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Full-Text Articles in Special Functions
Numerical Methods For Nonlinear Problems Using The Finite Element Method, Logan Price
Numerical Methods For Nonlinear Problems Using The Finite Element Method, Logan Price
Discovery Day - Daytona Beach
Numerical Methods for Nonlinear Problems Using the Finite Element Method is a computational mathematics capstone that builds and tests finite element method (FEM) workflows for nonlinear partial differential equations in FreeFEM++, with ParaView used for visualization. Two nonlinear model problems are used to demonstrate the approach. The first is a semilinear reaction-diffusion equation with a cubic nonlinearity. A manufactured solution is used so accuracy can be checked at a fixed final time, and refinement studies in both time step and mesh size are run while nonlinear iteration counts are tracked to show solver effort. The second problem is the steady …
Analytical And Numerical Solutions For The Hydrogen Atom, Kassidy Myers
Analytical And Numerical Solutions For The Hydrogen Atom, Kassidy Myers
Discovery Day - Daytona Beach
The Schrödinger equation is the foundational equation of non-relativistic quantum mechanics. The hydrogen atom is the simplest system for solving this equation, as it consists of only one proton and one electron. In this project, we work on the Schrödinger equation that models the spherically symmetric states of the hydrogen atom that depend only on the radial coordinate. We simplified and nondimensionalized the radial equation and solved the resulting equation using a power series (Frobenius) method. This approach revealed the physically meaningful solutions and led to quantized energy levels. In addition to finding the analytical solution, we numerically solve the …
Analytic Treatment Of Vortex States In Cylindrical Superconductors In Applied Axial Magnetic Field, Andrei Ludu, J. Van Deun, M. V, Milosevic, A. Cuyt, F. M. Peeters
Analytic Treatment Of Vortex States In Cylindrical Superconductors In Applied Axial Magnetic Field, Andrei Ludu, J. Van Deun, M. V, Milosevic, A. Cuyt, F. M. Peeters
Publications
We solve the linear Ginzburg–Landau GL equation in the presence of a uniform magnetic field with cylindrical symmetry and we find analytic expressions for the eigenfunctions in terms of the confluent hypergeometric functions. The discrete spectrum results from an implicit equation associated to the boundary conditions and it is resolved in analytic form using the continued fractions formalism. We study the dependence of the spectrum and the eigenfunctions on the sample size and the surface conditions for solid and hollow cylindrical superconductors. Finally, the solutions of the nonlinear GL formalism are constructed as expansions in the linear GL eigenfunction basis …