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There has been a great advancement in the study of fractional-order
nonlocal nonlinear boundary value problems during the last few
decades. The interest in the subject of fractional-order boundary
value problems owes to the extensive application of fractional
differential equations in many engineering and scientific
disciplines. Fractional-order differential and integral operators
provide an excellent instrument for the description of memory and
hereditary properties of various materials and processes, which
contributed significantly to the popularity of the subject and
motivated many researchers and modelers to shift their focus from
classical models to fractional order models. Some peculiarities of
physical, chemical or other processes happening inside the domain
cannot be formulated with the aid of classical boundary conditions.
This limitation led to the consideration of nonlocal and integral
conditions which relate the boundary values of the unknown function
to its values at some interior positions of the domain.The main
objective for writing this book is to present some recent results
on single-valued and multi-valued boundary value problems,
involving different kinds of fractional differential and integral
operators, and several kinds of nonlocal multi-point, integral,
integro-differential boundary conditions. Much of the content of
this book contains the recent research published by the authors on
the topic.
The main objective of this book is to extend the scope of the
q-calculus based on the definition of q-derivative [Jackson (1910)]
to make it applicable to dense domains. As a matter of fact,
Jackson's definition of q-derivative fails to work for impulse
points while this situation does not arise for impulsive equations
on q-time scales as the domains consist of isolated points covering
the case of consecutive points. In precise terms, we study quantum
calculus on finite intervals.In the first part, we discuss the
concepts of qk-derivative and qk-integral, and establish their
basic properties. As applications, we study initial and boundary
value problems of impulsive qk-difference equations and inclusions
equipped with different kinds of boundary conditions. We also
transform some classical integral inequalities and develop some new
integral inequalities for convex functions in the context of
qk-calculus. In the second part, we develop fractional quantum
calculus in relation to a new qk-shifting operator and establish
some existence and qk uniqueness results for initial and boundary
value problems of impulsive fractional qk-difference equations.
This book focuses on the recent development of fractional
differential equations, integro-differential equations, and
inclusions and inequalities involving the Hadamard derivative and
integral. Through a comprehensive study based in part on their
recent research, the authors address the issues related to initial
and boundary value problems involving Hadamard type differential
equations and inclusions as well as their functional counterparts.
The book covers fundamental concepts of multivalued analysis and
introduces a new class of mixed initial value problems involving
the Hadamard derivative and Riemann-Liouville fractional integrals.
In later chapters, the authors discuss nonlinear Langevin equations
as well as coupled systems of Langevin equations with fractional
integral conditions. Focused and thorough, this book is a useful
resource for readers and researchers interested in the area of
fractional calculus.
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