Numerical algorithms for non-linear singularly perturbed initial jumped problems

This study addresses singularly perturbed nonlinear initial-jump problems. The examined problem is first linearized using the quasilinearization method, followed by an asymptotic analysis of the linearized problem, demonstrating the solution's dependence on initial data. Subsequently, difference schemes are constructed on adaptive grids using linear basis functions, with quadrature rules applied to the residual term in integral form. The stability analysis and error estimation of the difference scheme are performed. The convergence rate of the numerical solution to the exact solution of the continuous problem is determined, and computer codes are implemented for a sample problem.