Asymptotic analysis of an ordinary differential equation and non-uniqueness for a semilinear partial differential equation

1986 ◽  
Vol 91 (3) ◽  
pp. 231-245 ◽  
Author(s):  
Fred B. Weissler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Analysis ◽  
Partial Differential ◽  
Semilinear Partial Differential Equation

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals About Some Possible Implementations of the Fractional Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 893 ◽  
Author(s):  
María Pilar Velasco ◽  
David Usero ◽  
Salvador Jiménez ◽  
Luis Vázquez ◽  
José Luis Vázquez-Poletti ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Calculus ◽  
Panoramic View ◽  
Partial Differential ◽  
Dirac Equations ◽  
Basic Equations

We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations.

Adaptive boundary control of a flexible-link flexible-joint manipulator under uncertainties and unknown disturbances

2021 ◽  
pp. 107754632110445
Author(s):  
Jiahao Zhu ◽  
Jian Zhang ◽  
Xiaobin Tang ◽  
Yangjun Pi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Trajectory Tracking ◽  
Sliding Mode ◽  
Flexible Link ◽  
Partial Differential ◽  
Flexible Joint ◽  
Unknown Disturbances ◽  
Flexible Joint Manipulator

In this article, we consider the trajectory tracking and vibration suppression of a flexible-link flexible-joint manipulator under uncertainties and external time-varying unknown disturbances. The coupled ordinary differential equation and partial differential equation model dynamic of the system is presented by employing the Hamilton principle. Using the singular perturbation theory, the dynamic is decomposed into a no-underactuated slow ordinary differential equation and fast partial differential equation subsystem, which solves the problem of the underactuated ordinary differential equation subsystem of the ordinary differential equation and partial differential equation cascade and reduces the analytical complexity. For the slow subsystem, to guarantee the trajectory tracking of the joint, an adaptive global sliding mode controller without gain overestimation is designed, which can guarantee the global stability of the slow system and reduce the chattering of the sliding mode control. For the fast subsystem, an adaptive boundary controller is developed to suppress the elastic vibration of the flexible link during the trajectory tracking. The stability of the whole closed-loop system is rigorously proved via the Lyapunov analysis method. Simulation results show the effectiveness of the proposed controller.

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