Computational Analysis of Nonlinear Models Arising from Van der Waals Fluids and Planck’s Radiation Law
DOI:
https://doi.org/10.53560/PPASA(63-3)723Keywords:
Derivative-free Iterative Methods, Hermite Interpolation, Systems of Nonlinear Equations, Van der Waals Equation, Planck’s Radiation Law, Computational Efficiency, Basins of AttractionAbstract
Nonlinear equations and systems of nonlinear equations arise in many physical and engineering models, such as equations of state for real gases and thermal radiation laws. Newton-type methods for these problems require derivatives or Jacobian matrices, which may be unavailable or expensive to evaluate. This study presents a derivative-free three-step iterative scheme, denoted PS6 (proposed sixth-order scheme), in which the first two steps use first-order divided differences and the derivative in the third step is approximated by a cubic Hermite interpolating polynomial. For scalar equations, PS6 converges with order six and uses four function evaluations per iteration, giving an efficiency index of . The scheme is extended to systems of nonlinear equations without evaluating Jacobian matrices, and its sixth-order convergence is proved for both cases by Taylor expansion. The computational efficiency index of PS6 exceeds those of four existing fourth-, fifth- and sixth-order schemes for every system size. In numerical experiments on models arising from the Van der Waals equation of state, Planck’s radiation law and particle interaction, and on two coupled nonlinear systems, PS6 required the same number of iterations as or fewer iterations than the compared methods, attained the smallest errors and residuals, and had the shortest central processing unit (CPU) time on both systems; the computational order of convergence agreed with the theoretical order. Basins of attraction for three test polynomials show that 71–92% of the initial points in the square [−1, 1] × [−1, 1] converge, mostly within three iterations. PS6 is therefore an accurate and efficient alternative for nonlinear problems in which derivatives are unavailable or costly.
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