For modelling molecular electronic and electrochemical boundary value problems (BVPs), we are faced with the solution of Schrodingers equation involving realistic models for potential energy functions. With the exception of a few canonical problems, there are currently no analytical methods available for obtaining closed-form solutions for the electron wavefunctions and their corresponding energy eigenvalues. One of the well-known techniques for obtaining approximate wavefunctions and energy states is the WKB approximation. The main drawbacks of the WKB method are the discontinuities at the so called turning points, where the total energy equals the potential energy and the restriction on exclusively solving for bound state solutions. To overcome these shortcomings, a novel accelerated numerical technique for solving time-independent Schrodingers equation is introduced, with applications to general potential functions. This method is based on the construction of an auxiliary BVP, which mimics significant features of the original BVP and possesses exact solutions. By construction, these solutions constitute a complete set of orthogonal problem-adapted analysing functions. This method provides numerical solutions with no discontinuities at the turning points.