A Homotopy Method with Adaptive Basis Selection for Computing Multiple Solutions of Differential Equations

2020 ◽  
Vol 82 (1) ◽  
Author(s):  
Wenrui Hao ◽  
Jan Hesthaven ◽  
Guang Lin ◽  
Bin Zheng
Keyword(s):  
Differential Equations ◽  
Multiple Solutions ◽  
Homotopy Method ◽  
Selection For ◽  
Adaptive Basis Selection

Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Yiliang Liu ◽  
Liang Lu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Nonlinear Boundary Conditions ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

AbstractIn this paper, we deal with multiple solutions of fractional differential equations with p-Laplacian operator and nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions. An example is given to illustrate our results.

On the use of ‘arc length’ and ‘defect’ for mesh selection for differential equations

2005 ◽  
Vol 1 (2) ◽  
pp. 47-52 ◽  
Author(s):  
W. H. Enright
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Method Of Lines ◽  
Arc Length ◽  
Mesh Selection ◽  
Mixed Order ◽  
Selection For

In this note we will consider mesh selection techniques for ordinary differential equations (ODEs) and for partial differential equations (PDEs) based on arc length and defect. We assume we are trying to approximate y(x) for x ϵ [0; T] in the ODE case and u(x; y) for (x; y) ϵ Ω in the PDE case. The two specific areas of application that we have in mind are the numerical solution of boundary value problems (BVPs) in ODEs and the numerical solution of PDEs when a method of lines (MOL) approach is employed. The approach we develop is applicable to a wider class of PDEs including mixed-order problems and 3D problems.

Forward and Inverse Position Kinematics for the RRSSR Parallel Robot With Hardware Validation

2016 ◽  
Author(s):  
Robert L. Williams ◽  
Ryan Lucas ◽  
J. Jim Zhu
Keyword(s):  
Analytical Solutions ◽  
Multiple Solutions ◽  
Parallel Robot ◽  
Ohio University ◽  
Hip Joints ◽  
Single Loop ◽  
Rigid Link ◽  
Simulation Control ◽  
Selection For ◽  
Active Joints

This paper presents forward and inverse position kinematics equations and analytical solutions for the 2-dof RRSSR Parallel Robot. Two ground-mounted perpendicular offset revolute (R) joints are actuated via servomotors, and the single-loop parallel robot consists of passive R-S-S (revolute-spherical-spherical) joints in between the active joints. A study of the multiple solutions in each case is presented, including means to select the appropriate solutions. This rigid-link parallel robot forms the hip joints of the Ohio University RoboCat walking quadruped. The methods of this paper are suitable to assist in design, simulation, control, and gait selection for the quadruped. RoboCat hardware has been built and used to help validate the examples and results of this paper.

A Particle Swarm Optimization Algorithm and its Application in Hydrodynamic Equations

2012 ◽  
Vol 510 ◽  
pp. 472-477
Author(s):  
Jian Hui Zhou ◽  
Shu Zhong Zhao ◽  
Li Xi Yue ◽  
Yan Nan Lu ◽  
Xin Yi Si
Keyword(s):  
Parameter Estimation ◽  
Particle Swarm Optimization ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Ordinary Differential Equations ◽  
Optimization Algorithm ◽  
Multiple Solutions ◽  
Particle Swarm Optimization Algorithm ◽  
Swarm Optimization ◽  
Estimation Problems

In fluid mechanics, how to solve multiple solutions in ordinary differential equations is always a concerned and difficult problem. A particle swarm optimization algorithm combining with the direct search method (DSPO) is proposed for solving the parameter estimation problems of the multiple solutions in fluid mechanics. This algorithm has improved greatly in precision and the success rate. In this paper, multiple solutions can be found through changing accuracy and search coverage and multi-iterations of computer. Parameter estimation problems of the multiple solutions of ordinary differential equations are calculated, and the result has great accuracy and this method is practical.

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