Initial-value technique for self-adjoint singular perturbation boundary value problems

Abstract

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results.

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

“…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).…”

An Intial-Value Technique for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

“…Mishra et al [17] studied an initial value technique for self adjoint singular perturbation boundary value problems. Natesan and Ramanujam [18] studied initial-value technique for singularly perturbed boundary-value problems for second-order ordinary differential equations arising in chemical reactor theory.…”

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].…”