Fractional Calculus. Bachelor Project Mathematics. faculty of mathematics and natural sciences. October Student: D.E. Koning

Transcription

1 fculty of mthemtics nd nturl sciences Frctionl Clculus Bchelor Project Mthemtics October 215 Student: D.E. Koning First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. H.L. Trentelmn

2 Abstrct This thesis introduces frctionl derivtives nd frctionl integrls, shortly differintegrls. After short introduction nd some preliminries the Grünwld-Letnikov nd Riemnn-Liouville pproches for defining differintegrl will be explored. Then some bsic properties of differintegrls, such s linerity, the Leibniz rule nd composition, will be proved. Therefter the definitions of the differintegrls will be pplied to few exmples. Also frctionl differentil equtions nd one method for solving them will be discussed. The thesis ends with some exmples of frctionl differentil equtions nd pplictions of differintegrls.

3 CONTENTS Contents 1 Introduction 4 2 Preliminries The Gmm Function The Bet Function Chnge the Order of Integrtion The Mittg-Leffler Function Frctionl Derivtives nd Integrls The Grünwld-Letnikov construction The Riemnn-Liouville construction The Riemnn-Liouville Frctionl Integrl The Riemnn-Liouville Frctionl Derivtive Bsic Properties of Frctionl Derivtives Linerity Zero Rule Product Rule & Leibniz s Rule Composition Frctionl integrtion of frctionl integrl Frctionl differentition of frctionl integrl Frctionl integrtion nd differentition of frctionl derivtive Exmples The Power Function The Exponentil Function The Trigonometric Functions Frctionl Liner Differentil Equtions The Lplce Trnsforms of Frctionl Derivtives Lplce Trnsform of the Riemnn-Liouville Differintegrl Lplce Trnsform of the Grünwld-Letnikov Frctionl Derivtive The Lplce Trnsform Method Exmples Applictions Economic exmple Concrete exmple Conclusions 29 9 References 31 Bchelor Project Frctionl Clculus 3

4 1 INTRODUCTION 1 Introduction Frctionl clculus explores integrls nd derivtives of functions. However, in this brnch of Mthemtics we re not looking t the usul integer order but t the non-integer order integrls nd derivtives. These re clled frctionl derivtives nd frctionl integrls, which cn be of rel or complex orders nd therefore lso include integer orders. In this thesis we refer to differintegrls if we re tlking bout the combintion of these frctionl derivtives nd integrls. So if we consider the function f(t) = 1 2 x2, the well-known integer first-order nd second-order derivtives re f (t) = x nd f (t) = 1, respectively. But wht if we would like to tke the 1 2 -th order derivtive or mybe even the th order derivtive? This question ws lredy mentioned in letter from the mthemticin Leibniz to L Hôpitl in Since then severl fmous mthemticins, such s Grünwld, Letnikov, Riemnn, Liouville nd mny more, hve delt with this problem. Some of them cme up with n pproch on how to define such differentition opertor. For very interesting more detiled history of Frctionl Clculus we refer to [1, p. 1-15] First in chpter 2 we shll give some bsic formuls nd techniques which re necessry to better understnd the rest of the thesis. Then in chpter 3 two definitions for differintegrl will be given. The Grünwld-Letnikov nd the Riemnn-Liouville pproch will be explored. These re the two most frequently used differintegrls. Afterwrds in chpter 4 some bsic properties of these differintegrls will be given nd proved. Then in chpter 5 we shll explore few exmples. In chpter 6 we will tke look t frctionl differentil equtions (FDE s). Therefore we lso need to explore the Lplce trnsforms of frctionl derivtives. Chpter 6 ends with some exmples of FDE s. Therefter chpter 7 dels with few pplictions of differintegrls which is followed by conclusion in chpter 8. Bchelor Project Frctionl Clculus 4

5 2 PRELIMINARIES 2 Preliminries In this section we shll give some bsic formuls nd techniques which re necessry to better understnd the rest of the thesis. We strt off with the Gmm function. 2.1 The Gmm Function The Gmm function plys n importnt role in Frctionl Clculus nd therefore it is mentioned in the Preliminries. Definition 2.1. Let z C, then we define the Gmm function s Γ(z) = e t t z 1 dt. This integrl converges for Re(z) > (the right hlf of the complex plne). One of the bsic properties of the Gmm function is Γ(z + 1) = zγ(z). (1) To prove this we integrte the formul for the Gmm function given in Definition 2.1 by prts Γ(z + 1) = e t t z = [ e t t z ] t= t= + z e t t z 1 dt, where the first term drops out nd the second term is equl to zγ(z), so identity (1) follows. We lso hve Γ(1) = 1 nd if we use identity (1) we get Γ(2) = 1 Γ(1) = 1 = 1! Γ(3) = 2 Γ(2) = 2 1! = 2! Γ(4) = 3 Γ(3) = 3 2! = 3!.. Γ(n + 1) = n Γ(n) = n (n 1)! = n! So by induction it follows tht Γ(n + 1) = n! for ll n N. 2.2 The Bet Function In some cses the Bet function is more fvorble thn the Gmm function. Since it is convenient to use it in frctionl derivtives of the Power function, we lso mention the Bet function here. Definition 2.2. Let z, w C, then we define the Bet function s B(z, w) = 1. τ z 1 (1 τ) w 1 dτ,. Bchelor Project Frctionl Clculus 5

6 2 PRELIMINARIES for Re(z) > nd Re(w) >. After we use the Lplce trnsform for convolutions the Bet function cn be expressed in terms of the Gmm function by B(z, w) = Γ(z)Γ(w) (2) Γ(z + w) nd it follows from (2) tht B(z, w) = B(w, z). (3) With the Bet function it is possible to obtin two useful results for the Gmm function π Γ(z)Γ(1 z) = sin(πz), (4) Γ(z)Γ(z ) = π2 2z 1 Γ(2z). (5) 2.3 Chnge the Order of Integrtion In section 4.4 bout the composition of differintegrls we will tke dvntge of chnging the order of n integrl. If we hve ny function f(t, τ, ξ) which is integrble with respect to τ nd ξ the chnge of order is given by the following formul τ f(t, τ, ξ) dξ dτ = 2.4 The Mittg-Leffler Function ξ f(t, τ, ξ) dτ dξ. (6) We know in integer-order differentil equtions the exponentil function e z plys n importnt role. This cn lso be written in its series form which is given by e z = z k Γ(k + 1). More generlly, we cn consider the expression z k E α,β (z) = Γ(αk + β), (7) where α, β C nd Re(α) >. We see tht in the specil cse of α = 1 nd β = 1 we hve E 1,1 (z) = e z. This generliztion is clled the Mittg-Leffler function nd the two-prmeter function is very useful in the frctionl clculus, especilly in frctionl differentil equtions, which we will discuss in section 6. Since the series for the Mittg-Leffler function (7) is uniformly convergent on ll compct subsets of C we cn differentite it term by term to get the following expression which is lso necessry lter on. Corollry 2.1. Let z C, α, β C, Re(α) > nd m N, then the m-times differentited Mittg-Leffler function is given by E (m) α,β (z) = (k + m)! k! z k Γ(αk + αm + β). Bchelor Project Frctionl Clculus 6

7 3 FRACTIONAL DERIVATIVES AND INTEGRALS 3 Frctionl Derivtives nd Integrls In fct the term Frctionl Clculus is not pproprite since it does not men the frction of ny clculus, nor the clculus of frctions. It is ctully the brnch of Mthemtics which generlizes the integer-order differentition nd integrtion to derivtives nd integrls of rbitrry order. If we look t the sequence of integer order integrls nd derivtives..., τ2 f(τ 1 ) dτ 1 dτ 2, f(τ 1 ) dτ 1, f(t), df(t) dt, d2 f(t) dt 2,... one cn see the derivtive of rbitrry order α s the insertion between two opertors in this sequence. It is clled frctionl derivtive nd throughout this thesis the following nottion is used: D α t f(t). For frctionl integrl the sme nottion is used, but with α <. Thus n integrl of order β cn be denoted by: D β t f(t). In this thesis we refer to this with the term differintegrl. The subscripts nd t re clled the terminls of the differintegrl nd they re the limits of integrtion. There hve been different pproches to define this differintegrl nd this section dels with the definitions of the differintegrls from Grünwld-Letnikov nd Riemnn-Liouville. 3.1 The Grünwld-Letnikov construction In this section we will derive formul for the so-clled Grünwld-Letnikov differintegrl. The proof is bsed on the forwrd difference derivtive given by f f(t + h) f(t) (t) = lim. h h If we pply this formul gin we get the well-known second-order derivtive f f(t + 2h) 2f(t + h) + f(t) (t) = lim h h 2. We cn generlize this formul for derivtive nd if we use the binomil coefficient ( ) n r = n! r!(n r)! we get for the n th -derivtive f (n) (t) = lim h r n ( 1)r( n r) f(t + (n r)h) h n. If we replce the integer n by p R we obtin the following definition. Definition 3.1. Let m be the smllest nturl number such tht p m, then we define the Grünwld-Letnikov differintegrl s D p 1 ( ) p f(t) = lim h h p ( 1) r f(t + (p r)h). r r<m Bchelor Project Frctionl Clculus 7

8 3 FRACTIONAL DERIVATIVES AND INTEGRALS Since we replced the integer n by the rel number p we lso hve to generlize the definition of the binomil coefficient. This cn be done using the multiplictive formul which gives ( ) p r = p(p 1)(p 2) (p r + 1), (8) r(r 1)(r 2) 1 where r N. When the substition h h is mde in Definition 3.1 we get the direct Grünwld-Letnikov differintegrl given by D p t f(t) = m lim h p h mh=t r= = lim h ( ) p ( 1) r f(t rh) r ( ) p t m ( ) p ( 1) r f m r r= ( t r t ). m When p = n this cn be seen s the n th -order derivtive nd if p = n it represents the n-fold integrl. The Grünwld-Letnikov nd the Riemnn-Liouville frctionl derivtive cn be relted to ech other. Therefore we need nother expression for the Grünwld- Letnikov derivtive of rbitrry order. This is given by the following formul. Corollry 3.1. D p t f(t) = m f (k) ()(t ) p+k Γ( p + k + 1) + 1 Γ( p + m + 1) (9) (t τ) m p f (m+1) (τ) dτ. In the lst formul the derivtives f (k) (t) for k = 1, 2,..., m + 1 hve to be continuous in the closed intervl [, t] nd m > p 1. The proof of Corollry 3.1 is pretty long. Therefore it won t be given in this thesis, but it cn be found in [2, p ]. 3.2 The Riemnn-Liouville construction Insted of beginning with the derivtive s in the Grünwld-Letnikov pproch, the Riemnn-Liouville strts with the integrl. The differintegrl is given by the following expression: D p t f(t) = ( ) m+1 d t (t τ) m p f(τ) dτ, (1) dt where m N stisfies (m p < m + 1). The expression for the Grünwld- Letnikov frctionl derivtive given in Corollry 3.1 cn be seen s specil cse of the lst formul. Corollry 3.1 cn be obtined from (1) by repetedly performing integrtion by prts nd differentition. The requirement of f(t) being integrble is sufficient condition since then the integrl given in (1) exists for t > nd it is possible to differentite it m + 1 times. We shll now show how to obtin the Riemnn-Liouville frctionl integrl nd therefter how to obtin the Riemnn-Liouville frctionl derivtive. Bchelor Project Frctionl Clculus 8

9 3 FRACTIONAL DERIVATIVES AND INTEGRALS The Riemnn-Liouville Frctionl Integrl The Riemnn-Liouville differintegrl is obtined by combining integer-order derivtives nd integrls. First we will generlize the definition of n integrl to get the Cuchy formul. If f(τ) is integrble in every finite intervl (, t) the integrl f ( 1) (t) = exists. Next we look t the two-fold integrl: f ( 2) (t) = f(τ) dτ τ1 dτ 1 f(τ) dτ = = f(τ) dτ τ dτ 1 (t τ)f(τ) dτ. If the lst expression is integrted we obtin the three-fold integrl of f(t) f ( 3) (t) = = = 1 2 τ1 τ2 dτ 1 dτ 2 f(τ) dτ τ1 dτ 1 (τ 1 τ)f(τ) dτ (t τ) 2 f(τ) dτ. Then, using induction, the Cuchy formul is derived f ( n) (t) = 1 Γ(n) (t τ) n 1 f(τ) dτ. (11) If we replce the integer n in the Cuchy formul (11) by the rel number p we obtin n integrl of rbitrry order. Definition 3.2. The Riemnn-Liouville frctionl integrl of order p R > is given by D p t f(t) = 1 (t τ) p 1 f(τ) dτ. Γ(p) The Riemnn-Liouville Frctionl Derivtive Now we will show how to obtin the Riemnn-Liouville frctionl derivtive. If we fix n 1 in formul (11) nd tke n integer k then it is possible to rewrite this expression s f (k n) (t) = 1 Γ(n) Dk (t τ) n 1 f(τ) dτ, (12) where D k represents k iterted integrtions if k nd k differentitions if k >. Formul (12) gives iterted integrls of f(t) when k = n 1, n 2,..., the function f(t) if k = n nd it gives the derivtives of order k n = 1, 2, 3,... of the function f(t) when k = n + 1, n + 2, n + 3,... Bchelor Project Frctionl Clculus 9

10 3 FRACTIONAL DERIVATIVES AND INTEGRALS If we replce the integer n in formul (12) by α R with k α > we obtin n expression for differentition of non-integer order Dt k α f(t) = 1 Γ(α) d k dt k (t τ) α 1 f(τ) dτ, (13) where < α 1. If we set p = k α we cn rewrite the lst expression nd obtin derivtive of rbitrry order. Definition 3.3. The Riemnn-Liouville frctionl derivtive of order p R > is given by d k D p 1 t f(t) = Γ(k p) dt k (t τ) k p 1 f(τ) dτ ( ) = dk dt k D (k p) t f(t), (k 1 p < k). In the lst equlity of Definition 3.3 we used the definition of the Riemnn- Liouville frctionl integrl given in Definition 3.2. If α = 1 we hve p = k 1 nd we del with the derivtive of integer order with order k 1 ( Dt k 1 f(t) = dk dt k ( = dk dt k ) D (k (k 1)) t f(t) ) Dt 1 f(t) = f (k 1) (t). Obviously, if we set p = k 1 nd t > nd use the zero rule given in (14), which will be proved in the next section, we obtin the usul derivtive of integer order k ( ) D p t f(t) = dk dt k Dt f(t) = dk f(t) dt k = f (k) (t). Bchelor Project Frctionl Clculus 1

11 4 BASIC PROPERTIES OF FRACTIONAL DERIVATIVES 4 Bsic Properties of Frctionl Derivtives In this section we will discover if some bsic properties, such s linerity, Leibniz s rule nd composition, still pply to differintegrls. 4.1 Linerity Linerity follows from just filling in the definitions of the frctionl derivtives nd integrls. If we use the expression of the Grünwld-Letnikov frctionl derivtive (9) we hve D p t ( ) λf(t) + µg(t) = lim h mh=t = λ lim h mh=t h p + µ lim h mh=t m r= m h p ( )( ) p ( 1) r λf(t rh) + µg(t rh) r r= h p ( ) p ( 1) r f(t rh) r m ( ) p ( 1) r g(t rh) r r= = λ D p t f(t) + µ D p t g(t). In this proof f(t) nd g(t) re functions for which the given opertor is defined nd λ, µ R re rel constnts. A similr proof cn be given for the frctionl integrl. A proof for the linerity of the Riemnn-Liouville differintegrl will lso be given. Using the frctionl integrl given in Definition 3.2 we hve ( ) D p t λf(t) + µg(t) = 1 ) (t τ) (λf(τ) p 1 + µg(τ) dτ Γ(p) = λ 1 Γ(p) + µ 1 Γ(p) = λ D p t (t τ) p 1 f(τ) dτ (t τ) p 1 g(τ) dτ f(t) + µ D p g(t). Agin, similr proof cn be given for the Riemn-Liouville derivtive. For exmple using the linerity of Riemnn-Liouville integrl which we hve just proved nd Definition Zero Rule It cn be proved tht if f(t) is continuous for t then we hve lim p D p t f(t) = f(t). The proof cn be found in [2, p ]. Hence, we define t D t f(t) = f(t). (14) Bchelor Project Frctionl Clculus 11

12 4 BASIC PROPERTIES OF FRACTIONAL DERIVATIVES 4.3 Product Rule & Leibniz s Rule If f nd g re functions we know the derivtive of their product is given by the product rule (f g) = f g + f g. This cn be generlized to (fg) (n) = n ( ) n f (k) g (n k), k which is lso known s the Leibniz rule. In the lst expression f nd g re n-times differentible functions. If f(τ) nd g(τ) nd their derivtives re continuous in [, t] it cn be proved tht the Leibniz rule for frctionl derivtives is given by the following expression D p t ( ) f(t)g(t) = m ( ) p f (k) (t) D p k t g(t), (15) k where gin the binomil coefficient is given by (8) nd m N stisfies (m p < m+1). The proof is firly long so it won t be given here, but cn be found in [2, p ]. If we know the frctionl derivtive of some function, sy g(t) nd we wnt to determine the frctionl derivtive of function which is product of g(t) nd nother function, sy f(t), the Leibniz s rule is very helpful. 4.4 Composition Frctionl integrtion of frctionl integrl The Riemnn-Liouville frctionl integrl given in Definition 3.2 hs the following importnt property ( ) ( ) D q t f(t) = D q t D p t f(t) = D p q t f(t), (16) D p t which is clled the composition rule for the Riemnn-Liouville frctionl integrls. Using the definition the proof is quite strightforwrd ( ) D p t D q t f(t) = 1 ( ) (t τ) p 1 Dτ q f(τ) dτ Γ(p) = 1 ( 1 τ ) (t τ) p 1 (τ ξ) q 1 f(ξ) dξ dτ Γ(p) Γ(q) = 1 Γ(p)Γ(q) τ (t τ) p 1 (τ ξ) q 1 f(ξ) dξ dτ. Chnging the order of integrtion using formul (6) gives ( D p t ) D q t f(t) = 1 Γ(p)Γ(q) f(ξ) ξ (t τ) p 1 (τ ξ) q 1 dτ dξ. We mke the substitution τ ξ t ξ = ζ from which it follows tht dτ = (t ξ)dζ nd the new intervl of integrtion is [, 1]. Now we re ble to rewrite the lst Bchelor Project Frctionl Clculus 12

13 4 BASIC PROPERTIES OF FRACTIONAL DERIVATIVES expression s ( D p t D q t f(t) ) = = 1 Γ(p)Γ(q) 1 Γ(p)Γ(q) 1 ) f(ξ) ((t ξ) p+q 1 (1 ζ) p 1 ζ q 1 dζ dξ ( ) f(ξ) (t ξ) p+q 1 B(p, q) dξ, where in the lst formul we used the Bet function given in Definition 2.2. If we now use identity (2) to express the Bet function in terms of the Gmm function we obtin ( ) D p t D q t f(t) 1 Γ(p)Γ(q) = Γ(p)Γ(q) Γ(p + q) 1 = (t ξ) p+q 1 f(ξ) dξ Γ(p + q) = D p q t f(t). f(ξ)(t ξ) p+q 1 dξ Frctionl differentition of frctionl integrl An importnt property of the Riemnn-Liouville frctionl derivtive is ( ) D q t f(t) = D p q t f(t), (17) D p t where f(t) hs to be continuous nd if p q, the derivtive D p q t f(t) exists. This property is clled the composition rule for the Riemnn-Liouville frctionl derivtives. We shll prove this property, but first we need nother property which ctully is specil cse of the previous one with q = p ( ) D p t f(t) = f(t), (18) D p t where p > nd t >. This implies tht the Riemnn-Liouville frctionl differentition opertor is the left inverse of the Riemnn-Liouville frctionl integrtion of the sme order p. We prove this in the following wy. First we consider the cse p = n N 1, then we hve ( ) Dt n Dt n f(t) = dn 1 t dt n (t τ) n 1 f(τ) dτ Γ(n) = d dt f(τ) dτ = f(t). For the non-integer cse we tke k 1 p < k nd use (16) to write ( ) Dt k f(t) = D (k p) t D p t f(t). Now using the definition of the Riemnn-Liouville differintegrl we obtin ( ) [ ( )] D p t D p t f(t) = dk dt k D (k p) t D p t f(t) = dk dt k [ Dt k f(t) ] = f(t). Bchelor Project Frctionl Clculus 13

14 4 BASIC PROPERTIES OF FRACTIONAL DERIVATIVES This completes the( proof. One ) note hs to be mde. The converse of (18) is not true, so D p t D p t f(t) f(t). The proof for this cn be found in [2, p. 7-71]. We won t give it here since it does not contribute to the proof of (17). So now we re ble to prove (17). We consider two cses. First we ll del with q p. Then we hve ( ) [ ( )] D p t D q t f(t)) = D p t D p t D (q p) t f(t) = D p q t f(t). This follows directly from (16) nd (18). Now we will consider the second cse in which we hve p > q. Using Definition 3.3 nd gin (16) we see tht D p t ( ) [ D q t f(t) = dk dt k ( = dk dt k ( D (k p) t = D p q t f(t). D p q k t f(t) So in both cses we proved eqution (17). )] D q t f(t) ) ( = dk dt k ) D (k (p q)) t f(t) Frctionl integrtion nd differentition of frctionl derivtive Their re two more possibilities when we re deling with composition of differintegrls, i.e. the frctionl integrtion of derivtive nd the frctionl differentition of frctionl derivtive. Both compositions re not useful contributions to this thesis so we shll not give their definitions nd proofs here. Bchelor Project Frctionl Clculus 14

15 5 EXAMPLES 5 Exmples This section dels with some exmples of frctionl derivtives nd integrls. First we will tke look t the Power function nd therefter explore the Exponentil function nd Trigonometric functions. 5.1 The Power Function The Power function is very importnt in Mthemtics since mny functions cn be derived from n infinite power series. First we will use the Riemnn- Liouville frctionl integrl given in Definition 3.2 to compute the integrl of order p R > of the power function (t ) β. Plugging this into the eqution gives D p t (t ) β = 1 (t τ) p 1 (τ ) β dτ. Γ(p) If we mke the substitution τ t = ξ from which it follows tht dτ = (t )dξ nd the new intervl of integrtion is [, 1], we cn rewrite the lst expression s D p t (t ) β (t )β+p = Γ(p) 1 (1 ξ) p 1 ξ β dξ (t )β+p = B(p, β + 1) Γ(p) Γ(β + 1) = Γ(β + p + 1) (t )β+p, (19) where in the lst eqution we mde use of (2) to write the Bet function in terms of the Gmm function. It follows tht β > 1. Next we will compute the derivtive of order r R > of the sme power function (t ) β using the Riemnn-Liouville frctionl derivtive given in Definition 3.3. Agin filling in f(t) = (t ) β gives ( Dt r (t ) β = dk dt k D (k r) t (t ) ). β Now we re ble to use the integrl of the power function we hve just computed in (19). If we replce the order p by k r > we cn rewrite the lst expression s Dt r (t ) β Γ(β + 1) = Γ(β + k r + 1) = d k Γ(β + 1) Γ(β r + 1) (t )β r, (t )β+k r dtk (2) with β > 1. The following two exmples cn clrify this using concrete numbers. First we would like to derive the hlf-derivtive of the function f(x) = x, so in the lst Bchelor Project Frctionl Clculus 15

16 5 EXAMPLES expression we set t = x, =, β = 1 nd r = 1 2. Then we obtin D 1 2 t (x ) 1 Γ(1 + 1) = Γ( )(x )1 1 2 D 1 2 t x = Γ(2) x Γ( 3 2 = 2 π. 2 )x1 In our next exmple we would like to know the derivtive of order 3 4 of the function f(x) = x 2, so gin in formul (2) we set t = x, =, but now β = 2 nd r = 3 4. This gives us D 3 4 t (x ) 2 = Γ(2 + 1) Γ( )(x )2 3 4 D 3 4 t x 2 = Γ(3) Γ(2 1 4 )x x The Exponentil Function Another frequently used function in Mthemtics is the exponentil function. We shll use the Weyl frctionl integrl, which is formlly equl to the Riemnn- Liouville frctionl integrl given in Definition 3.2, to compute the integrl of order p R > of the function f(t) = e λt, where λ C. This Weyl differintegrl, which cn be found in [3, p. 8], pplies to periodic functions where the integrl is equl to zero over period. If we use the Weyl differintegrl we do not hve to mke the restriction of setting Re(λ) >. So using the Weyl frctionl integrl nd setting equl to gives us This expression cn be rewritten s D p t e λt = 1 (t τ) p 1 e λτ dτ. Γ(p) D p t e λt = λ 1 p 1 t ( ) p 1e λ(t τ) λτ dτ. Γ(p) If we mke the substitution ξ = λ(t τ) it follows tht ξ goes from nd λdτ = dξ so dτ = λ 1 dξ. Now we cn rewrite the lst expression s D p t e λt = λ 1 p 1 Γ(p) = λ 1 p 1 Γ(p) p eλt = λ Γ(p) ξ p 1 e λt ξ λ 1 dξ ξ p 1 e λt ξ λ 1 dξ ξ p 1 e ξ dξ. Now using the Gmm function given in Definition 2.1 we get D p t e λt p eλt = λ Γ(p) Γ(p) = λ p e λt. Bchelor Project Frctionl Clculus 16

17 5 EXAMPLES The frctionl derivtive of order p R > cn be obtined in the sme wy but now using Definition 3.3 nd is given by D p t e λt = λ p e λt. So ctully we hve for ll p R. D p t e λt = λ p e λt (21) 5.3 The Trigonometric Functions In this exmple we would like to explore the differintegrl of the sine nd cosine functions. We re ble to use the lst exmple since we cn write the trigonometric functions in terms of the exponentil function in the following wy sin(t) = eit e it cos(t) = eit + e it. 2i 2 First we will explore the Weyl differintegrl of order p R of the sine function ( D p t sin(t) = D p e it e it ) t. 2i If we now use the linerity of the Weyl differintegrl, which follows directly from the linerity of the Riemn-Liouville differintegrl given in Section 4.1 since they re formlly equl, the lst expression cn be rewritten s D p t sin(t) = 1 ( ) D pt e it D pt e it. 2i If we now use the expression for the differintegrl of the exponentil function (21) given in the lst exmple we obtin D p t sin(t) = 1 ) (i p e it ( i) p e it = 1 ) (e i π2 p e it e i π2 p e it 2i 2i = 1 ) (e i(t+ π2 p) e i(t+ π2 p) = sin(t + π 2i 2 p). The differintegrl for the cosine function cn be obtined in the sme wy nd is given by D p t cos(t) = cos(t + π 2 p). Bchelor Project Frctionl Clculus 17

18 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS 6 Frctionl Liner Differentil Equtions Frctionl differentil equtions re generliztion of differentil equtions. They cn be solved by severl methods of which the Lplce trnsform is one. We shll explore this method, but first give some bsic properties of the Lplce trnsform, which re necessry to understnd the rest of this chpter. 6.1 The Lplce Trnsforms of Frctionl Derivtives First the definition of the Lplce trnsform itself is given. Definition 6.1. We define the Lplce trnsform of function f(t) for t R nd s C s the function F (s) such tht F (s) = L{f(t); s} = For this integrl to exist we must hve e st f(t) dt. e αt f(t) M for ll t > T, where M nd T re positive constnts. The originl function f(t) cn be recovered from the Lplce trnsform. Definition 6.2. The inverse Lplce trnsform f(t) where t R >, s C nd F (s) is the Lplce trnsform is given by f(t) = L 1 {F (s); t} = c+ c e st F (s) ds. In Definition 6.2 c = Re(s) > c nd c lies in the right hlf plne of the bsolute convergence of the Lplce integrl given in Definition 6.1. An importnt property of the Lplce trnsform is tht it is liner opertor, i.e. L{f(t) + g(t); s} = L{f(t); s} + L{g(t); s}, L{cf(t); s} = cl{f(t); s}, (22) where L{f(t); s} nd L{g(t); s} hve to exist nd c is constnt. For nother useful property of the Lplce trnsform we first hve to define the convolution of two functions. Definition 6.3. The convolution of two functions f(t) nd g(t) is defined s (f g)(t) = f(t τ)g(τ) dτ = f(τ)g(t τ) dτ. If f(t) nd g(t) re equl to zero for t < nd F (s) nd G(s) exist, the Lplce trnsform of this convolution is equl to the product of the Lplce trnsform of those functions. This property is given in the following theorem. Bchelor Project Frctionl Clculus 18

19 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Theorem 6.4. The Lplce trnsform of the convolution of two functions f(t) nd g(t) is given by L{f(t) g(t); s} = F (s)g(s). If we integrte the Lplce integrl (Definition 6.1) by prts we obtin nother necessry property. Corollry 6.1. The Lplce trnsform of the derivtive of integer order n is given by n 1 n 1 L{f n (t); s} = s n F (s) s n k 1 f (k) () = s n F (s) s k f (n k 1) () Lplce Trnsform of the Riemnn-Liouville Differintegrl First we shll explore the Lplce trnsform of the Riemnn-Liouville frctionl integrl. Using Definition 3.2 nd setting the lower terminl equl to zero we get D p t f(t) = 1 (t τ) p 1 f(τ) dτ. Γ(p) If we use the definition for convolution (Definition 6.3) nd define the function g(t) = t p 1, the lst expression cn be rewritten s D p t f(t) = 1 Γ(p) tp 1 f(t) = 1 1 g(t) f(t) = (g f)(t). (23) Γ(p) Γ(p) If we now tke look t the Lplce trnsform of g(t) nd therefore use the definition of the Lplce trnsform given in Definition 6.1 we hve G(s) = L{g(t); s} = L{t p 1 ; s} = t p 1 e st dt. If we mke the substitution st = r it follows tht dt = 1 s dr nd we cn rewrite the lst expression s G(s) = 1 s p r p 1 e r dr = s p r p 1 e r dr = Γ(p)s p, (24) where in the lst equlity we used the definition of the Gmm function given in Definition 2.1. Now it s possible to define the Lplce trnsform of the Riemnn-Liouville frctionl integrl. First using (23) we get { } L{ D p 1 t f(t); s} = L (g f)(t); s. Γ(p) Using the Lplce trnsform of convolution given in Theorem 6.4 nd the linerity of of the Lplce trnsform (22), the lst expression cn be rewritten s L{ D p t f(t); s} = 1 G(s)F (s). Γ(p) If we now use (24) we obtin for the Lplce trnsform of the Riemnn-Liouville integrl of order p > L{ D p t f(t); s} = 1 Γ(p) Γ(p)s p F (s) = s p F (s). (25) Bchelor Project Frctionl Clculus 19

20 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Next we shll explore the Lplce trnsform of the Riemnn-Liouville frctionl derivtive. As suggested in [2] we shll write this frctionl derivtive in the following form D p t f(t) = g (n) (t), from which it follows tht g(t) = D (n p) 1 t f(t) = Γ(n p) (t τ) n p 1 f(τ) dτ, (26) for n 1 p < n. If we use the Lplce trnsform of n integer-order derivtive given in Corollry 6.1 we cn write n 1 L{ D p t f(t); s} = L{g (n)(t) ; s} = s n G(s) s k g (n k 1) (). (27) To rewrite this lst expression we will evlute G(s) nd g (n k 1) (t). First we mke use of the Lplce trnsform of the Riemnn-Liouville frctionl integrl given in (25) to write G(s) = L{g(t); s} = L{ D (n p) t f(t); s} = s (n p) F (s). (28) Now we will explore g (n k 1) (t) by tking the (n k 1) th -derivtive of g(t) given in (26). Also using the Riemnn-Liouville frctionl derivtive formul given in Definition 3.3 enbles us to write g (n k 1) (t) = dn k 1 dt n k 1 D (n p) t f(t) = D p k 1 t f(t). (29) So substituting the lst two equtions in (27) gives the expression for the Lplce trnsform of the Riemnn-Liouville frctionl derivtive of order p > for n 1 p < n. n 1 [ L{ D p t f(t); s} = s n s (n p) F (s) s k n 1 [ = s p F (s) s k D p k 1 t f(t) So using the lst expression for the cse n = 1 we obtin ] D p k 1 t f(t) t= ], t= (3) L{ D p t f(t); s} = s p F (s) D p 1 t f(), (31) where p < 1. For n = 2 we hve 1 p < 2 nd it follows from (3) tht L{ D p t f(t); s} = s p F (s) D p 1 t f() s D p 2 t f(). (32) We shll see tht these specil cses re helpful in solving some simple frctionl differentil equtions which will be treted in the exmples t the end of this chpter. Bchelor Project Frctionl Clculus 2

21 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Lplce Trnsform of the Grünwld-Letnikov Frctionl Derivtive In this prt we will explore the Lplce trnsform of the Grünwld-Letnikov frctionl derivtive. Actully we ve lredy done most of the work nd it s bsiclly using definitions. Agin, s in the Riemnn-Liouville cse, we set the lower terminl equl to zero. First we shll consider the cse p < 1. Using the definition of the Grünwld-Letnikov frctionl derivtive given in Corollry 3.1 we hve D p t f(t) = f()t p Γ(1 p) + 1 Γ(1 p) (t τ) p f (τ) dτ, where f(t) is bounded ner t =. Using the Lplce trnsform of the function given in (24), the lplce trnsform for convolutions given in Theorem 6.4 nd the Lplce trnsform of the integer-order derivtive given in Corollry 6.1 we obtin L{ D p t f(t); s} = f() s 1 p + 1 ( ) s 1 p sf (s) f() = s p F (s). (33) In the cse of p > 1 the functions in the sum of Corollry 3.1 cn not be integrted in the clssicl sense. However, it cn be proved tht under the ssumption tht m p < m + 1 the Lplce trnsform of the Grünwld- Letnikov frctionl derivtive given in (33) still holds in the sense of generlized functions. 6.2 The Lplce Trnsform Method Before we continue we lso need the Lplce trnsform of very importnt function for liner frctionl differentil equtions consisting of two terms. We need to explore the Lplce trnsform of the following function { } L t αm+β 1 E (m) α,β (tα ); s. (34) If we look more closely we cn see this function is combintion of the power function nd the differentited Mittg-Leffler function given in Corollry 2.1. Evluting this Mittg-Leffler function in t α yields E (m) α,β (tα ) = (k + m)! k! (t α ) k Γ(αk + αm + β) = Substuting this expression in (34) gives { } { L t αm+β 1 E (m) α,β (tα ); s = L t αm+β 1 (k + m)! k! (k + m)! k! k t αk Γ(αk + αm + β). k t αk } Γ(αk + αm + β) ; s. Using the linerity of the Lplce trnsform (22) we cn rewrite the lst expression s { } L t αm+β 1 E (m) α,β (tα ); s = (k + m)! k k! Γ(αk + αm + β) L{tαk+αm+β 1 ; s} (35) Bchelor Project Frctionl Clculus 21

22 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Now we wnt to inspect L{t αk+αm+β 1 ; s} from the lst eqution. We ve lredy determined the Lplce trnsform of the power function in (24) which gve us the following equlity So in this cse we hve L{t p 1 ; s} = Γ(p)s p. L{t αk+αm+β 1 ; s} = Γ(αk + αm + β)s (αk+αm+β) = Substituting this in (35) gives us { } L t αm+β 1 E (m) α,β (tα ); s = = Γ(αk + αm + β) s αk+αm+β. (k + m)! k Γ(αk + αm + β) k! Γ(αk + αm + β) s αk+αm+β (k + m)! k k! s αk+αm+β = = s αm β (k + m)! k! (k + m)! k! ) k. ( s α To further rewrite the lst expression we look t the series ( ) k (k + m)! k! s α = = k s αk+αm+β ( (k + m)(k + m 1) (k + 1) k=m = dm dt m ( k(k 1) (k m + 1) k=m ( ) k s α. s α s α ) k m ) k (36) Since the first m terms dispper fter differentition we cn rewrite the lst expression s ( ) k (k + m)! k! s α = dm dt m ( ) k s α = dm So substituting this in (36) we finlly obtin { } L t αm+β 1 E (m) α,β (tα ); s dt m 1 1 s α = = s αm β m! (1 s α ) m+1 = m! (1 s α ) m+1. m! sα β (s α. (37) ) m+1 The following tble shows some specil cses of expression (37) nd lso the Lplce trnsform of the Power function given in (24). Bchelor Project Frctionl Clculus 22

23 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Tble 1: Useful Lplce trnsforms F (s) f(t) = L 1 {F (s)} 1 s α 1 t α 1 Γ(α) s α t α 1 E α,α (t α ) s α s(s α +) E α ( t α ) s(s α +) 1 E α ( t α ) 1 s α (s ) t α E 1,α+1 (t) s α β s α t β 1 E α,β (t α ) In Tble 1 L 1 is the inverse Lplce trnsform given in Definition Exmples In this section we shll explore some exmples of simple liner frctionl differentil equtions. Exmple 1 given by Let s sy we would like to solve the frctionl differentil eqution D 1 3 t f(t) = c 1 f(t), (38) where c 1 is constnt. Since p = 1 3 < 1 we will use the Lplce trnsform of the Riemnn-Liouville frctionl derivtive for n = 1 given in (31) to tke the Lplce tnsform t both sides of the lst eqution. If we lso use the linerity of the Lplce trnsform (22) this gives L{ D 1 3 t f(t)} = L{c 1 f(t)} = c 1 L{f(t)} s 1 3 F (s) D t f() = c 1 F (s) s 1 3 F (s) D 2 3 t f() = c 1 F (s). We see tht D 2 3 t f() is the vlue of D 2 3 t f(t) evluted t t =. If we ssume this vlue exists we cn set D 2 3 t f() equl to c 2 to obtin If we solve this for F(s) we get s 1 3 F (s) c2 = c 1 F (s). F (s) = c 2 s 1 3 c 1. If we look t Tble 1 we see this is specil cse of (37) with α = 1 3, β = 1 3 nd = c 1, so the solution is given by { } f(t) = L 1 c2 = c s 1 2 t E c, 1 (c 1t ) = c2 t 2 3 E 1 3, 1 (c 1t ). 1 In this exmple we ssumed D 2 3 t f() exists nd it s vlue is equl to c 2. To Bchelor Project Frctionl Clculus 23

24 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS prove this ssumption ws correct we will first tke the Lplce trnsform of D 2 3 t f(t) using the Lplce trnsform of the Riemnn-Liouville integrl given in (25). This gives Since we just found F (s) = get L{ D 2 3 t f(t)} = s 2 3 F (s). c2 s 1 3 c 1 we cn substitute this in the lst eqution to L{ D 2 3 t f(t)} = c 2s 2 3. s 1 3 c 1 Tking the inverse Lplce trnsform of both sides yields { D 2 3 t f(t) = L 1 c2 s 2 } 3. s 1 3 c 1 If we tke look t Tble 1 gin we see tht this is the vlue of F (s) with α = 1 3, β = 1 nd = c 1, so this implies the lst eqution is equl to D 2 3 t f(t) = c 2 t 1 1 E 1 3,1 (c 1 t 1 3 ) = c2 E 1 3,1 (c 1 t 1 3 ). If we evlute this expression t t = we hve s desired. D 2 3 t f() = c 2 E 1 3,1 (c ) = c2 E 1 3,1 () = c 2, Exmple 2 given by Now we would like to solve the frctionl differentil eqution D t f(t) =. Since 1 p = < 2 we will use the Lplce trnsform of the Riemnn-Liouville frctionl derivtive for n = 2 given in (32) to tke the Lplce trnsform of both sides. This gives s 19 L{ D t } = 12 F (s) D t f() s D t f() = s F (s) D 7 12 t f() s D 5 12 t f() =. Just s in exmple 1 we ssume D 7 12 t f() nd D 5 12 t f() exist nd we set them equl to c 3 nd c 4 respectively. Then the lst eqution becomes If we solve this for F (s) we obtin s F (s) c3 c 4 s =. F (s) = c 3 s c 4s. s Bchelor Project Frctionl Clculus 24

25 6 FRACTIONAL LINEAR DIFFERENTIAL EQUATIONS Agin using Tble 1 we get the solution { } f(t) = L 1 c3 s { = L 1 c3 s { } + L 1 c4 s s } } + L 1 { c4 s 7 12 = c 3t 7 12 ( ) + c 4t 5 12 ( ). Γ Γ Exmple 3 In this exmple we will generlize the problem given in Exmple 1 nd we re given n initil vlue. So we would like to solve 7 12 D p t f(t) = c 1 f(t), f() = c 2 (39) with p < 1 nd c 1 constnt. Agin we will use the Lplce trnsform of the Riemnn-Liouville frctionl derivtive for n = 1 given in (31) to tke the Lplce tnsform t both sides of the lst eqution. If we lso use the linerity of the Lplce trnsform (22) this gives L{ D p t f(t)} = c 1 L{f(t)} s p F (s) D p 1 t f() = c 1 F (s). Agin ssuming tht D p 1 t f() exists nd setting it equl to c 3 gives s p F (s) c 3 = c 1 F (s). If we solve the lst expression for F (s) we get F (s) = c 3 s p c 1. Mking use of Tble 1 we find the solution { } f(t) = L 1 c3 s p = c 3 t p 1 E p,p (c 1 t p ). c 1 To find the vlue of c 3 we will use the initil vlue f() = c 2. Since we hve it follows tht lim t p 1 E p,p (c 1 t p ) = 1, t + f() = c 3 1 = c 3. So the intitil vlue f() = c 2 is equl to c 3 nd we cn rewrite the solution of the frctionl liner differentil eqution s f(t) = c 2 t p 1 E p,p (c 1 t p ). Bchelor Project Frctionl Clculus 25

26 7 APPLICATIONS 7 Applictions Frctionl Clculus is used in mny problems, for exmple in engineering, physics, economics, biologicl processes, etc. Mny models cn be represented by frctionl differentil equtions nd therefore it is incresingly used in these brnches. It brings new possibilities, nmely frctionl derivtives cn describe memory effects, so it is possible to evlute the influence of the pst on the behvior of the system t present time. One of the first to use Frctionl Clculus for problem ws the Norwegin mthemticin Niels Henrik Abel. In 1823 he pplied it in the formultion of his solution for the Tutochrone Problem. The ide of this problem is to find the curve of frictionless wire which lies in the (x, y)-plne such tht the time required for prticle to slide down to the lowest point of the curve is independent of where the prticle is plced. Since then Frctionl Clculus hs been pplied to mny other problems such s the frctionl conservtion of mss, the groundwter flow problem, the frctionl dvection dispersion eqution, time-spce frctionl diffusion eqution models, structurl dmping models, cousticl wve equtions for complex medi, the frctionl Schrödinger eqution in quntum theory nd mny more. Although it would be nice to discuss some of these problems, their solutions go beyond the difficulty level of this thesis. Therefore we only mentioned some models nd problems nd leve it to the reder to further explore these pplictions of Frctionl Clculus if desired. We will tret one simple economic exmple to show how frctionl clculus cn be implemented in commonly used model. 7.1 Economic exmple Let s sy customer buys product for price eb. The customer does not py instntly for the product, but chooses to py off in y months. The interest rte of the seller is r% per month. The monthly pyment the customer is chrged is denoted by em. If we define f(τ) to be the remining debt t the end of the τ th month, it cn be shown tht we hve f(τ) = b(1 + r) τ m [ (1 + r) τ 1 ]. (4) r At τ = y the customer should hve pyed off his product so then we must hve f(y) =. Now we re ble to solve (4) for m which gives m = b(1 + r)y r (1 + r) y 1. (41) Usully this problem cn be solved using the following differentil eqution f (τ) rf(τ) = m. (42) If we wnt to pproximte this with frctionl differentil eqution we rewrite the lst formul nd consider D p t f(τ) rf(τ) = m, with < p 1. (43) Bchelor Project Frctionl Clculus 26

27 7 APPLICATIONS As we hve shown in section we cn solve this frctionl differentil eqution by tking the Lplce trnsform on both sides. So using the Lplce trnsform of the Riemnn-Liouville frctionl derivtive for n = 1 (31) nd the linerity of the Lplce trnsform (22) we obtin L{ D p t f(τ)} L{rf(τ)} = L{m} s p F (s) D p 1 t f() rf (s) = m s. As before, in section 6.2.1, we ssume D p 1 t f() exists nd cll it c. Now we re ble to solve for F (s) nd obtin F (s) = c s p r m s(s p r). Using Tble 1 we tke the inverse Lplce trnsform on both sides nd get We hve lredy seen tht nd we lso hve f p (τ) = cτ p 1 E p,p (rτ p ) mτ p E p,p+1 (rτ p ). (44) lim τ p 1 E p,p (rτ p ) = 1, τ + lim τ p E p,p+1 (rτ p ) =. τ + Therefore, if we evlute expression (44) in τ = we get f p () = c. Since f p (τ) denotes the remining debt t the end of month τ, the lst expression cn be seen s the debt t the beginning, which is equl to the price of the product eb. So we hve b = f p () = c nd we cn rewrite (44) s f p (τ) = bτ p 1 E p,p (rτ p ) mτ p E p,p+1 (rτ p ). (45) To clrify this we shll give n exmple using conrete numbers Concrete exmple Suppose we hve customer who wnts to buy cr. This cr costs e2,. He hs to py it bck in 5 yers nd the interest rte of the cr slesmn is 14% per yer. This mens we hve the following vlues b = 2, ; r = ; y = 6; m = b(1 + r)y r (1 + r) y 1 = This llows us to rewrite expression (45) s ( ) ( ) f p (τ) = 2, τ p 1 E p,p 12 τ p τ p E p,p+1 12 τ p. (46) Bchelor Project Frctionl Clculus 27

28 7 APPLICATIONS Note tht for p = 1 we hve the integer order first derivtive which implies we re deling with norml differentil eqution. Evluting the lst expression in p = 1 gives ( f 1 (τ) = 2, ) τ [( ) τ 1] Indeed if we set τ = y = 6 we obtin f 1 (6), so the remining debt fter 6 months (5 yers) is pproximtely zero. Vlues of p 1 ner 1 re bit hrder to compute, but it turns out in these cses it tkes less time to py off the cr which mens y is smller. This is due to the intervls between pyments becoming shorter nd therefore the interest rte will be lower. Bchelor Project Frctionl Clculus 28

29 8 CONCLUSIONS 8 Conclusions This thesis introduced the concept of Frctionl Clculus; the brnch of Mthemtics which explores frctionl integrls nd derivtives. We first gve some bsic techniques nd functions, such s the Gmm function, the Bet function nd the Mittg-Leffler function, which were necessry to understnd the rest of this pper. Therefter we proved the construction of the Grünwld-Letnikov nd the Riemnn- Liouville method to define differintegrl. Therefore we used the forwrd difference derivtive nd the Cuchy formul for repeted integrtion respectively. Altough these differintegrls do not look the sme, we sw tht the Grünwld- Letnikov differintegrl ws specil cse of the Riemnn-Liouville differintegrl nd therefore give the sme result under some specil conditions. Then we checked if some bsic rules of differentition nd integrtion still hold for these differintegrls. We proved they re both liner nd gve n expression for the Leibniz rule for frctionl derivtives. We lso explored the composition of frctionl integrls nd frctionl derivtives. After giving the frmework of differintegrls we were ble to mke use of it. We explored exmples of some frequently used functions, nmely the Power function, the Exponentil function nd the Trigonometric functions. Next we studied Frctionl Liner Differentil Equtions. First we hd to give some bsics bout the Lplce trnsform, since we were bout to use this method for solving these differentil equtions. Then we pplied the Lplce trnsform to the Riemnn-Liouville nd Grünwld-Letnikov differintegrl. After evluting the Lplce trnsform of very useful function, nmely combintion of the Power function nd the Mittg-Leffler function, we were ble to explore some simple exmples. At lst we briefly discussed some pplictions of Frctionl Clculus nd exmined commonly used economic model using frctionl differentil equtions. This thesis did not cover everything relted to Frctionl Clculus. There hve been mny more pproches to define differintegrl. For exmple the Cputo, Hdmrd nd Miller-Ross differintegrls re lso frequently used. However the Grünwld-Letnikov nd the Riemnn-Liouville differintegrl re the most common so we decided to leve it there since the other differintegrls would not hve been very useful ddition to this thesis. In ddition there re mny more methods for solving frctionl liner differentil equtions. Besides the Lplce trnsform we could lso hve used the Fourier trnsform, the method of reduction to Volterr integrl eqution, the power series method or the trnsformtion to n ordinry differentil eqution. Since the purpose ws to give some brief introduction to frctionl differentil equtions nd their solutions we decided to explore only one method. Some people dvocte differintegrls should be implemented in stndrd Mthemtics nd replce the integer order derivtives nd integrls. According to them they provide more possibilities since differintegrls cover derivtives nd Bchelor Project Frctionl Clculus 29

30 8 CONCLUSIONS integrls of rbitrry order, nd therefore lso integer order derivtives nd integrls. Although I gree to some extent with this, I don t think Frctionl Clculus is necessry for ordinry Mthemtics, since these extr possibilities re not relly commonly used dditions. Besides, mny definitions for differintegrl exist so which one should we use in generl? I lso think tht the formuls re pretty wkwrd, definitely for first yer students. It would be lot hrder to compute just simple integer order derivtive or integrl. Though it is very interesting subject nd definitely worth reserching, I believe it should be left s n exotic brnch of Mthemtics. Bchelor Project Frctionl Clculus 3

31 9 REFERENCES 9 References [1] Keith B. Oldhm nd Jerome Spnier. The Frctionl Clculus; Theory nd Applictions of Differentition nd Integrtion to Arbitrry Order. Acdemic Press, Sn Diego, Cliforni, USA, [2] I. Podlubny. Frctionl Differentil Equtions. Acdemic Press, Sn Diego, Cliforni, USA, [3] B. Ross. Frctionl Clculus nd Its Applictions. Springer-Verlg Berlin Heidelberg, Germny, Bchelor Project Frctionl Clculus 31