134,623 research outputs found
Approximate Solutions of Polynomial Equations
AbstractIn this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x) ∈Q [ x ], with its constant term equal to 0, such that F(x, y) =∑j=0ncj(y−g(x))jfor some rational numbers cj, hence, F(x, g(x) +a) ∈Q for all a∈Q, or there are at most t distinct polynomials g1(x),⋯ , gt(x), t≤n, such that F(x, gi(x)) ∈Q for 1 ≤i≤t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x),⋯ , gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y)
Spectra generated by a confined softcore Coulomb potential
Analytic and approximate solutions for the energy eigenvalues generated by a
confined softcore Coulomb potentials of the form a/(r+\beta) in d>1 dimensions
are constructed. The confinement is effected by linear and harmonic-oscillator
potential terms, and also through `hard confinement' by means of an
impenetrable spherical box. A byproduct of this work is the construction of
polynomial solutions for a number of linear differential equations with
polynomial coefficients, along with the necessary and sufficient conditions for
the existence of such solutions. Very accurate approximate solutions for the
general problem with arbitrary potential parameters are found by use of the
asymptotic iteration method.Comment: 17 pages, 2 figure
Regular polynomial interpolation and approximation of global solutions of linear partial differential equations
We consider regular polynomial interpolation algorithms on recursively
defined sets of interpolation points which approximate global solutions of
arbitrary well-posed systems of linear partial differential equations.
Convergence of the 'limit' of the recursively constructed family of
polynomials to the solution and error estimates are obtained from a priori
estimates for some standard classes of linear partial differential equations,
i.e. elliptic and hyperbolic equations. Another variation of the algorithm
allows to construct polynomial interpolations which preserve systems of linear
partial differential equations at the interpolation points. We show how this
can be applied in order to compute higher order terms of WKB-approximations of
fundamental solutions of a large class of linear parabolic equations. The error
estimates are sensitive to the regularity of the solution. Our method is
compatible with recent developments for solution of higher dimensional partial
differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo,
and has obvious applications to mathematical finance and physics.Comment: 28 page
Localized Method Of Approximate Particular Solutions For Solving Optimal Control Problems Governed By PDES
In this thesis, the method of approximate particular solutions(MAPS) and localized method of approximate particular solutions(LMAPS) with polynomial basis, and radial basis functions are proposed and applied on the optimal control problems(OCPs) governed by partial differential equations(PDEs).
This study proceeds in several steps. First, polynomial basis and radial basis functions are used to globally approximate solutions for the PDEs which have been combined into a single matrix system numerically from the optimality conditions of the OCPs. Secondly, polynomial and radial basis functions are used to locally approximate particular solutions for the same matrix system numerically. We use these approaches to two types of problems, a smooth and singular problem. The first example numerically experiments on a square domain and the second example on an L-shaped disc domain. These approaches are tested and compared. The results show our proposed method for solving optimal control problems governed by partial differential equations works
Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations
We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods
Hybrid Chebyshev Polynomial Scheme for Solving Elliptic Partial Differential Equations
We propose hybrid Chebyshev polynomial scheme (HCPS), which couples the Chebyshev polynomial scheme and the method of fundamental solutions into a single matrix system. This hybrid formulation requires solving only one system of equations and opens up the possibilities for solving a large class of partial differential equations. In this work, we consider various boundary value problems and, in particular, the challenging Cauchy-Navier equation. The solution is approximated by the sum of the particular solution and the homogeneous solution. Chebyshev polynomials are used to approximate a particular solution of the given partial differential equation and the method of fundamental solutions is used to approximate the homogeneous solution. Numerical results show that our proposed approach is efficient, accurate, and stable
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