Grünwald-type approximations and boundary conditions for one-sided fractional derivative operators

Transcription

1 Grünwald-type approximations and boundary conditions for one-sided fractional derivative operators Haris Sankaranarayanan a tesis submitted for te degree of Doctor of Pilosopy at te University of Otago, Dunedin, New Zealand. October 31, 2014

2

3 Abstract Te focus of tis tesis is two-fold. Te first part investigates iger order numerical scemes for one-dimensional fractional-in-space partial differential equations in L 1 R. Te approximations for te space fractional derivative operators are constructed using a sifted Grünwald-Letnikov fractional difference formula. Rigorous error and stability analysis of te Grünwald-type numerical scemes for space-time discretisations of te associated Caucy problem are carried out using Fourier multiplier teory and semigroup teory. Te use of a transference principle facilitates te generalisation of te results from te L 1 -setting to any function space were te translation semi group is strongly continuous. Furtermore, te results extend to te case wen te fractional derivative operator is replaced by te fractional power of a semi group generator on an arbitrary Banac space. Te second part is dedicated to te study of certain fractional-in-space partial differential equations associated wittruncated Riemann-Liouville and first degree Caputo fractional derivative operators on Ω := [0,1]. Te boundary conditions encoded in te domains of te fractional derivative operators dictate te inclusion or exclusion of te end points of Ω. Elaborate tecnical constructions and detailed error analysis are carried out to sow convergence of Grünwald-type approximations to fractional derivative operators on X = C 0 Ω and L 1 [0,1]. Te wellposedness of te associated Caucy problem on X is establised using te approximation teory of semigroups. Te culmination of te tesis is te result wic sows convergence in te Skorood topology of te well understood stocastic processes associated wit Grünwald-type approximations to te processes governed by te corresponding fractional-in-space partial differential equations. iii

4

5 Acknowledgements I extend my deepest gratitude to my Matematics Gurus my supervisors, Dr. Boris Baeumer and Dr. Miály Kovács, for teir excellent tutelage over te last five years. I tank Boris for initiating and training me in te dark, magical art of problem solving, for teacing me to see te bigger picture and for is time, patience and energy. I tank Misi for enligtning me on abstract tecnical concepts, for teacing me to write wit surgical precision and for literally watcing over me as a big broter. I am also grateful to bot of tem for many lunc treats and for part of te financial assistance to attend te ANZIAM conference. I would like to tank my co-supervisor Prof. Jörg Frauendiener for is magnanimity and for is words of wisdom. I would also like to tank te Head of te Department, Prof. Ricard Barker as well as all te academic staff at te department for teir constant encouragement. I am especially grateful to Prof. Jon Clark and Dr. Austina Clark for giving me te privelage of lecturing te Modern Algebra course part-time. Special tanks to te Department Manager, Mrs. Lenette Grant for all te elp, including several tutoring opportunities over te last five years. I would like to acknowledge te PD scolarsip offered by te University of Otago and te financial assistances given by CSIRO, ANZIAM and NZ-ANZIAM to attend te ANZIAM conference. Special tanks to te wonderful support staff, Cris, Greg, Marguerite, and especially, Leanne for being a great listener, and for putting tings into perspective on several occasions. It is also a real pleasure to tank fellow PD student Aidin wo sared te teacing responsibilities wit me for tree consecutive Summer Scool Mat160 courses. I tank my v

6 current work mates, including Cris. S, Danie, Ilija, Leon, Paula, Ricard and Zara from Matematics, Adrian and wife Janet, Levi and Sam from Pysics as well as my past work mates, Cris. L, Fabien and wife Yuri and George and partner Kati, for teir camaraderie. I am extremely grateful to Rick and Nicola Commodore Motel, Dunedin for teir ospitality and generosity, and for te cozy room tat elped me rest and rejuvenate during te crucial, last one and alf years of my PD study. Special tanks to my Kiwi surrogate family, including Mark, Debbie and Bob, Anne and Derek, Jo and Tony, Mary Ann and Budge, and all my Kiwi friends, including Balakrisna, Juliana and Murali, and Daisy, Mundy, Cidleys, and not to forget big Gerry for many reminders of baby steps, 1 plus 1 equals 2. I tank Abdulmannan and family Momtaj, Hassan and Mridula for treating me as teir own. I also tank my dear friends, Dev and Anu, for always being tere for me. I am extremely grateful to my brave parents for believing in me, for encouraging me to return to studies and for supporting me every step of te way, including financially. I am eternally indebted to my younger broter Dr. Balaji and my sister-in-law Sripriya for teir love, affection and financial support, and most importantly, I congratulate tem on teir best result, little bundle of joy, Sanju. vi

7 Dedicated to my loving parents, Vasanta and Sanku. vii

8

9 Contents Introduction 1 1 Grünwald-type approximations of fractional derivative operators on te real line Fractional derivative operators on R Grünwald-type approximations Bound for multiplier norms Carlson-type inequality Carlson-type inequality for periodic multipliers Grünwald periodic multiplier operators Construction of iger order Grünwald-type approximations Consistency of iger order Grünwald-type approximations Semigroups generated by Grünwald-type approximations Semigroups generated by periodic multipliers approximating fractional derivative operators First-order Grünwald-type approximation Examples of second order stable Grünwald-type approximations Application to fractional powers of operators Numerical results Boundary conditions for fractional-in-space partial differential equations Extension of a finite state Markov process to a Feller process on [0,1] Construction of te transition operators One-sided fractional derivative operators wit different combinations of boundary conditions Fractional integral operators and fractional derivatives Fractional derivative operators on a bounded interval encoding various boundary conditions Properties of te one-sided fractional derivative operators on a bounded interval ix

10 4 Grünwald-type approximations for fractional derivative operators on a bounded interval Grünwald-type approximations for fractional derivative operators on C 0 Ω and L 1 [0,1] Adjoint formulation and boundary weigts Semigroups and processes associated wit Grünwald-type approximations and fractional derivative operators on X Examples of Grünwald scemes for Caucy problems on L 1 [0,1] Detailed proof of Proposition Construction of approximate power functions Proof of Proposition for te case X = C 0 Ω Proof of Proposition for te case X = L 1 [0,1] A Properties of Grünwald coefficients 191 B Reminder on function spaces, distributions and transforms 193 B.1 Function spaces B.2 Fourier and Laplace Transforms B.3 Distributions C Reminder on operator teory, semigroups and multipliers 199 C.1 Operator teory C.2 Semigroups C.3 Multipliers D MATLAB codes for Grünwald scemes 207 References 211 x

11 List of Tables 2.1 Maximum error beaviour for second and tird order Grünwald approximations Relations satisfied by te constants a,b,c,d for BC Constants r,s,t for ϕ given by Boundary conditions for C 0 Ω Boundary conditions for L 1 [0,1] Corresponding fractional derivative operators on C 0 Ω and L 1 [0,1] Boundary conditions and boundary weigts for L 1 [0,1] Corresponding fractional derivative operators on C 0 Ω and L 1 [0,1] Boundary conditions for C 0 Ω Corresponding fractional derivative operators on C 0 Ω and L 1 [0,1] Boundary conditions for L 1 [0,1] xi

12

13 List of Figures 2.1 L 1 -error for different initial conditions f i and a first and second order sceme. Note te less tan first order convergence for a bad initial condition; i.e. one tat is not in te domain of A. Also note te less tan second order convergence for a second order sceme but first order convergence for te first order sceme for an initial condition tat is in te domain of A but not in te domain of A Time evolution of te numerical solution to te Caucy problem associated wit Dc 1.5,DD and initial value u 0 given by Time evolution of te numerical solution to te Caucy problem associated wit Dc 1.5,DN and initial value u 0 given by Take note of te build up at te rigt boundary Time evolution of te numerical solution to te Caucy problem associated wit Dc 1.5,ND and initial value u 0 given by Time evolution of te numerical solution to te Caucy problem associated wit Dc 1.5,NN and initial value u 0 given by Time evolution of te numerical solution to te Caucy problem associated wit D 1.5,ND and initial value u 0 given by Time evolution of te numerical solution to te Caucy problem associated wit D 1.5,NN and initial value u 0 given by xiii

14

15 Introduction Te fundamentals of fractional calculus and teir applications ave been treated by several autors, see for example [43, 46, 51, 82, 84, 86, 90, 95] and te references terein. Even toug fractional derivatives ave existed as long as teir integer order counterparts only in recent decades ave fractional derivative models become exciting new tools in te study of practical problems in displines as diverse as pysics [13, 19, 20, 26, 71, 72, 73, 80, 81, 96, 110], finance [69, 89, 93], biology [5, 6] and ydrology [3, 15, 16, 17, 97, 98, 99]. Tis observation, tat fractional derivative models are becoming increasingly popular among te wider scientific community, is te main motivation to study numerical scemes for fractional partial differential equations. From a purely matematical perspective, fractional partial differential equations can be tougt of as generalizations of te corresponding classical partial differential equations. On occasions owever, fractional partial differential equations do arise naturally as better teoretical models for practical problems in diverse scientific disciplines. For instance, in geopysical sciences [99], te autors employ te Lévy-Gnedenko generalised central limit teorem [41] and fractional conservation of mass arguments [98, 107] to derive fractional-in-space as well as fractional-in-time advection-dispersion equations. According to te autors, tese fractional advection-dispersion equations provide better models for te motion of an ensemble of particles on Eart s surface as measured by te concentration or mass in space and time. In general, particle transport penomena may involve random states of motion as well as rest. Terefore, jump lengt and waiting time between motion of a particle can be viewed as random variables [15, 98, 99]. It is well known tat if te probability density function tat describes te jump lengt decays at least as fast as an exponential distribution, ten jump lengt distribution as finite mean and variance. Furter, assuming tat te waiting time distribution as finite mean, te concentration or mass in tis case may be adequately described by te classical advection-dispersion equation Fokker Planck equation. Te associated stocastic process, te so-called 1

16 Brownian motion wit drift, is governed by te classical advection-dispersion equation wose fundamental solution is te Gaussian density [41]. Te normal scaling of dispersion Fickian or Boltzmann scaling, described by te standard deviation, is proportional to t2. 1 Te term anomalous diffusion is given to diffusion penomena tat cannot be adequately described by te classical advection-dispersion equations. One suc scenario is wen jump lengt follows an infinite-variance distribution. In addition to infinitevariance jump lengt distribution, assuming a finite-mean waiting time distribution, te autors in [99] sow tat te associated stocastic process Lévy motion is Markovian and is governed by fractional-in-space advection-dispersion equation wose fundamental solution is a Lévy α-stable density. Te diffusion penomena is referred to as super-diffusion because of te faster tan normal t 1 2 scaling, as te scaling of dispersion in tis case is proportional to t 1 α were 1 < α < 2 is te order of te fractional-in-space derivative used in te model. Fractional-in-space advection-dispersion equations arise as natural models wen te velocity variations are eavy tailed. On te oter and, assuming infinite-mean waiting time distribution, te associated stocastic process Lévy motion subordinated to an inverse Lévy process is sown to be non-markovian and is governed by a fractional-in-time advection-dispersion equation wose fundamental solution is a subordinated Lévy α-stable density. Te diffusion penomena inerits te name sub-diffusion in tis case since te scaling of dispersion is proportional to t γ 2 were 0 < γ < 1 is te order of te fractional-in-time derivative used in te model. A compreensive review of random walk and oter teoretical models for anomalous sub-diffusion and super-diffusion as well as evidence of te occurence of anomalous dynamics in various fields suc as biology, geopysics, pysics and finance can be found in [80, 81]. Te connection of fractional calculus wit probability teory tat we ave briefly outlined above is interesting in its own rigt. Evidently, tis provides an insigt into te stocastic processes governed by fractional partial differential equations. More importantly, tis link also provides new tools from probability teory tat can be used in te searc for numerical solutions for fractional partial differential equations. For instance, Feller investigated te semigroups generated by a certain pseudo differential operator and identified te underlying stocastic processes [40]. As it turns out, tese processes are governed by a certain diffusion equation obtained by replacing te secondorder space derivative by te pseudo differential operator in te classical diffusion equation [44, 45, 94]. Te fundamental solutions of tis diffusion equation generate all 2

17 te Lévy stable densities wit index α 0,2]. In [44, 45], tis diffusion equation is revisited using a random walk model tat employs te Grünwald-Letnikov difference sceme were te autors refer to te equation as Lévy-Feller diffusion equation and te processes governed by tem as Lévy-Feller processes. Teoretical and numerical metods as well as connections wit stocastic processes for various types of fractional partial differential equations ave been investigated by several autors, see for example [3,7,9,28,70,71,72,74,79,97]andtereferencesterein. In[76], teautorsprovide an in-dept treatment on te connection of fractional calculus to stocastic processes from a probabilistic perspective. Inevitably, two crucial issues ave to be addressed in te study of fractional partial differential equations. Firstly, te existence and uniqueness of solutions; tat is, weter te associated Caucy problem is well-posed in te function space framework cosen for study. Secondly, te consistency and stability of numerical scemes used to solve te fractional partial differential equations. Te latter is particularly important since te inclusion of an external forcing function and/or te imposition of boundary conditions, especially in practical applications, could make te task of finding analytical solutions elusive [109]. Te well-posedness of te Caucy problem associated wit certain fractional-inspace partial differential equations on bounded domains in te L 2 -setting as been investigated by many autors. By constructing appropriate function spaces and demonstrating equivalence to fractional Sobolev spaces, variational solutions to te steady state fractional advection-dispersion equations on bounded domains Ω R n were investigated in [38, 39]. Tese autors employ te Lax-Milgram Lemma to sow te existence and uniqueness of solutions in L 2 Ω. Te autors in [34] study a very general class of non-local diffusion problems on bounded domains of R d using non-local vector calculus and non-local, non-linear conservation laws developed in [33, 35]. According to te autors, certain fractional derivative models for anomalous diffusion are special cases of teir non-local diffusion model. In particular, te autors claim tat te fractional Laplacian and te symmetric version wit α = 2s of te more general asymmetric fractional derivative operator of [70] are special cases of teir non-local operator. Moreover, tey sow te well-posedness of steady-state volumeconstrained diffusion problems in L 2 Ω, Ω R n. To our knowledge, te issue of te well-posedness of te Caucy problem associated wit fractional-in-space partial differential equations on bounded domains in function spaces oter tan L 2 Ω as not been completely resolved. 3

18 Several works ave addressed te need for numerical metods to solve different types of fractional partial differential equations. For instance, iger order linear multi-step metods to solve Abel-Volterra integral equations, of wic fractional differential equations form a sub-class [31], were made popular by [63] in te 1980s using convolution quadratures and fast fourier transforms [49]. Since ten, linear multi-step metods ave been used by many autors, see for example [61, 64, 65, 66], to numerically solve fractional integral equations and fractional partial differential equations. A review and some applications of tese metods can be found in [67]. Algoritms as well as te difficulties encountered wile implementing suc numerical scemes were discussed in [31, 32]. Matrix metods for approximating fractional integrals and derivatives ave been investigated in [87, 88]. In [109], a fractional weigted average finite difference metod along wit von Neumann stability analysis of te numerical scemes was carried out. An implicit numerical sceme for time fractional diffusion equation based on finite difference approximations was developed in [59]. In [62], te autors investigate computationally efficient numerical metods for fractional-in-space diffusion equation wit insulated ends obtained by replacing te second order space derivative in te classical diffusion equation by a Caputo fractional derivative of order 1 < α < 2. Tey use an explicit finite difference metod and te metod of lines to obtain numerical solutions. Stability and convergence of te explicit finite difference numerical metod along wit its scaling restriction were discussed. A similar metod was also used in [100] combined wit Grünwald-Letnikov difference sceme for space discretisation to solve a fractional Fokker-Planck equation. In [75, 77, 78, 101, 102], te autors develop a fractional Crank-Nicolson sceme using a sifted Grünwald formula to solve fractional-in-space partial differential equations. In very recent works [103, 112], a tird order numerical metod using a weigted and sifted Grünwald difference sceme for space fractional diffusion equations in one and two dimensions was developed. Te autors carry out te analysis of numerical stability and convergence wit respect to discrete L 2 -norm. Space fractional derivative operators are non-local. Tus, tey can be used to caracterise influences from a distance, for example super-diffusion penomena [99]. In tis tesis, we are particularly interested in fractional-in-space partial differential equations wic can be used to model suc non-local beaviour in space. Our numerical approximations for tespace fractional derivative operators are also constructed using a sifted Grünwald formula [44, 45, 77, 112] and so trougout tis tesis we refer to tem as Grünwald-type approximations. 4

19 In our study of fractional-in-space partial differential equations, we make an attempt to address te following fundamental issues tat we believe are lacking in te literature. Construction of iger order numerical approximations for fractional derivative operators in L 1 R and in oter function spaces. Stability and smooting properties of iger order numerical approximations tat yield optimal convergence rate wit minimal regularity of initial data for space-time discretisations of te abstract Caucy problem associated wit fractional derivative operators. Truncation of fractional derivative operators on a bounded interval Ω R for combinations of boundary conditions suc as Diriclet, Neumann etc. tat yield well-defined operators wit desirable properties. Te question of well-posedness of te abstract Caucy problems associated wit fractional derivative operators wose domains encode various boundary conditions in function spaces oter tan L 2 Ω, in particular, L 1 Ω and C 0 Ω. Construction of approximations for truncated fractional derivative operators wose associated stocastic processes can be easily identified and understood. Convergence of te stocastic processes associated wit te approximation operators to te corresponding stocastic processes associated wit te truncated fractional derivative operators. Tesis outline: In te first part of tis tesis, in Capters 1 and 2, we explore convergence wit error estimates for iger order Grünwald-type approximations of semigroups generated first by a fractional derivative operator on L 1 R and ten, using a transference principle, by fractional powers of group or semigroup generators on arbitrary Banac spaces. Te main motivation for te investigation of iger order scemes are te works of Meerscaert, Sceffler and Tadjeran [75, 77, 78, 101, 102]. In tese articles, te autors explored consistency and stability of numerical scemes for fractional-in-space partial differential equations using a Grünwald formula wit non-negative integer sift to approximate te fractional derivative operator. In particular, in [102], tey sowed consistency if te order of te spatial derivative is less or equal to 2. Tey obtained specific error term expansion for f C 4+n R, were n is te number of error terms, as 5

20 well as proved stability of teir fractional Crank-Nicolson sceme, using Gersgorin s Teorem to determine te spectrum of te Grünwald matrix. Ricardson extrapolation was ten employed to obtain second order convergence in space. Tis consistency result was extended for a Grünwald formula wit any sift p R in [4, Proposition 4.9] were te autors sowed tat for all f X α R := {f L 1 R : g L 1 R wit ĝk = ik α ˆfk,k R}, te first order Grünwald sceme A α 1 1,pfx = Γ α α m=0 Γm α fx m p 1 Γm+1 converges in L 1 R to te fractional derivative operator f α as 0+. Here ĝk = eikx gxdxdenotestefouriertransformofg L 1 Randforf X α R,f α = g iff ik α ˆfk = ĝk for all k R. In Section 1.5 we improve tis result furter and develop iger order Grünwald-type approximations Ãα. In Corollary we give a consistency error estimate of te form Ãα f f α C L1 n f α+n L1 R 2 R for an n-t order sceme. Using a Carlson-type inequality for periodic multipliers developed in Section Teorem we investigate te stability and smooting of Grünwald-type approximation scemes Ãα. Te main tool is Teorem wic gives a sufficient condition for multipliers associated wit difference scemes approximating te fractional derivative operator to lead to stable scemes wit desirable smooting. In particular, we sow in Proposition 2.1.2, tat stability for a numerical sceme using 1 to solve te Caucy problem associated wit fractional derivative operator f α were 2q 1 < α < 2q +1,q N can only be acieved for a unique sift p in te Grünwald formula. Tat is, it is necessary tat p = q for 1 q+1 A α,p to generate bounded semigroups on L 1 R were te bound is uniform in. Furtermore, in Teorem 2.1.6, we prove stability and smooting of a second order sceme. Developing te teory in L 1 allows in Section 2.2 te transference of te teory to fractional powers A α of te generator A of a strongly continuous semi-group G on a Banac space X,, noting tat fx m p in 1 will read as Gm pf [4]. Te abstract Grünwald approximations wit te optimal sifts generate analytic semigroups, uniformly in, as sown in Teorem Tis is te main property 6

21 needed in Corollary to sow tat te error between S α tf = e t 1q+1 A α f and a fully discrete approximation u n obtained via a Runge-Kutta metod wit stage order s, order r s+1, and an N +1 order Grünwald approximation is bounded by S α tf u n C n r f + N+1 log t AN+1 f, > 0, t = nτ. In error estimates, te smooting of te numerical sceme is used in an essential way to reduce te regularity requirements on te initial data. Furter, tis yields error estimates of te numerical approximation scemes applied to Caucy problem associated wit fractional derivative operators in spaces were te translation semigroup is strongly continuous, suc as L p R, 1 p <, BUCR, C 0 R, etc. Using te abstract setting we can also conclude tat te consistency error estimate 2 olds in tose spaces, wit te L 1 norm replaced by te appropriate norm. Section 2.3 marks te conclusion of te first part of tis tesis wit results of some numerical experiments, including a tird order sceme, tat igligt te efficiency of te iger order scemes as well as te sarpness of te error estimates depending on te smootness of te initial data. Te results from te first part of tis tesis ave been accepted for publication in Transactions of te American Matematical Society and available online [8]. Let us turn our attention to te second part of tis tesis. Te Fokker-Planck equation of a Lévy stable process on R is a fractional-in-space partial differential equation. Te space fractional derivative operator is non-local wit infinite reac. In te second part of tis tesis, in Capters 3 and 4, we investigate truncated Riemann- LiouvilleandfirstdegreeCaputofractionalderivativeoperatorsoforder1 < α < 2ona bounded interval, Ω := [0,1]. Te interval Ω may or may not contain its end points depending on te boundary conditions encoded by te domain of te truncated fractional derivative operator under consideration. We sow convergence in te Skorood topology of easily identifiable finite state sub-markov processes to a sub-markov process governed by te Fokker-Planck equation on Ω associated wit te truncated fractional derivative operators. Observe tat te fractional derivative operators tat we consider below on function spaces defined on te interval Ω are one-sided. Te approac employed in [34] applies only to te symmetric fractional derivative operators as mentioned earlier and terefore do not extend to one-sided fractional derivative operators. However, te boundary conditions tat we consider can be interpreted as special cases of te volume constraints employed in [34] and related works one-dimensional mass constraints. 7 α

22 Te stage is set in Section 3.1 were we discuss te general teoretical framework. Here we exploit te fact tat te convergence, uniformly for t [0,t 0 ], of Feller semigroups on C 0 Ω implies convergence of te corresponding processes in te Skorood topology. To do tis, we turn a finite state sub-markov processes into a Feller process by creating parallel copies of te finite state processes wose transition matrices interpolate continuously. Te main idea beind te construction of tese continuous interpolation matrices is te division of te interval [0,1] into n + 1 grids of equal lengt so tat te Feller process can jump between grids only in multiples of. Te transition operators on X = C 0 Ω or L 1 [0,1]. are constructed using tese interpolation matrices. Tese transition operators are ten employed in Capter 4 to construct te Grünwald-type approximations for te fractional derivative operators on X. In Section 4.3, we sow tat te Grünwald transition approximation operators are te generators of te backward or forward semigroups associated wit te extended finite state sub-markov processes and tus identify te processes associated wit te Grünwald approximation operators. Te central objects of study, te one-sided fractional derivative operators, are introduced in Section Te one-sided fractional derivative operators are denoted in general by A,BC : DA,BC X X wose domains DA,BC encode a particular combination of boundary conditions denoted by BC. Te boundary conditions tat we consider are Diriclet, Neumann and Neumann*, were te latter appears naturally in te adjoint formulation of te fractional derivative operators in L 1 [0,1] wit a rigt Neumann boundary condition. We consider functions of te form f = I α g +ap α +bp α 1 +cp α 2 +dp 0, g X 3 as candidates for te domain of te fractional derivative operators, were a,b,c,d R are determined by te boundary conditions and p β = xβ. Te crucial point to note Γβ+1 ere is te structure of te domains of te fractional derivative operators. Tat is, te domains are defined as te range of te corresponding fractional integral operators I α, supplemented by a linear combination of some particular power functions wit constant weigts tat encode te regularity as well as te boundary conditions BC satisfied by te functions in te domain. In Sections 3.4 and 4.1, well-posedness of te associated one-dimensional fractionalin-space partial differential equations is establised using te approximation teory of 8

23 semigroups [2, 37, 52, 85]. Tat is, we sow tat te fractional derivative operators A, BC generate strongly continuous contraction semigroups on X. To do tis, in Section 3.4, we sow tata, BC are densely defined, closed operators and tat rgλi A are dense in X for some λ > 0. To make use of te Lumer-Pillips Teorem we furter require tat te operators A, BC are dissipative wic is establised using te convergence property of te Grünwald-type approximations, Proposition TeGrünwald-typeapproximation operatorsg areconstructedincapter4using te general teory for numerical scemes developed in Sections 3.1 and 3.2. For te numerical sceme, te boundary conditions BC are encoded into te generic n n sifted Grünwald matrix were = 1, n N given by n+1 G n n = 1 α b l 1 G0 α 0 0. G1 α , 4 b l n 1 Gn 2 α G1 α G0 α b n b r n 1 b r 1 using te boundary weigts b l i,b r i and b n. Te n n sifted Grünwald matrices G n n play te role of te transition rate matrices of te underlying finite state sub-markov processes. In Section 4.2, we first discuss te adjoint formulation of te abstract Caucy problem on X associated wit te fractional derivative operators. In doing so, we list te corresponding fractional derivative operators on X tat are approximated by te Grünwald transition operators constructed using tese boundary weigts. Following tat we conjecture te pysical interpretation of te stocastic processes tat would give rise to tese different boundary conditions BC and discuss our reasons beind te coice of te boundary weigts b l i,b r i, and b n tat appear in te generic Grünwald matrix 4.1 in te L 1 [0,1] case. In Section 4.4 we provide some examples of numerical solutions to te Caucy problem associated wit te fractional derivative operators A,BC on L 1 [0,1] and te initial value u 0 L 1 [0,1]. In Section 4.3, we prove te key result, Proposition 4.3.2, tat te Grünwald transition operators converge to te respective fractional derivative operators on X. Tat is, for eac f DA,BC we sow tat tere exist sequences f X suc tat f f and G f Af in X for eac of te fractional derivative operators A,BC. Tis as it turns out involves detailed error analysis employing elaborate constructions of appoximations for te power functions p β tat appear in 3 above. Tis result is 9

24 essential firstly to sow tat te fractional derivative operators A, BC are dissipative. Using Proposition 4.3.2, we also conclude tat te semigroups generated by te operators A,BC are te strong and uniform for t in compact intervals limit of te semigroups generated by te Grünwald transition operators using te Trotter-Kato Teorem. As a consequence, te underlying Feller processes associated wit Grünwald approximations converge in te Skorood topology to te Feller processes governed by te corresponding fractional-in-space partial differential equations. Tis identifies te processes governed by te fractional-in-space partial differential equations wit boundary conditions BC as limits of processes wose boundary beaviour is perfectly understood. 10

25 Capter 1 Grünwald-type approximations of fractional derivative operators on te real line In tis capter we study Grünwald-type approximations for te fractional derivative operators on R. We explore convergence and conduct a detailed error analysis using Fourier multiplier teory. Following tat, combining Grünwald formulae wit different sifts and step sizes, iger order Grünwald-type approximations are constructed for te fractional derivative operators on R. We sow convergence of te iger order approximations to te fractional derivative operators wit optimal convergence rate under minimal regularity assumptions. 1.1 Fractional derivative operators on R Te error analysis of iger order Grünwald-type numerical approximations of fractional derivative operators on R is carried out using multiplier teory. To facilitate tis, we define te fractional derivatives of L 1 R-functions in te Fourier or Laplace space depending on te support of te function under consideration. To define te fractional derivative operator of order α R + using Fourier transform if f L 1 R and Laplace transform if f L 1 R +, let us begin wit te following two spaces. Definition Let α R + and z α := z α e iαargz be as in Remark B.3.4. Ten, we define te following two spaces: 1. X α R := {f L 1 R : g L 1 R wit ĝk = ik α ˆfk, k R}, 11

26 were ˆfk and ĝk denote te Fourier transforms of f and g, respectively, given by B X α R + := {f L 1 R + : g L 1 R + wit ĝz = z α ˆfz, Rez 0}, were ˆfz and ĝz denote te Laplace transforms of f and g, respectively, given by B.6. Here is te formal definition of te fractional derivative operator on R tat we use in tis tesis. Definition For f X α R, if g L 1 R and ik α ˆfk = ĝk, k R, ten we define f α := g. Alongsimilarlinesforf X α R +, wedefinef α := g, ifg L 1 R + and z α ˆfz = ĝz for Rez 0. To keep te notation simple, we denote te norms in bot tese spaces by f α, tat is, we set f α := f α, for f X L1 R αr and f α := f α, for f X L1 R + αr To connect te above definition of fractional derivatives on R wit te standard definitions of fractional derivatives found in te literature, we list te definitions of te Riemann-Liouville fractional integrals and derivatives. Definition Let α > 0 and f L 1 R, ten te so called left-sided Riemann- Liouville fractional integral if a = 0 or Liouville fractional integral if a = of order α, is defined by ai α xfx := x a x s α 1 fs ds, x > a Γα were te lower limit of te integral a R is fixed or a =, wile te upper limit x R is variable, see [86, p. 65] and [95, p. 33 and 94]. 12

27 Tis definition of fractional integral is well-defined for any piecewise continous function f L 1 a,b, owever, for our purposes we take L 1 R as te domain of definition. Definition For α > 0, let n = α denote te least integer greater tan α, ten te so called Riemann-Liouville fractional derivative, D α : DD α L 1 R is defined by D α fx := D n I n α x fx wit domain DD α := { f L 1 R : I n α x f W n,1 R }, were D n denotes te integer order derivative of order n on R wit respect to variable upper limit of te fractional integral and te Sobolev space W n,1 R is given in Definition B.1.1. Remark Let f DD α, ten D α fk = ik α ˆfk [95, p ]. Tus, f α = D α f a.e.; tat is, te function f α defined uniquely in Definition is equal to te Riemann-Liouville fractional derivative D α f given by almost everywere by te uniqueness of te Fourier transform of L 1 R-functions. 1.2 Grünwald-type approximations In te Grünwald-Letnikov approac to fractional calculus, te fractional derivative of arbitrary order α > 0 is defined as te limit of te corresponding fractional difference quotient [86, 95], n adxfx α = lim α 0 mfx m n=x a m=0g α, 1.2 were a and x are te lower and upper terminals, respectively and Gm α = 1 m α m are given by A.1. Podlubny [86, p.63] demonstrates te equivalence of te Riemann- Liouville and te Grünwald-Letnikov definitions of te fractional derivative under te assumption tat f is m+1-times differentiable were α < m+1. Tus, in numerical scemes, it is natural to use te Grünwald-Letnikov formula 1.2 wit a fixed step size to approximate te fractional derivative operator. Here is te formal definition of te sifted Grünwald formula tat we use to approximate te fractional derivative operator on R, also see [44], [77], [105]. 13

28 Definition Let f L 1 R and > 0, ten te p-sifted Grünwald formula is given by A α,pfx := 1 α Gmfx m p, α 1.3 m=0 were te sift p R and te properties of te Grünwald coefficients, Gm α = 1 m α m can be found in Appendix A. Observe te sift p used in te argument of te function f compared to 1.2 above. Meerscaert et al. in [77] used tis Grünwald formula, a modified version of 1.2, wit a non-negative integer sift p, to numerically approximate te Riemann-Liouville fractional derivative. Te autors also proved stability of teir numerical sceme for space-fractional advection dispersion equation wit 1 < α < 2, using te Grünwald formula for space discretisation wit sift p = 1 and te implicit Euler metod for time discretisation. In te same article in Remark 2.5, and in [102] in Remark 3.2, te autors mention tat te Grünwald formula wit no sift or any sift p yields first order consistency of numerical scemes for fractional-in-space partial differential equations. However, tey attribute te best performance of te numerical scemes to te optimal sift p obtained by minimising p α. In [44], te autors refer to te 2 optimal sift as a clever sift using wic yields a consistent approximation for te Riemann-Liouville fractional derivative for sufficiently smoot functions. We sow in Capter 2, tat more is true of tese remarks, tat in fact te numerical scemes using te Grünwald formula wit integer sift are stable if and only if te sift p is optimal. Remark We make a clarification of te convention tat we adopt at tis juncture. WenweapplytesiftedGrünwaldformula1.3toafunctionf L 1 R +, wealways assume implicitly tat f is extended to L 1 R by setting fx = 0 for x < 0. To keep matters simple, we will also refer to tis extended function as f. In te case wen te sift p > 0 we regard A α,p as an operator on L 1R. However, in te case wen p 0, wit tis convention, one can verify tat te support of A α,p f is contained in R+ and ence A α,p can be regarded as an operator on L 1R +. Indeed, for p 0 if x < 0, ten for all m N, x m p x+p x < 0 wic, in view of our convention, implies tat fx m p = 0 and ence A α,pfx = 0. Te following result of Tadjeran et al. [102], tat te numerical scemes tat employ te sifted Grünwald formula yields second order consistency is our main motivation to explore iger order scemes. Let te Sobolev space W 1,3+n R be given by Definition B.1.1 and 1 < α < 2. Ten, for f W 1,3+n R, te autors sowed tat te error term expansion for te numerical approximation of te Riemann-Liouville fractional 14

29 derivative of order α by te non-negative integer sifted Grünwald formula is given by n 1 A α,pfx D α fx = al D α+l fx l +O n, l=1 were te constants a l are independent of f, and x. In te same paper, te autors proved stability of teir fractional Crank-Nicolson sceme by determining te spectrum of te Grünwald matrix associated wit te sifted Grünwald formula using Gersgorin s Teorem. Moreover, teir fractional Crank-Nicolson sceme was sown to be consistent wit a second order in time and first order in space local truncation errors. Furtermore, tey obtained a second order local truncation error in space employing te Ricardson extrapolation metod. Tis result tat te Grünwald approximation for te fractional derivative is consistent was furter refined by Baeumer et. al. in [4, Proposition 4.9] to all f X α R wit any sift p R were X α is given in Definition 1.1.1, tat is, A α,p f Dα f in L 1 R as 0+. A proof for te unsifted case, p = 0 wit α > 0 and f X α R + can be found in [106, Teorem 13]. We generalise tese results in Teorem under minimal regularity assumptions. Tat is, for f X α+β R te convergence rate of te first order Grünwald-type approximation for te fractional derivative operator in L 1 R can be furter fine-tuned to te order β, 0 < β 1 as 0+. If p 0, ten te same convergence rate olds in L 1 R + for f X α+β R +. Following tat, in Section 1.5, we construct iger-order Grünwald-type approximations for te fractional derivative operator on R, by combining Grünwald formulae wit different sifts p and accuracy suc tat te lower order error terms cancel out. We ten conduct a detailed error analysis using Fourier multipliers and conclude tis capter wit te main result, Corollary on te consistency of te iger order scemes. To begin wit, we sow tat te sifted Grünwald formula given in Definition 1.2.1, maps L 1 R into L 1 R and derive an explicit formula for its Fourier transform. Lemma Let f L 1 R, α R +, p R, > 0 be fixed, and te sifted Grünwald formula A α,p be given by 1.3, ten A α,pf L 1 R. Moreover, its Fourier transform is given by A α,p fk = ω α,p ik f α k, 1 e αe were ω α,p z = z zp. z 15

30 Proof. Take note tat te series m=0 Gα m is absolutely convergent, see A.10. Tus, te iterated integral m=0 R G α mfx m p dx Gm f α L1 R < m=0 and te use of Fubini s teorem [92, p. 141], is justified. Hence, A α,p f = 1 L 1 G α R R mfx m p α dx m=0 1 G α m fx m p dx α 1 α m=0 R Gm f α L1 R <. m=0 Terefore, te Fourier transform of te sifted Grünwald formula exists and using Fubini s teorem once again, we ave A α,p fk = α m=0 G α m R e ikx fx m pdx α = α 1 m e ikm p ˆfk m m=0 Te Fourier transform of te Grünwald formula can be written in te following product form using te Binomial series A.4, A α,p fk = α α 1 m e ikm p ˆfk m m=0 = α e ikp 1 e ik α ˆfk 1 e = α ik α ik α e ikp ˆfk ik = ω α,p ik ik α ˆfk = ω α,p ik f α k, 1.4 were we ave used Definition in te last line and introduce te special function αe ω α,p z = zp. 1 e z z Remark Let α R + suc tat 2q 1 < α < 2q +1 were q N and let ψz = 1 q+1 α e pz 1 e z α

31 Note tat te second line of 1.4 above sows tat for any f L 1 R te Grünwald formula 1.3 can be expressed in te multiplier notation of Appendix C.3 as A α,p = T 1 q+1 ψ α,,p were te Grünwald multiplier is given by ψ α,,p k := ψik = 1 q+1 α e ikp 1 e ik α = 1 q+1 ω α,p ik ik α. 1.6 Tis fact tat te Grünwald formula can be viewed as a multiplier operator naturally leads us in Section 1.3 to study inequalities tat estimate multiplier norms. Remark Te Grünwald multiplier in Remark above, involves te function ω α,p. In te error analysis of te Grünwald scemes, as we will see later, te function ω α,p : C C, were α R + and p R, plays a very important role. Hence, for easy reference we give it a special name, omega function, 1 e z α ω α,p z := e zp z and study tis function in detail in Section Bound for multiplier norms In tis section we study Carlson-type inequalities tat bound L 1 Fourier multiplier norms. We refer to Appendix C.3 for te definition and results relating to L 1 - multipliers. Tese inequalities are not only crucial in our error analysis, but also important in teir own rigt. Firstly, tey prove to be particulary useful in scenarios were only te multipliers are known explicitly wile te corresponding measures even wen tey exist may not be known. Secondly, tese Carlson-type inequalities elp bound multiplier operator norms by solely exploiting te properties of te multipliers and as a consequence elp to sow tat, in fact, te multipliers under consideration are L 1 -multipliers. Tirdly, tese inequalities elp estimate te L 1 -norms of functions, defined as te inverse Fourier transforms of functions in L r for 1 < r 2 troug te L r -norms of te Fourier transform and its derivative. Lastly, as we will see in te applications in Capter 2, it turns out tat it is essential to consider L r -spaces for r 2. In some situations te Carlson-type inequality, given in Proposition below, is not directly applicable. One suc scenario is wen te decay of te multiplier at infinity 17

32 is insufficient for it to be in L r but its derivative as more tan necessary decay to be in L r. In tis situation, we employ a partition of unity, and derive a similar result, tereby enancing te reac of te Carlson-type inequality. We do tis in Section 1.3.1, were we study inequalities for multipliers wic along wit teir generalised first derivatives belong to L r R, for 1 < r 2. Anoter scenario were te Carlsontype inequality cannot be applied is for periodic multipliers. In Section 1.3.2, we study similar inequalities for periodic multipliers Carlson-type inequality Te first inequality tat we consider is a special case of a more general Carlson-type inequality, see [60, Teorem 5.10, p.107]. We give a simple proof ere to keep our discussions self-contained. Te case r = 2 is usually referred to in te literature as Carlson-Beurling Inequality, and can be found in [1, p.429] and [18, 24, 36]. Let W r,1 R, r 1, denote te Sobolev space of L r R-functions given in Definition B.1.1 and FL 1 denote te ring of Fourier transforms of L 1 -functions as in Remark B.3.2. Te following result yields a sarp bound for te L 1 -multiplier operator norm. Proposition Carlson-type inequality. If ψ W r,1 R, 1 < r 2, ten tere exists ξ L 1 suc tat ˆξ = ψ; tat is, ψ FL 1. Moreover, tere exists a constant Cr > 0, independent of ξ and ψ, suc tat were 1 r + 1 s = 1. ξ L1 Cr ψ 1 s Lr ψ 1 r Lr, Proof. Let ψ W r,1 R, 1 < r 2 and set ψ k := ψ k, k R. Ten, define te function ξ := 2π ψ 1 L s R, were = 1, see Remark B.2.2. First, note tat r s ξ L1 = 2π ψ 1 = L1 1 2π ˆψ and ˆψ x = ixˆψx. L1 Moreover, recall te Hausdorff-Young-Titcmars inequality given by B.1 ˆψ 2π 1 1 s ψ Lr,ψ L r, 1 r 2, Ls r + 1 s = 1. If ψ 0 tere is noting to prove. So let us assume tat ψ 0, ten using Hölder s ψ Lr inequality in te second line, B.1 in te tird and setting v = we ave r 1 1/r ψ Lr ξ L1 = 1 ˆψ = 1 ˆψx dx+ 1 2π L1 2π x v x >v x xˆψx dx 21 r v 1 r ˆψ +v 1 s ˆψ r 1 1/r 2π Ls Ls 18

33 2 1 r 2π 1 s 1 v 1 r ψ Lr +v 1 s ψ Lr r 1 1/r = 2 π 1 rr 1 1 r 2 ψ 1 s L r ψ 1 r Lr. Terfore, ξ L 1 and so ˆξ exists. Tus, ψk = 1 ψ ˆ k = ˆξk for almost all k R by te inversion formula for te Fourier transform. Since, ψ and ˆξ are continuous, tis olds for all k R. Remark In fact, te preceding proof sows tat ψ is an L 1 -multiplier and te Carlson-type inquality can be rewritten in multiplier notation of Appendix C.3 as were Cr is independent of ψ. T ψ BL1 = ξ L1 Cr ψ 1 s Lr ψ 1 r Lr, In some cases, te multiplier as insufficient decay to be in L r wile its generalised derivative migt ave more tan necessary decay to be in L r. In suc scenarios, partition of unity turns out to be an excellent tool to bound te multiplier norm, see [21, 22, 50]. Te following result is a version of Carlson-type inequality employing a partition of unity. Corollary Let ψ : R C and θ j be suc tat j Z θ jx = 1 for all x R. If θ j ψ W r,1 R, 1 < r 2, for all j and j θ jψ 1 s Lr θ j ψ 1 r Lr <, were 1 r + 1 s = 1, ten ψ FL 1 ; tat is, tere exists ξ L 1 R and a constant Cr independent of ξ and ψ, suc tat ˆξ = ψ and 2π ξ L1 Cr j θ j ψ 1 s Lr θ j ψ 1 r Lr. Proof. By design, ψx = j θ jxψx for almost all x. Let ξ j L 1 R be suc tat ˆξ j = θ j ψ by Proposition By assumption, te partial sums n n j j L1 j= nξ L 1 j= n ξ Cr θ j ψ 1 s Lr θ j ψ 1 r Lr <. j Tus, te series j ξ j converges to some ξ L 1 and ξ L1 Cr j θ j ψ 1 s Lr θ j ψ 1 r Lr. Hence, te use of Fubini s teorem is justified and ˆξ = j ˆξ j = ψ. 19

34 1.3.2 Carlson-type inequality for periodic multipliers For periodic multipliers a Carlson-type inequality is not directly applicable as tese are not Fourier transforms of L 1 -functions. In [22] a suitable smoot cut-off function η wit compact support was used were η = 1 in a neigborood of [ π,π], to estimate te multiplier norm of a periodic multiplier ψ by te non-periodic one ηψ. For te multiplier norm of ηψ te above Carlson-type inequality can be ten used. However, we prove a result similar to Proposition for periodic multipliers wic along wit teirderivativesavelocall r bounds. Tismakesteintroductionofacut-offfunction superfluous and ence simplifies te tecnicalities in later estimates. Let W r,1 per[ π, π] below denote te Sobolev space of 2π-periodic functions as in Definition B.1.1. Teorem Let ψ W r,1 per[ π,π], 1 < r 2, ten ψ is an L 1 -multiplier and tere is C > 0, independent of ψ, suc tat T ψ BL1 a 0 +Cr ψ 1 s Lr[ π,π] ψ 1 r Lr[ π,π], were = 1 and a r s 0 = 1 π ψxdx, denotes te 2π π 0t Fourier coefficient of ψ. Proof. Since ψ L r [ π,π], it follows using Hölder s inequality tat ψ L 1 [ π,π] and so we can define te k t Fourier coefficient of ψ, a k = 1 2π π π e ikx ψxdx, k Z. First, note tat a 0 1 π ψx dx < and using integration by parts and te 2π π fact tat ψ is absolutely continuous, we ave tat ika k are te Fourier coefficients of ψ. Next, recall te Hausdorff-Young inequality for Fourier series, see [23, p. 177], k= a k s 1 s and Bellman s inequality, see [14] and [60, p. 25], αβ+α β b k Cα,β k=1 2π 1 r ψ Lr[ π,π], 1 < r 2, 1 r + 1 s = 1 k=1 b α k α 1 k β bk β, α,β > 1, b k 0, k N. k=1 On setting, α = β = s and b k = a k, we ave 1 a k Cr a k s s 2 k=1 k=1 k=1 1 s s 1 s ika k s Cr ψ 1 s Lr[ π,π] ψ 1 r. Lr[ π,π] 20