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A Novel Multistep Iterative Technique for Models in Medical Sciences with Complex Dynamics.

Sania Qureshi^1,2, Amanullah Soomro¹, Asif Ali Shaikh¹

¹Mehran University of Engineering and Technology, Department of Basic Sciences and Related Studies, Jamshoro 76062, Pakistan.

Computational and Mathematical Methods in Medicine

|August 8, 2022

Summary

This study introduces a novel iterative method for solving nonlinear equations in medical science, offering faster convergence. The technique efficiently solves complex medical models, visualizing results with polynomiographs.

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Area of Science:

Numerical analysis
Applied mathematics
Biomedical modeling

Background:

Nonlinear equations are prevalent in various scientific fields, including medicine.
Existing iterative methods for solving these equations have limitations in convergence speed and efficiency.

Purpose of the Study:

To develop and present a novel, efficient three-step iterative technique for solving nonlinear equations.
To enhance the convergence rate compared to existing methods.
To apply the technique to models in medical science.

Main Methods:

A three-step iterative technique was developed by combining Newton's method with an existing two-step method.
The method requires three function and two first derivative evaluations per iteration.
The technique was applied to scalar and vector models in population growth, blood rheology, and neurophysiology.

Main Results:

The proposed method demonstrates faster convergence than many existing techniques.
Successful application to diverse medical science models, including population dynamics and physiological processes.
Complex-valued polynomials were visualized using polynomiographs to illustrate convergence basins.

Conclusions:

The novel iterative technique provides an efficient and faster approach for solving nonlinear equations in medical science.
The method's effectiveness is validated through its application to various biomedical models.
Polynomiographs offer valuable insights into the convergence behavior of the iterative method.