Books in the Fractional Calculus in Applied Sciences and Engineering series

This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to find high-precision solutions to fractional-order differentiations and differential equations. Numerical algorithms and their implementations are proposed to analyze multivariable fractional-order control systems. Through high-quality MATLAB programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems. ContentsIntroduction to fractional calculus and fractional-order controlMathematical prerequisitesDefinitions and computation algorithms of fractional-order derivatives and IntegralsSolutions of linear fractional-order differential equationsApproximation of fractional-order operatorsModelling and analysis of multivariable fractional-order transfer function MatricesState space modelling and analysis of linear fractional-order SystemsNumerical solutions of nonlinear fractional-order differential EquationsDesign of fractional-order PID controllersFrequency domain controller design for multivariable fractional-order SystemsInverse Laplace transforms involving fractional and irrational OperationsFOTF Toolbox functions and modelsBenchmark problems for the assessment of fractional-order differential equation algorithms

This book extends classical Hermite-Hadamard type inequalities to the fractional case via establishing fractional integral identities, and discusses Riemann-Liouville and Hadamard integrals, respectively, by various convex functions. Illustrating theoretical results via applications in special means of real numbers, it is an essential reference for applied mathematicians and engineers working with fractional calculus. ContentsIntroductionPreliminariesFractional integral identitiesHermite-Hadamard inequalities involving Riemann-Liouville fractional integralsHermite-Hadamard inequalities involving Hadamard fractional integrals

This book presents fractional difference, integral, differential, evolution equations and inclusions, and discusses existence and asymptotic behavior of their solutions. Controllability and relaxed control results are obtained. Combining rigorous deduction with abundant examples, it is of interest to nonlinear science researchers using fractional equations as a tool, and physicists, mechanics researchers and engineers studying relevant topics. ContentsFractional Difference EquationsFractional Integral EquationsFractional Differential EquationsFractional Evolution Equations: ContinuedFractional Differential Inclusions

Complexity increases with increasing system size in everything from organisms to organizations. The nonlinear dependence of a system's functionality on its size, by means of an allometry relation, is argued to be a consequence of their joint dependency on complexity (information). In turn, complexity is proven to be the source of allometry and to provide a new kind of force entailed by a system's information gradient. Based on first principles, the scaling behavior of the probability density function is determined by the exact solution to a set of fractional differential equations. The resulting lowest order moments in system size and functionality gives rise to the empirical allometry relations. Taking examples from various topics in nature, the book is of interest to researchers in applied mathematics, as well as, investigators in the natural, social, physical and life sciences. ContentsComplexityEmpirical allometryStatistics, scaling and simulationAllometry theoriesStrange kineticsFractional probability calculus