104,787 research outputs found
Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations
This paper develops a new framework for designing and analyzing convergent
finite difference methods for approximating both classical and viscosity
solutions of second order fully nonlinear partial differential equations (PDEs)
in 1-D. The goal of the paper is to extend the successful framework of
monotone, consistent, and stable finite difference methods for first order
fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs
such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency,
generalized monotonicity, and stability are introduced; among them, the
generalized monotonicity and consistency, which are easier to verify in
practice, are natural extensions of the corresponding notions of finite
difference methods for first order fully nonlinear Hamilton-Jacobi equations.
The main component of the proposed framework is the concept of "numerical
operator", and the main idea used to design consistent, monotone and stable
finite difference methods is the concept of "numerical moment". These two new
concepts play the same roles as the "numerical Hamiltonian" and the "numerical
viscosity" play in the finite difference framework for first order fully
nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent
and monotone finite difference methods are proposed for second order fully
nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also
are proved to be stable and the second class contains Godunov-like methods.
Numerical results are also presented to gauge the performance of the proposed
finite difference methods and to validate the theoretical results of the paper.Comment: 23 pages, 8 figues, 11 table
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
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