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Strongly coupled (or cross-diffusion) systems of parabolic and elliptic partial differential equations appear in many physical applications. This book presents a new approach to the solvability of general strongly coupled systems, a much more difficult problem in contrast to the scalar case, by unifying, elucidating and extending breakthrough results obtained by the author, and providing solutions to many open fundamental questions in the theory. Several examples in mathematical biology and ecology are also included. Contents Interpolation Gagliardo-Nirenberg inequalities The parabolic systems The elliptic systems Cross-diffusion systems of porous media type Nontrivial steady-state solutions The duality RBMO(μ)-H1(μ)| Some algebraic inequalities Partial regularity

Control theory. --- Coupled mode theory. --- Differential equations, Partial. --- Differential equations, Parabolic. --- Differential equations, Elliptic. --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- Parabolic differential equations --- Parabolic partial differential equations --- Partial differential equations --- Coupled modes, Theory of --- Coupled systems --- Oscillations --- Vibration --- Wave-motion, Theory of --- Dynamics --- Machine theory

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This book suggests a new common approach to the study of resonance energy transport based on the recently developed concept of Limiting Phase Trajectories (LPTs), presenting  applications of the approach to significant nonlinear problems from different fields of physics and mechanics. In order to highlight the novelty and perspectives of the developed approach, it places the LPT concept in the context of dynamical phenomena related to the energy transfer problems and applies the theory to numerous problems of practical importance. This approach leads to the conclusion that strongly nonstationary resonance processes in nonlinear oscillator arrays and nanostructures are characterized either by maximum possible energy exchange between the clusters of oscillators (coherence domains) or by maximum energy transfer from an external source of energy to the chain. The trajectories corresponding to these processes are referred to as LPTs. The development and the use of the LPTs concept are motivated by the fact that non-stationary processes in a broad variety of finite-dimensional physical models are beyond the well-known paradigm of nonlinear normal modes (NNMs), which is fully justified either for stationary processes or for nonstationary non-resonance processes described exactly or approximately by the combinations of the non-resonant normal modes. Thus, the role of LPTs in understanding and analyzing of intense resonance energy transfer is similar to the role of NNMs for the stationary processes. The book is a valuable resource for engineers needing to deal effectively with the problems arising in the fields of mechanical and physical applications, when the natural physical model is quite complicated. At the same time, the mathematical analysis means that it is of interest to researchers working on the theory and numerical investigation of nonlinear oscillations.

Engineering. --- Solid state physics. --- Thermodynamics. --- Heat engineering. --- Heat transfer. --- Mass transfer. --- Vibration. --- Dynamical systems. --- Dynamics. --- Nanotechnology. --- Vibration, Dynamical Systems, Control. --- Solid State Physics. --- Engineering Thermodynamics, Heat and Mass Transfer. --- Energy transfer. --- Coupled mode theory. --- Coupled modes, Theory of --- Coupled systems --- Oscillations --- Vibration --- Wave-motion, Theory of --- Energy storage --- Force and energy --- Transport theory --- Construction --- Industrial arts --- Technology --- Molecular technology --- Nanoscale technology --- High technology --- Cycles --- Mechanics --- Sound --- Mass transport (Physics) --- Thermodynamics --- Heat transfer --- Thermal transfer --- Transmission of heat --- Energy transfer --- Heat --- Mechanical engineering --- Chemistry, Physical and theoretical --- Dynamics --- Physics --- Heat-engines --- Quantum theory --- Solids --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Statics

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In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications.

Caputo fractional derivative --- fractional differential equations --- hybrid differential equations --- coupled hybrid Sturm–Liouville differential equation --- multi-point boundary coupled hybrid condition --- integral boundary coupled hybrid condition --- dhage type fixed point theorem --- linear fractional system --- distributed delay --- finite time stability --- impulsive differential equations --- fractional impulsive differential equations --- instantaneous impulses --- non-instantaneous impulses --- time-fractional diffusion-wave equations --- Euler wavelets --- integral equations --- numerical approximation --- coupled systems --- Riemann–Liouville fractional derivative --- Hadamard–Caputo fractional derivative --- nonlocal boundary conditions --- existence --- fixed point --- LR-p-convex interval-valued function --- Katugampola fractional integral operator --- Hermite-Hadamard type inequality --- Hermite-Hadamard-Fejér inequality --- space–fractional Fokker–Planck operator --- time–fractional wave with the time–fractional damped term --- Laplace transform --- Mittag–Leffler function --- Grünwald–Letnikov scheme --- potential and current in an electric transmission line --- random walk of a population --- fractional derivative --- gradient descent --- economic growth --- group of seven --- fractional order derivative model --- GPU --- a spiral-plate heat exchanger --- parallel model --- heat transfer --- nonlinear system --- stochastic epidemic model --- malaria infection --- stochastic generalized Euler --- nonstandard finite-difference method --- positivity --- boundedness --- n/a --- coupled hybrid Sturm-Liouville differential equation --- Riemann-Liouville fractional derivative --- Hadamard-Caputo fractional derivative --- Hermite-Hadamard-Fejér inequality --- space-fractional Fokker-Planck operator --- time-fractional wave with the time-fractional damped term --- Mittag-Leffler function --- Grünwald-Letnikov scheme