An efficient pseudospectral method for numerical solution of nonlinear singular initial and boundary value problems arising in astrophysics

Abstract

In the manuscript, a pseudospectral method is developed for approximate and efficient solution of nonlinear singular Lane-Emden-Fowler initial and boundary value problems arising in astrophysics. In the proposed method, the Gauss pseudospectral method is utilized to reduce the problem to the solution of a system of algebraic equations. Furthermore, the Gauss pseudospectral method is developed for finding the first zero of the solution of this equation that gives the radius of the star, in which the numerous properties of the star such as mass, central pressure, and binding energy can be computed through their relations to this solution. The main advantage of the proposed method is that good results are obtained even by using a small number of discretization points and the rate of convergence is high. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

Direct pseudo-spectral method for optimal control of obstacle problem - an optimal control problem governed by elliptic variational inequality

“…In problem (24), the decision variables are grouped in the matrices Y and U. Traditionally, we prefer the decision variables to be placed in a vector.…”

“…where the decision variables are the elements of matrices Y and U. By looking at the constraints of problem (24), we see that the constraints (24b) and (24c) enforce non-positivity and the constraints (24d) enforce that the right-hand sides of the non-positive constraints (24b) and (24b) are complementary to each other. This type of constraints is called complementarity constraints [39].…”

“…The PS approaches were initially applied to the problems in fluid dynamics [23]. Also, the applications of PS methods to solve engineering problems [24][25][26] and optimal control problems [27][28][29] have become popular because of their computational feasibility and efficiency. Some varieties of PS methods are implemented successfully for optimal control problems [27,30] even for non-smooth [31] and bang-bang [32] problems.…”

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

“…60 In our simulations, we continued the iteration of fsolve until TolFun and TolX become less than 10 −12 . Using this solver, we can adjust the accuracy of the solution based on the two parameters TolFun and TolX, where the former specifies the termination tolerance on the function value and the later specifies the termination tolerance on the variables.…”

“…Using this solver, we can adjust the accuracy of the solution based on the two parameters TolFun and TolX, where the former specifies the termination tolerance on the function value and the later specifies the termination tolerance on the variables. 60 In our simulations, we continued the iteration of fsolve until TolFun and TolX become less than 10 −12 .…”