In this article, the numerical solutions for the flow of heat transfer for an incompressible Maxwell fluid on a stretching sheet channel are presented in this study. By applying appropriate transformations, the system of governing partial differential equations is transformed into a system of ordinary differential equations. A successive linearization method (SLM) is used to describe and solve the resulting nonlinear equations numerically using MATLAB software. The main goal of this paper is to compare the results of solving the velocity and temperature equations in the presence of β1 changes through SLM for introducing it as a precise and appropriate method for solving nonlinear differential equations. Tables with the numerical results are created for comparison. This contrast is important because it shows how precisely the successive linearization method can resolve a set of nonlinear differential equations. Non-Newtonian parameters on the flow field, like mixed convection, Hartman, Deborah, and Prandtl numbers, are explored and illustrated graphically. Apart from that, a great deal of agreement has been seen between the current results and the published data that have been evaluated and compared in a limited way.