In this paper, we are concerned about numerical solutions to ODEs and PDEs that are used to model and describe real-life problems. Usually, these equations are approximated numerically because it is convenient and on-hand to be calculated on computing devices. The Burger-Huxley partial differential equations model the interaction between reactions, diffusion, and convection besides other phenomenas in liquid crystals. Numerically, the Burger-Huxley equations have been solved using the quadrature technique, homotopy perturbation, finite-difference, and B-spline quasi-interpolation methods. Many existing numerical methods have ill-conditioned matrices. Our aim is to develop a numerical algorithm based on the use of splines and their derivatives without requiring the solution of the resulting system that might be ill-conditioned. The method will be applied to initial value-boundary value problems. Special technique will be improved for the Burger-Huxley equations. It is anticipated that the approximate solution of the IV-BV problems has significant accuracy and efficiency.