Numerical Approximation of Variational Problems

2017 ◽  
pp. 115-135
Author(s):  
Pablo Pedregal
Keyword(s):  
Numerical Approximation ◽  
Variational Problems

scholarly journals Direct method for solution variational problems by using Hermite polynomials

2021 ◽  
Vol 39 (6) ◽  
pp. 223-237 ◽  
Author(s):  
Ayatollah Yari ◽  
Mirkamal Mirnia
Keyword(s):  
Optimal Control ◽  
Numerical Approximation ◽  
Optimal Control Problems ◽  
Direct Method ◽  
Hermite Polynomials ◽  
Variational Problems ◽  
Computational Method ◽  
Control Problems ◽  
Operational Matrices

‎In this approach‎, ‎one‎ computational method is presented for numerical approximation of variational problems‎. ‎This method with variable ‎coeffici‎ents is based on Hermite polynomials‎. ‎The properties of Hermite polynomials with the operational matrices of derivative and integration are used to reduce optimal control problems to the solution of linear algebraic equations‎. ‎Illustrative examples are included to demonstrate the validity and applicability of the technique‎.  

Exponentially Accurate Rayleigh–Ritz Method for Fractional Variational Problems

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Dongling Wang ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Exponential Decay ◽  
Numerical Approximation ◽  
Exponential Convergence ◽  
Ritz Method ◽  
Variational Problems ◽  
Basis Functions ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Fractional Variational Problems ◽  
Sturm Liouville

A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

Numerical approximation of non-homogeneous, non-convex vector variational problems

2001 ◽  
Vol 89 (3) ◽  
pp. 425-444 ◽  
Author(s):  
Ernesto Aranda ◽  
Pablo Pedregal
Keyword(s):  
Numerical Approximation ◽  
Variational Problems

Numerical approximation of young measuresin non-convex variational problems

2000 ◽  
Vol 84 (3) ◽  
pp. 395-415 ◽  
Author(s):  
Carsten Carstensen ◽  
Tomáš Roubíček
Keyword(s):  
Numerical Approximation ◽  
Variational Problems ◽  
Convex Variational Problems

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

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