scholarly journals Solving Nonholonomic Systems with the Tau Method

2019 ◽  
Vol 24 (4) ◽  
pp. 91 ◽  
Author(s):  
Alexandra Gavina ◽  
José M. A. Matos ◽  
Paulo B. Vasconcelos
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Numerical Procedure ◽  
Necessary Optimality Conditions ◽  
Transversality Condition ◽  
Nonholonomic Systems ◽  
Numerical Approach ◽  
Ease Of Use ◽  
Tau Method ◽  
Control Functions

A numerical procedure based on the spectral Tau method to solve nonholonomic systems is provided. Nonholonomic systems are characterized as systems with constraints imposed on the motion. The dynamics is described by a system of differential equations involving control functions and several problems that arise from nonholonomic systems can be formulated as optimal control problems. Applying the Pontryagins maximum principle, the necessary optimality conditions along with the transversality condition, a boundary value problem is obtained. Finally, a numerical approach to tackle the boundary value problem is required. Here we propose the Lanczos spectral Tau method to obtain an approximate solution of these problems exploiting the Tau toolbox software library, which allows for ease of use as well as accurate results.

scholarly journals An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sh. Rezapour ◽  
B. Azzaoui ◽  
B. Tellab ◽  
S. Etemad ◽  
H. P. Masiha
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Caputo Fractional Derivatives ◽  
Control Functions ◽  
Existence Of Positive Solutions ◽  
Semilinear Differential Equation ◽  
The Given ◽  
Existence Criteria

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.

Numerical Approach for Solving Reynolds Equation With JFO Boundary Conditions Incorporating ALE Techniques

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Bernhard Schweizer
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Reynolds Equation ◽  
Boundary Value ◽  
Main Idea ◽  
Numerical Approach ◽  
Arbitrary Lagrangian Eulerian ◽  
Fluid Films ◽  
Ale Formulation ◽  
Hydrodynamic Journal Bearing

Calculating the fluid flow and pressure field in thin fluid films, lubrication theory can be applied, and Reynolds fluid film equation has to be solved. Therefore, boundary conditions have to be formulated. Well-established mass-conserving boundary conditions are the Jakobsson–Floberg–Olsson (JFO) boundary conditions. A number of numerical techniques, which have certain advantages and certain disadvantages, have been developed to solve the Reynolds equation in combination with JFO boundary conditions. In the current paper, a further method is outlined, which may be a useful alternative to well-known techniques. The main idea is to rewrite the boundary value problem consisting of the Reynolds equation and the JFO boundary conditions as an arbitrary Lagrangian–Eulerian (ALE) problem. In the following, an ALE formulation of the Reynolds equation with JFO boundary conditions is derived. Based on a finite element implementation of the governing boundary value problem, numerical examples are presented, and pressure fields are calculated for a plain hydrodynamic journal bearing with an axial oil groove.

Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation

2019 ◽  
Vol 27 (5) ◽  
pp. 745-758 ◽  
Author(s):  
Dmitry V. Lukyanenko ◽  
Maxim A. Shishlenin ◽  
Vladimir T. Volkov
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Analysis ◽  
Boundary Value ◽  
Numerical Approach ◽  
Singularly Perturbed ◽  
Advection Equation ◽  
Unknown Boundary ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem ◽  
Time Periodic

Abstract In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method.

A new approach for the optimal control of time-varying delay systems with external persistent matched disturbances

2017 ◽  
Vol 24 (19) ◽  
pp. 4505-4512 ◽  
Author(s):  
Amin Jajarmi ◽  
Mojtaba Hajipour ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Necessary Optimality Conditions ◽  
Delay Systems ◽  
Time Varying ◽  
Time Varying Delay ◽  
Augmented System ◽  
Varying Delay

The aim of this study is to develop an efficient iterative approach for solving a class of time-delay optimal control problems with time-varying delay and external persistent disturbances. By using the internal model principle, the original time-delay model with disturbance is first converted into an augmented system without any disturbance. Then, we select a quadratic performance index for the augmented system to form an undisturbed time-delay optimal control problem. The necessary optimality conditions are then derived in terms of a two-point boundary value problem involving advance and delay arguments. Finally, a fast iterative algorithm is designed for the latter advance-delay boundary value problem. The convergence of the new iterative technique is also investigated. Numerical simulations verify that the proposed approach is efficient and provides satisfactory results.

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