Browsing by Author "Soomro, Amanullah"

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Citation - WoS: 3
Citation - Scopus: 5
Variable Stepsize Construction of a Two-Step Optimized Hybrid Block Method With Relative Stability
(de Gruyter Poland Sp Z O O, 2022) Baleanu, Dumitru; Qureshi, Sania; Soomro, Amanullah; Shaikh, Asif Ali
Several numerical techniques for solving initial value problems arise in physical and natural sciences. In many cases, these problems require numerical treatment to achieve the required solution. However, in today's modern era, numerical algorithms must be cost-effective with suitable convergence and stability features. At least the fifth-order convergent two-step optimized hybrid block method recently proposed in the literature is formulated in this research work with its variable stepsize approach for numerically solving first- and higher-order initial-value problems in ordinary differential equations. It has been constructed using a continuous approximation achieved through interpolation and collocation techniques at two intra-step points chosen by optimizing the local truncation errors of the main formulae. The theoretical analysis, including order stars for the relative stability, is considered. Both fixed and variable stepsize approaches are presented to observe the superiority of the latter approach. When tested on challenging differential systems, the method gives better accuracy, as revealed by the efficiency plots and the error distribution tables, including the machine time measured in seconds.
Citation - WoS: 17
Citation - Scopus: 19
A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior
(Mdpi, 2021) Qureshi, Sania; Soomro, Amanullah; Hincal, Evren; Baleanu, Dumitru; Shaikh, Asif Ali; Tassaddiq, Asifa
There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.