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This two-volume monograph presents new methods of construction of global asymptotics of solutions to nonlinear equations with small parameter. These allow one to match the asymptotics of various properties with each other in transition regions and to get unified formulas for the connection of characteristic parameters of approximate solutions. This approach underlies modern asymptotic methods and gives a deep insight into crucial nonlinear phenomena in the natural sciences. These include the outset of chaos in dynamical systems, incipient solitary and shock waves, oscillatory processes in crystals, engineering applications, and quantum systems. Apart from being of independent interest, such approximate solutions serve as a foolproof basis for testing numerical algorithms. This first volume presents asymptotic methods in oscillation and resonance problems described by ordinary differential equations, whereby the second volume will be devoted to applications of asymptotic methods in waves and boundary value problems. Contents Asymptotic expansions and series Asymptotic methods for solving nonlinear equations Nonlinear oscillator in potential well Autoresonances in nonlinear systems Asymptotics for loss of stability Systems of coupled oscillators

Oscillations. --- Cycles --- Fluctuations (Physics) --- Vibration --- Nonlinear equations. --- approximate solutions. --- global asymptotics. --- small parameter.

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This Special Edition contains new results on Differential and Integral Equations and Systems, covering higher-order Initial and Boundary Value Problems, fractional differential and integral equations and applications, non-local optimal control, inverse, and higher-order nonlinear boundary value problems, distributional solutions in the form of a finite series of the Dirac delta function and its derivatives, asymptotic properties’ oscillatory theory for neutral nonlinear differential equations, the existence of extremal solutions via monotone iterative techniques, predator–prey interaction via fractional-order models, among others. Our main goal is not only to show new trends in this field but also to showcase and provide new methods and techniques that can lead to future research.

Research & information: general --- Mathematics & science --- fourth-order differential equations --- neutral delay --- oscillation --- ψ-Caputo fractional derivative --- Cauchy problem extremal solutions --- monotone iterative technique --- upper and lower solutions --- third-order differential equation --- boundary value problem --- existence --- sign conditions --- mixed type nonlinear equation --- hilfer operator --- mittag–leffler function --- spectral parameter --- solvability --- equations of the pseudo-elliptic type of third order --- energy estimate --- analog of the Saint-Venant principle --- even-order differential equations --- Dirac delta function --- distributional solution --- Laplace transform --- power series solution --- integro-differential equation --- mixed type equation --- small parameter --- spectral parameters --- Caputo operators of different fractional orders --- inverse problem --- one value solvability --- Rosenzweig–MacArthur model --- fractional derivatives --- threshold harvesting --- distributed-order fractional calculus --- basic optimal control problem --- Pontryagin extremals --- fourth-order differential equations --- neutral delay --- oscillation --- ψ-Caputo fractional derivative --- Cauchy problem extremal solutions --- monotone iterative technique --- upper and lower solutions --- third-order differential equation --- boundary value problem --- existence --- sign conditions --- mixed type nonlinear equation --- hilfer operator --- mittag–leffler function --- spectral parameter --- solvability --- equations of the pseudo-elliptic type of third order --- energy estimate --- analog of the Saint-Venant principle --- even-order differential equations --- Dirac delta function --- distributional solution --- Laplace transform --- power series solution --- integro-differential equation --- mixed type equation --- small parameter --- spectral parameters --- Caputo operators of different fractional orders --- inverse problem --- one value solvability --- Rosenzweig–MacArthur model --- fractional derivatives --- threshold harvesting --- distributed-order fractional calculus --- basic optimal control problem --- Pontryagin extremals