comparative study of numerical methods for singularly perturbed boundary turning point problems with mixed boundary conditions

A comparative study on numerical methods for singularly perturbed boundary turning point problems (spbtpps) featuring discontinuous source terms are examined. the study involves developing and analyzing two specific numerical techniques: the finite difference method and a hybrid difference method incorporating a shishkin-type mesh. this approach demonstrates notable capabilities, exhibiting almost first-order and second-order convergence for the finite difference and hybrid difference methods, respectively. numerical results are given  to support the theoretical findings.