The destruction of spherical symmetry in non-linear variational problems

1986 ◽  
pp. 249-275 ◽  
Author(s):  
Yu. I. Sapronov
Keyword(s):  
Spherical Symmetry ◽  
Variational Problems ◽  
Non Linear

scholarly journals Condensations and Cavities

1983 ◽  
Vol 104 ◽  
pp. 217-217
Author(s):  
F. Occhionero ◽  
P. Santangelo ◽  
N. Vittorio
Keyword(s):  
Spherical Symmetry ◽  
Linear Growth ◽  
Non Linear ◽  
Unified Algorithm ◽  
Hubble Flow

We present a unified algorithm which describes the non-linear growth 1) of condensations surrounded by cavities or 2) of cavities surrounded by condensations (i.e. ridges of higher density) in the Hubble flow. The main idealization is that of pressureless spherical symmetry (Tolman-Bondi solution); overall algebraic details and results for problem 1) are given in previous work (Occhionero, et al., 1981 a and b); results for problem 2) will be given elsewhere (Occhionero, et al., 1982).

scholarly journals Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 224 ◽  
Author(s):  
Harendra Singh ◽  
Rajesh Pandey ◽  
Hari Srivastava
Keyword(s):  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Variational Problems ◽  
Algebraic Equations ◽  
The Third ◽  
Fractional Variational Problems ◽  
Nonlinear Algebraic Equations ◽  
Non Linear ◽  
Fourth Kind

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

Uniqueness theorems and the maximum-minimum principle for a type of non-linear partial differential equations

1938 ◽  
Vol 34 (4) ◽  
pp. 527-533
Author(s):  
W. H. J. Fuchs ◽  
P. Weiss
Keyword(s):  
Differential Equations ◽  
Minimal Surfaces ◽  
Minimum Principle ◽  
Linear Equations ◽  
Variational Problems ◽  
Elliptic Type ◽  
Principle Of Superposition ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problemk = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.

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