ON EMBEDDING FAMILY OF NUMERICAL SCHEME FOR SOLVING NON-LINEAR EQUATIONS WITH ENGINEERING APPLICATIONS - Shams, M.; Kausar, N.; Ozbilge, E.; Ozbilge, E. (TURKIC WORLD MATHEMATICAL SOC, 2024)
The solution of non-linear equations is one of the most important and frequent problems in numerous engineering and scientific disciplines. Numerous real-world issues can be described using non-linear equations in a variety of scientific fields, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Iterative techniques must be used in order to solve such nonlinear problems. The majority of numerical methods required the first or higher derivative of the functions, whose computational cost is large and diverges if the slope of the functions at the beginning or some intermediate points approaches zero. To prevent this, we develop numerical methods that utilize the parameter embedding, also known as Homotopy methods, to find the root of nonlinear equations. Convergence analysis shows that the proposed family of methods' order of convergence is two. To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real- world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical methods already existing in literature.
Keywords: Embedding parameter; Roots; Homotopy; Dynamical Plane; Convergence order; ITERATIVE METHODS; CONVERGENCE; Mathematics