Numerical methods for second-order nonlinear perturbed boundary value problems

Nonlinear problems with singular perturbation are mathematical models that emerge in the fields of mathematics, physics and other scientific disciplines. The inherent difficulties associated with both nonlinearity and singular perturbation often preclude the attainment of an exact solution. For this reason, approximate methods are employed. In light of the rapid changes in solution observed in problems with singular perturbation, it is evident that the derivatives in these regions will also grow to infinity. In classical difference schemes where approximate derivatives are employed in lieu of the exact derivatives, the numerical algorithm is unstable. In this study, difference schemes are established in boundary-solid compatible and piecewise regular networks. Computer codes are written for the established difference schemes and computer support is provided for the solution of the problem. Due to the nonlinear term, each value of each solution in the difference scheme is also calculated approximately with Newton Raphson iteration.