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Variational Methods
Published in Matthew N.O. Sadiku, Computational Electromagnetics with MATLAB®, 2018
The Rayleigh–Ritz method is the direct variational method for minimizing a given functional. It is direct in that it yields a solution to the variational problem without recourse to the associated differential equation [17]. In other words, it is the direct application of variational principles discussed in the previous sections. The method was first presented by Rayleigh in 1877 and extended by Ritz in 1909. Without loss of generality, let the associated variational principle be I(Φ)=∫SF(x,y,Φ,Φx,Φy)dS
Numerical Methods for Modeling of Nanosystems
Published in Alexander V. Vakhrushev, Computational Multiscale Modeling of Multiphase Nanosystems, 2017
In contrast to the perturbation theory, variational methods for finding solutions of the Schrödinger equation are based on the simplification of the wave function rather than on the simplification of the Hamiltonian of the system. The main idea of the variational method is that any system tends to the state with minimal energy, which, in its turn, depends on the wave function. If we substitute approximated or “trial” functions instead of a wave function (which is unknown a priori) in the integral for the system total energy, that function which will give the minimal value of the energy can be considered to be the solution of the Schrödinger equation in a certain approximation. It is obvious that it is impossible to select such trial function arbitrarily; it is necessary to use optimization methods. Therefore the Schrödinger equation solution is replaced by the search of the extreme points of the functional of the system energy with regard to the variations (the virial) of the trial function. The task indicated above is reduced to two main problems:the selection of the class of trial functions;the selection of such trial function in the selected class, which provides the minimal total energy of the system.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
The finite element method (FEM), like the finite difference method, is a numerical procedure based on discretization of the solution domain. This method overcomes the disadvantages associated with the variational method by providing a systematic procedure for discretizing the solution domain into simply shaped sub-regions, called finite elements. This is followed by deriving the approximate function or solution over each of these elements, using one of the previously mentioned variational method of approximation such as Rayleigh-Ritz or Galerkin method. The total solution over the whole domain is then generated by linking together or assembling the individual element solutions satisfying the continuity at the inter-element boundaries. Hence, the finite element method can be viewed as a piece-wise or element-wise application of the variational methods. Also, a finite element method is named based on the type of variational method being used such as the Rayleigh-Ritz finite element method or the Galerkin finite element method.
Effects of applied magnetic field and pressure on the diamagnetic susceptibility and binding energy of donor impurity in GaAs quantum dot considering the non-parabolicity model’s influence
Published in Philosophical Magazine, 2023
Ibrahim Maouhoubi, Redouane En-nadir, Kamal El bekkari, Izeddine Zorkani, Abdallah Ouazzani Tayebi Hassani, Anouar Jorio
Within the effective-mass approximation, the Hamiltonian for a shallow-donor impurity (D) placed inside of a symmetrical QDisk made out GaAs surrounded by a wide band-gap semiconductor that forces the particle to remain confined inside the GaAs material (infinite potential barrier) is given as follows: where r0 and are the electron-impurity position and relative dielectric constant of the material, respectively. is the magnetic-field potential and is the confining potential barrier in polar coordinates. m* is the effective electron mass given as follows. For a uniform magnetic field, we can write (for a symmetric gauge ), where . The introduction of the impurity into the system makes it impossible to analytically solve the Schrodinger equation, therefore, it is necessary to use the variational method. Equation (1) is given in atomic unit as:
A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation
Published in Journal of Modern Optics, 2018
In this paper, we have shown a possible connection between the semi-inverse variational method and the evaluation of solutions of a given quantum system. Specially, we have introduced the potential of Coulomb as a reference to obtain simultaneously some bound energies and the associated quantum states. Furthermore, we have proposed some trial forms for the solutions for several configurations, and we have demonstrated that the trial Lagrangian build by this approach has provided an exact aspect of solutions that are the same as those obtained by the expansion method [41]. We feel that the semi-inverse variational method as treated here will be useful to study more complex systems such that the Klein–Gordon Hamiltonian. A significant development must be provided in this direction in order to obtain the appropriate solutions. The details will be given elsewhere and will be reported in our future research work.
Investigating reduction of uplift forces by longitudinal drains with underlined canals
Published in ISH Journal of Hydraulic Engineering, 2018
Farzin Salmasi, Rahman Khatibi, Bahram Nourani
In this study, Laplace’s equation is solved numerically using Seep/w (2007) software, by Geo-studio software package, together with using the appropriate boundary conditions. The numerical solution is implemented using the Finite-Element Method (FEM), which is a numerical approach for solving mathematical and engineering problems and their applications often lead to a system of algebraic equations. FEM is an approximation and requires the discretization of points over the solution domain by subdividing it into finite elements and reformulating the equations using variational methods into a system of equation solvable through the matrix techniques.