the unctions deined on ractal sets. The idea o local ractional calculus has been a subject o interest not only among mathematicians but also among physicists and engineers [1-15]. There are many deinitions o local ractional derivatives and local ractional integrals (also called ractal calculus) [1-13]. Hereby we write down Gao-Yang-Kang deinitions as ollows. Gao-Yang- Kang local ractional derivative is denoted by [7-8,1-11,14] d lim, (1.1) d 1 o order in the interval, with integral o. Gao-Yang-Kang local ractional 1 1 ab is denoted by [7,9-11,14] jn1 b aib t dt lim tj tj 1 a 1 t j, (1.2) 1
with tj tj 1 tj t ma t1, t2, t j,..., where or j,..., N 1, t j, t j is a 1 partition o the intervalab, and t a, t N b. Yang presented that there eists the relation [14] (1.3) and with,or, and,. Then continuous on the intervalab,, denoted by, is called local ractional C a b. (1.4) In this paper, our attempt is to model an engineering problem in ractal-time space based on local ractional calculus. 2 preliminaries 2.1 Notation Deinition 1 There always eists the relation where is constant. y y is called Hölder unction o eponent o., or any y,, (2.1) Deinition 2 There always eists [14] or any a b o,,, (2.2) which is called continuous, or local ractional continuous on the intervalab,. Remark 1. From (2.1) and (2.2) this case is called unity local ractional continuity o. Form (1.3) this deinition is called local ractional continuity o standard local ractional continuous), by denoted by For ab, lim, we have the notation (or called, (2.3), C a b. (2.4) 2
2.2 Generalized local ractional Taylor ormula with local ractional derivative o one-variable unction Proposition 1 Suppose that C a, b, then [4] a b I ( ba) 1, a b. (2.5) Theorem 2 Suppose that k, k1 C a, b, or 1, then we have k k k k 1 k 1 k ( ) I [ ] [ ] I with a b, where k1 times k 1 D... D. Proo. From (2.5) we have k 1 times k 1... k 1, (2.6) I I I k1 n1 n 1 I [ ] I k dt (2.7) Successively, it ollows rom (2.9) that 1 1 k k k and I (2.8) I I. (2.9) k k k k 1 (2.1) I I k k k k k 1 1 k 1 I (2.11) k 1 1 2 I 12 1 (2.12) k 2 k k ( ) k 1 (2.13) 3
Hence we have the result. Theorem 3 (Generalized mean value theorem or local ractional integrals) Suppose that C a, b, Ca, b, we have Proo. Taking k 1in (2.6), we deduce the result. ( ) 1, a b. (2.14) Theorem 4 (Generalized local ractional Taylor ormula) Suppose that k1 C a, b, or k,1,..., nand 1, then we have k n1 k n1 k n n (2.15) 1 1 1 k, ab, with a b Proo. Form (2.6), we get k1 times k 1, where D... D. k k k1 k1 k I [ ] [ ] I a Successively, it ollows rom (2.16) that k k n k1 k1 [ ] [ ] n1 [ n1 k I I I By using (2.5) and (2.18) we have n ( ) k k ( ) k 1. (2.16) (2.1 k. k k 1 (2.18) 1 1 (2.19) n1 n1 n n 1 I I dt a 7) 1 n n 1 ai n1 ai n 1 n1 n1 1 n 1 (2.2) (2.21) (2.22) 4
, ab, with a b. Combing the ormulas (2.22) and (2.18) in (2.16), we have the result. Hence, the proo o the theorem is completed. 2.3 Local ractional continuity o two-variable unction Deinition 3 There eists the relation,, y y (2.23). Then,. The unction, with 1and y y 2, or, 1, 2 and, 1, 2, called local ractional continuous at y E continuous in the region E, denoted by y is y is local ractional C E. (2.24) Deinition 4 Setting y, CE, local ractional partial derivative o, espect to, is denoted by with, y y,, y lim y,, y 1 y,, y. Similarly, local ractional partial derivative o, enoted by with, y y, y, y y lim y y y y y y, y, 1 y, y,. The second derivatives are denoted by 2,, y y 2,, y y y y y y y with r (2.25) y with respect to y, is d 2 2 yy, (2.26), (2.27), (2.28) 5
2,, y y y y 2,, y y y y 2 y, (2.29) 2 y. (2.3) Remark 2. Suppose that and 2 are local ractional continuous in a ractal region R 2 y y o the y -plane, then. (2.31) 2 2 y y For local ractional partial derivative o high order, we get n n... (2.32) n y y y and n m mn...... n m. (2.33) y y y y In similar manner, we get generalized local ractional Taylor series or two-variable unction as ollows:, n Suppose that y, CE with y E., n C n i E, then y y y y n in n i n i, (2.34) n i y i ni,, 2.4 Applications o approimation o unctions From (2.34) we have the ollowing relation n in n i ni y, y y, y n i R i y n i (2.35) where its reminder is n i n Rn y y y y i ni, n i (2.36) i n i and 6 n
Rn n lim. (2.37) Now taking the ormula (2.15) into account we arrive at this relation which yields In similar manner, rom (2.34) we get which deduces E E k k. (2.38) k 1 k 1 n k k (2.39) k k k1k k1k E y. (2.4) n k n k k1k k1k E y. (2.41) 2.5 Useul results Here the ollowing ormulas hold: d d (1 k) (1 k 1 ) k k1, (2.42) de k ke k ; (2.43) d de E. (2.44) d 3 Application to governing equation in engineering in ractal space In this section some models or governing equations in ractal space are suggested. We start with typical models with local raction derivative in engineering. 3.1 Some typical models with local ractional derivatives in engineering Model 1. Local ractional transient heat conduction equation The transient heat conduction equation in ractal spaces can be described by the equation 7
, yt 2 y t 2 t with the initial and boundary conditions y, t, ; y, t k hy, t y, L; y, t y, t i, t,, 1, (3.1) (3.2) where is the ractal thermal diusivity and k is the thermal conductivity o the ractal wall material. Model 2. Local ractional wave equation The wave motion in ractional space can be described by the equation 2 y t, yt,, 1, (3.3) 2 t with the initial and boundary conditions lim yt, lim yt, yt, t t Model 3. Local ractional diusion equation The diusion equation in ractional space can be described by the equation 2, yt, y t 2 a t with the boundary-value problem 2, lim yt, lim yt, yt, t t. (3.4) t,, 1, (3.5) 3.2 Application to local ractional relaation equation The relaation equation in ractional space can be described by the equation y t t c y t. (3.6), c, t,1, (3.7) 8
with the initial value y t t y. Its solution can be easily obtained, which reads From (2.44), taking k y t y E c t. (3.8) c implies that d E c t c E c t (3.9) d It ollows rom (3.9) that a special solution to equation (3.7) Hence we have the ollowing relation y t E c t. (3.1) y t me c t. (3.11) Taking initial value condition into account in (3.11), we obtain the solution o equation (3.7), which is The solution depends upon ractal dimension. Given any pointt t orm E y t y E c t. (3.12), we have the local ractional Taylor-Yang epansion o c t 1 k and we always get the Hölder relation or anyt 1, t 2. 1k k k c E c t t t k y t in the (3.13) 1 2 1 2 E ct E ct mt t, (3.14) Hence, rom (3.13) and (3.14) its solution is local ractional continuous and eists ractal property. 4 Conclusions The suggested governing equations with local ractional derivative are easy to be used or any ractal process and we get the solution to relaation equation in ractional space. However, solutions to transient heat conduction equation, wave equation and diusion equation in ractal space are much needed. Maybe, Yang-Laplace transorms [14] in ractal space get the result. 9
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