scholarly journals Exact solutions of Mathieu’s equation

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
Derek J. Daniel
Keyword(s):  
Closed Form ◽  
Recurrence Relations ◽  
Periodic Potential ◽  
Analytic Solution ◽  
Research Work ◽  
Approximation Schemes ◽  
Order Ordinary Differential Equation ◽  
Experimental Physics ◽  
Linearly Independent ◽  
Mathieu’S Equation

Abstract Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.

Eulerian Multi-Fluid Model of Air Blast Atomization

2006 ◽  
Author(s):  
Srimani Bhamidipati ◽  
Mahesh Panchagnula ◽  
John Peddieson
Keyword(s):  
Closed Form ◽  
Recurrence Relations ◽  
Model Problem ◽  
Fluid Model ◽  
Closed Form Solution ◽  
Fluid Phase ◽  
Form Solution ◽  
Air Blast ◽  
Carrier Fluid ◽  
Selection Of

The application of fully Eulerian "multi-fluid" models to air blast atomization is discussed. Such models envision the system as consisting one carrier fluid phase and multiple drop phases, each having a discrete size. A model problem is formulated which allows a general closed form solution in terms of recurrence relations. This closed form solution is employed to produce representative results. A selection of these is used to illustrate interesting aspects of the predictions.

An investment model with entry and exit decisions

2000 ◽  
Vol 37 (2) ◽  
pp. 547-559 ◽  
Author(s):  
J. Kate Duckworth ◽  
Mihail Zervos
Keyword(s):  
Dynamic Programming ◽  
Closed Form ◽  
Production Scheduling ◽  
Analytic Solution ◽  
Investment Model ◽  
Programming Approach ◽  
Entry And Exit ◽  
Dynamic Programming Approach ◽  
Exit Decisions ◽  
Parameter Values

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form analytic solution that can take qualitatively different forms depending on parameter values.

scholarly journals Quantum Calcium-Ion Interactions with EEG

Sci ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 7 ◽  
Author(s):  
Lester Ingber
Keyword(s):  
Closed Form ◽  
Path Integral ◽  
Stochastic Systems ◽  
Analytic Solution ◽  
Short Term Memory ◽  
Mathematical Physics ◽  
Modest Improvement ◽  
Classical Physics ◽  
Arbitrary Time ◽  
Eeg Data

Background: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. Objective: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. Conclusions: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

Some Properties of R. G. D. Allen's Treatment of Kalecki's 1935 Model of Business Cycles

2002 ◽  
Vol 24 (4) ◽  
pp. 463-474
Author(s):  
John V. Baxley ◽  
John C. Moorhouse
Keyword(s):  
Business Cycles ◽  
Business Cycle ◽  
Mathematical Models ◽  
Closed Form ◽  
Analytic Solution ◽  
John Maynard Keynes

More than sixty-five years have passed since Michal Kalecki (1935) published one of the first formal mathematical models of business cycles. His paper presents a closed-form analytic solution. This characteristic, among others, sets Kalecki's work apart from that of contemporary literary business cycle theorists such as Friedrich A. Hayek (1935) and John Maynard Keynes (1936).

scholarly journals Closed-Form Analytic Solution of Cloud Dissipation for a Mixed-Layer Model

2017 ◽  
Vol 74 (8) ◽  
pp. 2525-2556 ◽  
Author(s):  
Bengu Ozge Akyurek ◽  
Jan Kleissl
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Mixed Layer ◽  
Analytic Solution ◽  
Cloud Base ◽  
Layer Model ◽  
Stratocumulus Clouds ◽  
Initial And Boundary Conditions ◽  
Cloud Thickness ◽  
Mixed Layer Model

Abstract Stratocumulus clouds play an important role in climate cooling and are hard to predict using global climate and weather forecast models. Thus, previous studies in the literature use observations and numerical simulation tools, such as large-eddy simulation (LES), to solve the governing equations for the evolution of stratocumulus clouds. In contrast to the previous works, this work provides an analytic closed-form solution to the cloud thickness evolution of stratocumulus clouds in a mixed-layer model framework. With a focus on application over coastal lands, the diurnal cycle of cloud thickness and whether or not clouds dissipate are of particular interest. An analytic solution enables the sensitivity analysis of implicitly interdependent variables and extrema analysis of cloud variables that are hard to achieve using numerical solutions. In this work, the sensitivity of inversion height, cloud-base height, and cloud thickness with respect to initial and boundary conditions, such as Bowen ratio, subsidence, surface temperature, and initial inversion height, are studied. A critical initial cloud thickness value that can be dissipated pre- and postsunrise is provided. Furthermore, an extrema analysis is provided to obtain the minima and maxima of the inversion height and cloud thickness within 24 h. The proposed solution is validated against LES results under the same initial and boundary conditions.

scholarly journals DUGDALE-MAUGIS ADHESIVE NORMAL CONTACT OF AXISYMMETRIC POWER-LAW GRADED ELASTIC BODIES

2018 ◽  
Vol 16 (1) ◽  
pp. 9 ◽  
Author(s):  
Emanuel Willert
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Power Law ◽  
Analytic Solution ◽  
Normal Contact ◽  
Elastic Bodies ◽  
Tabor Parameter

A closed-form general analytic solution is presented for the adhesive normal contact of convex axisymmetric power-law graded elastic bodies using a Dugdale-Maugis model for the adhesive stress. The case of spherical contacting bodies is studied in detail. The known JKR- and DMT-limits can be derived from the general solution, whereas the transition between both can be captured introducing a generalized Tabor parameter depending on the material grading. The influence of the Tabor parameter and the material grading is studied.

scholarly journals Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

2018 ◽  
Vol 70 (2) ◽  
Author(s):  
Ivana Kovacic ◽  
Richard Rand ◽  
Si Mohamed Sah
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Geometric Nonlinearity ◽  
Elliptic Type ◽  
Viscous Damping ◽  
Stiffness Coefficient ◽  
Periodic Forcing ◽  
Order Ordinary Differential Equation ◽  
Mathieu’S Equation ◽  
Stability Chart

This work is concerned with Mathieu's equation—a classical differential equation, which has the form of a linear second-order ordinary differential equation (ODE) with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation, or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.

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