Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF1DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament.

A phase-fitting technique can be used to eliminate the phase-lag and its first, second, third, and fourth derivatives. The new method is known as the economical method because it employs the highest possible algebraic order ( AOR ) with the fewest possible function evaluations ( FEvs ). The one-of-a-kind approach can be stated as the equation PF 4 DPFN 142 SPS . The proposed method is an infinitely periodic P-Stable approach. Many problems having periodic and/or oscillating solutions are susceptible to the proposed technique. This innovative method was utilized to solve the tough problem of Schrödinger-type coupled differential equations in quantum chemistry. Because the new approach only costs 5 FEvs to complete each stage, we call it an economic algorithm . This allows us to achieve a 14 AOR , which is a big improvement over the current situation.

The phase lag and its first, second, third, fourth, and fifth derivatives are all accounted for in a phase-fitting method. The new system uses the fewest possible function evaluations per integration step ( FEvs ) to achieve the highest possible algebraic order ( AOR ) and is hence P-stable. PF 5 DPFN 2 SPS is the symbol for the new method. The proposed method can be used for many different problems with periodic and/or oscillating solutions. We used the unique approach to the well-known problem in quantum chemistry posed by Schrödinger-type coupled differential equations. The new method is part of the economic algorithms since it uses 5 FEvs at each stage to achieve a 12 AOR .

Using a phase-fitting method can get rid of the phase lag together with its first and second derivatives. The new technique has been dubbed the economical method since it uses the maximum algebraic order (AOR) while needing the minimum number of function evaluations (FEvs). This original strategy is represented by the formula PF2DPFN142SPS. The proposed approach is the endlessly periodic P-Stable method. The proposed procedure can be applied to many problems where periodic and/or oscillating solutions are present. The challenging problem of Schrödinger-type coupled differential equations in quantum chemistry was addressed by adopting this unique approach. The new strategy is called an economic algorithm since a 14AOR can be achieved with only a 5FEvs at each stage.

A phase-fitting, first and second derivatives phase-fitting method is produced. The new algorithm is singularly P-Stable and belongs to the economic algorithms. The new method is symbolized as PF2DPFN2SPS. It can be used to any problem with periodical and/or oscillating solutions. We chosen to be applied to a well known problem of Quantum Chemistry. The new scheme is an economic one because 5 function evaluations per step are used in order an algebraic order (AOR) of 12 to be achieved.