Variable Mesh Finite Difference Method for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

Abstract

A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameteruniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.Mathematics subject classification: 45Exx, 65L10, 65L12, 65L50, 65L70.

Fourth-order stable central difference method for self-adjoint singular perturbation problems

“…Tables 1, 2 and 4 clearly indicate that the proposed scheme is more efficient than the methods given in (Patidar and Kadalbajoo, 2003;;Kumar and Kadalbajoo, 2008;Kadalbajoo and Kumar, 2010). Table 3 (Table 5).…”

“…Detailed discussions on the theory of asymptotical and numerical solutions of singular perturbation problems have been published (Boglave, 1981;Kadalbajoo and Kumar, 2008;Mishra et al, 2009;Gupta and Pankaj, 2011). So, the treatment of singularly perturbed problems presents severe difficulties that have to be addressed to ensure accurate numerical solutions (Roos et al, 1996;Kadalbajoo and Kumar, 2010). …”

Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems

“…If we take h ≥ ε, the existing classical numerical methods produce oscillatory solution and pollute the solution in the entire interval, because of boundary layer behavior. In connection to this, there are some numerical methods suggested by various authors for solving self-adjoint singular perturbation problems, namely, initial value technique [5], quintic spline method [6], nonpolynomial spline functions method [7], difference scheme using cubic spline [8], finite difference method with variable mesh [9], fitted mesh B-spline collocation method [10], higher order numerical methods [11].…”