Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

Abstract

The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

“…Since it is generally difficult to find closed-form solutions for BVPs, many researchers have attempted to develop methods to find approximate or numerical solutions for BVPs. Well-known methods involve the shooting method [1], finite difference methods [2][3][4] and spectral methods [5][6][7]. In some real-life situations, the shooting method produces numerically sensitive systems of algebraic equations, which must be solved using other numerical methods [8].…”