Search results for "Solutions"
showing 10 items of 757 documents
Types and Multiplicity of Solutions to Sturm–Liouville Boundary Value Problem
2015
We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.
Nonlocal Third Order Boundary Value Problems with Solutions that Change Sign
2014
We investigate the existence and the number of solutions for a third order boundary value problem with nonlocal boundary conditions in connection with the oscillatory behavior of solutions. The combination of the shooting method and scaling method is used in the proofs of our main results. Examples are included to illustrate the results.
A Simple Indicator Based Evolutionary Algorithm for Set-Based Minmax Robustness
2018
For multiobjective optimization problems with uncertain parameters in the objective functions, different variants of minmax robustness concepts have been defined in the literature. The idea of minmax robustness is to optimize in the worst case such that the solutions have the best objective function values even when the worst case happens. However, the computation of the minmax robust Pareto optimal solutions remains challenging. This paper proposes a simple indicator based evolutionary algorithm for robustness (SIBEA-R) to address this challenge by computing a set of non-dominated set-based minmax robust solutions. In SIBEA-R, we consider the set of objective function values in the worst c…
Error Estimates for a Class of Elliptic Optimal Control Problems
2016
In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible …
Analytic solutions of the diffusion-deposition equation for fluids heavir than atmospheric air
2008
A steady-state bi-dimensional turbulent diffusion equation was studied to find the concentration distribution of a pollutant near the ground. We have considered the air pollutant emitted from an elevated point source in the lower atmosphere in adiabatic conditions. The wind velocity and diffusion coefficient are given by power laws. We have found analytical solutions using or the Lie Group Analysis or the Method of Separation of Variables. The classical diffusion equation has been modified introducing the falling term with non-zero deposition velocity. Analytical solutions are essential to test numerical models for the great difficulty in validating with experiments.
An Interactive Evolutionary Multiobjective Optimization Method: Interactive WASF-GA
2015
In this paper, we describe an interactive evolutionary algorithm called Interactive WASF-GA to solve multiobjective optimization problems. This algorithm is based on a preference-based evolutionary multiobjective optimization algorithm called WASF-GA. In Interactive WASF-GA, a decision maker (DM) provides preference information at each iteration simple as a reference point consisting of desirable objective function values and the number of solutions to be compared. Using this information, the desired number of solutions are generated to represent the region of interest of the Pareto optimal front associated to the reference point given. Interactive WASF-GA implies a much lower computational…
Symmetric locally free resolutions and rationality problems
2022
We show that the birationality class of a quadric surface bundle over $\mathbb{P}^2$ is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over $\mathbb{P}^2$ with discriminant curves of fixed degree. In particular, we construct explicit models of these bundles for some discriminant data. Among others, we obtain various birational models of a nodal Gushel-Mukai fourfold, as well as of a cubic fourfold containing a plane. Finally, we prove stable irrationality of several types of quadric surface bundles.
Gradient regularity for elliptic equations in the Heisenberg group
2009
Abstract We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485–544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C 1 , α -regularity for p-harmonic functions in the Heisenberg group for …
Discontinuous solutions of linear, degenerate elliptic equations
2008
Abstract We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.
Numerical study of the transverse stability of the Peregrine solution
2020
We generalise a previously published approach based on a multi-domain spectral method on the whole real line in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. It is shown that the Peregrine solution is unstable against all…