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

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