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

234 Pages·2014·1.34 MB·English

Gru¨nwald-type approximations
and boundary conditions for
one-sided fractional derivative
operators
Harish Sankaranarayanan
a thesis submitted for the degree of
Doctor of Philosophy
at the University of Otago, Dunedin,
New Zealand.
October 31, 2014

Abstract
The focus of this thesis is two-fold. The first part investigates higher order
numerical schemes for one-dimensional fractional-in-space partial differen-
tial equations in L (R). The approximations for the (space) fractional
1
derivative operators are constructed using a shifted Gru¨nwald-Letnikov
fractional difference formula. Rigorous error and stability analysis of the
Gru¨nwald-type numerical schemes for space-time discretisations of the as-
sociated Cauchy problem are carried out using (Fourier) multiplier the-
ory and semigroup theory. The use of a transference principle facilitates
the generalisation of the results from the L -setting to any function space
1
where the translation (semi) group is strongly continuous. Furthermore,
the results extend to the case when the fractional derivative operator is
replaced by the fractional power of a (semi) group generator on an arbi-
trary Banach space. The second part is dedicated to the study of certain
fractional-in-spacepartialdifferentialequationsassociatedwith(truncated)
Riemann-Liouville and first degree Caputo fractional derivative operators
on Ω := [(0,1)]. The boundary conditions encoded in the domains of the
fractional derivative operators dictate the inclusion or exclusion of the end
points of Ω. Elaborate technical constructions and detailed error analy-
sis are carried out to show convergence of Gru¨nwald-type approximations
to fractional derivative operators on X = C (Ω) and L [0,1]. The well-
0 1
posedness of the associated Cauchy problem on X is established using the
approximation theory of semigroups. The culmination of the thesis is the
result which shows convergence in the Skorohod topology of the well under-
stood stochastic processes associated with Gru¨nwald-type approximations
to the processes governed by the corresponding fractional-in-space partial
differential equations.
iii

Acknowledgements
I extend my deepest gratitude to my Mathematics Gurus (my supervisors),
Dr. Boris Baeumer and Dr. Miha´ly Kova´cs, for their excellent tutelage
over the last five years. I thank Boris for initiating and training me in the
(dark, magical) art of problem solving, for teaching me to see the bigger
picture and for his time, patience and energy. I thank Mishi for enlightning
me on abstract technical concepts, for teaching me to write with surgical
precision and for (literally) watching over me as a big brother. I am also
grateful to both of them for many lunch treats and for part of the financial
assistance to attend the ANZIAM conference.
I would like to thank my co-supervisor Prof. J¨org Frauendiener for his
magnanimity and for his words of wisdom. I would also like to thank the
Head of the Department, Prof. Richard Barker as well as all the academic
staff at the department for their constant encouragement. I am especially
grateful to Prof. John Clark and Dr. Austina Clark for giving me the priv-
elage of lecturing the Modern Algebra course (part-time). Special thanks
to the Department Manager, Mrs. Lenette Grant for all the help, includ-
ing several tutoring opportunities over the last five years. I would like to
acknowledge the PhD scholarship offered by the University of Otago and
the financial assistances given by CSIRO, ANZIAM and NZ-ANZIAM to
attend the ANZIAM conference.
Special thanks to the wonderful support staff, Chris, Greg, Marguerite,
and especially, Leanne for being a great listener, and for putting things
into perspective on several occasions. It is also a real pleasure to thank
(fellow PhD student) Aidin who shared the teaching responsibilities with
me for three consecutive Summer School Math160 courses. I thank my
v
current work mates, including Chris. S, Danie, Ilija, Leon, Paula, Richard
and Zahra from Mathematics, Adrian (and wife Janet), Levi and Sam from
Physics as well as my past work mates, Chris. L, Fabien (and wife Yuri)
and George (and partner Kathi), for their camaraderie.
I am extremely grateful to Rick and Nicola (Commodore Motel, Dunedin)
for their hospitality and generosity, and for the cozy room that helped me
rest and rejuvenate during the crucial, last one and half years of my PhD
study. Special thanks 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 Balakrishna, Juliana and Murali, and Daisy,
Mundy, Chidleys, and not to forget big Gerry for many reminders of baby
steps, 1 plus 1 equals 2.
I thank Abdulmannan and family (Momtaj, Hassan and Mridula) for treat-
ing me as their own. I also thank my dear friends, Dev and Anu, for always
being there for me. I am extremely grateful to my brave parents for believ-
ing in me, for encouraging me to return to studies and for supporting me
every step of the way, including financially. I am eternally indebted to my
younger brother (Dr.) Balaji and my sister-in-law Sripriya for their love,
affection and financial support, and most importantly, I congratulate them
on their best result, little bundle of joy, Sanju.
vi
Dedicated to my loving parents, Vasantha and Sanku.
vii

Contents
Introduction 1
1 Gru¨nwald-type approximations of fractional derivative operators on
the real line 11
1.1 Fractional derivative operators on R . . . . . . . . . . . . . . . . . . . . 11
1.2 Gru¨nwald-type approximations . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Bound for multiplier norms . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Carlson-type inequality . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 Carlson-type inequality for periodic multipliers . . . . . . . . . . 20
1.4 Gru¨nwald (periodic) multiplier operators . . . . . . . . . . . . . . . . . 21
1.5 Construction of higher order Gru¨nwald-type
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 Consistency of higher order Gru¨nwald-type
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Semigroups generated by Gru¨nwald-type approximations 43
2.1 Semigroups generated by periodic multipliers approximating fractional
derivative operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.1 First-order Gru¨nwald-type approximation . . . . . . . . . . . . 46
2.1.2 Examples of second order stable Gru¨nwald-type approximations 49
2.2 Application to fractional powers of operators . . . . . . . . . . . . . . . 51
2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Boundary conditions for fractional-in-space partial differential equa-
tions 61
3.1 Extension of a finite state Markov process to a Feller process on [0,1] . 61
3.2 Construction of the transition operators . . . . . . . . . . . . . . . . . 63
3.3 One-sided fractional derivative operators with different combinations of
boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Fractional integral operators and fractional derivatives . . . . . 70
3.3.2 Fractional derivative operators on a bounded interval encoding
various boundary conditions . . . . . . . . . . . . . . . . . . . . 75
3.4 Properties of the one-sided fractional derivative operators on a bounded
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
ix
4 Gru¨nwald-type approximations for fractional derivative operators on
a bounded interval 93
4.1 Gru¨nwald-type approximations for fractional
derivative operators on C (Ω) and L [0,1] . . . . . . . . . . . . . . . . 93
0 1
4.2 Adjoint formulation and boundary weights . . . . . . . . . . . . . . . . 96
4.3 Semigroups and processes associated with
Gru¨nwald-type approximations and fractional
derivative operators on X . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.4 Examples of Gru¨nwald schemes for
Cauchy problems on L [0,1] . . . . . . . . . . . . . . . . . . . . . . . . 116
1
4.5 Detailed proof of Proposition 4.3.2 . . . . . . . . . . . . . . . . . . . . 120
4.5.1 Construction of approximate power functions . . . . . . . . . . 120
4.5.2 Proof of Proposition 4.3.2 for the case X = C (Ω) . . . . . . . . 128
0
4.5.3 Proof of Proposition 4.3.2 for the case X = L [0,1] . . . . . . . 161
1
A Properties of Gru¨nwald coefficients 191
B Reminder on function spaces, distributions and transforms 193
B.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
B.2 Fourier and Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . 194
B.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
C Reminder on operator theory, semigroups and multipliers 199
C.1 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
C.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
C.3 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
D MATLAB codes for Gru¨nwald schemes 207
References 211
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