Narrow Results

This paper develops optimal family of fourth-order iterative techniques in order to find a single root and to generalize them for simultaneous finding of all roots of polynomial equation. Convergence study reveals that for single root finding methods, its optimal convergence order is 4, while for simultaneous methods, it is 12. Computational cost and numerical illustrations demonstrate that the newly developed family of methods outperformed the previous methods available in the literature.

Based on the finite volume method (FVM), a numerical scheme is constructed to simulate the unsteady convection–diffusion transport problem. New expressions are obtained for interface approximation of the field variable, subsequently, these newly obtained interface expressions are used to develop the numerical scheme. Convection-dominant and diffusion-dominant phenomena are simulated by taking different values of convective velocity $(V)$ and diffusion coefficient $(k)$. This newly proposed numerical scheme gives second order of convergence along space and time. Experiments are carried out to test the new proposed upwind approach. Numerical results produced by the proposed approach are compared with the conventional finite volume method, step-wise approach FVM and quadratic upwind interpolation finite volume approach. This comparative study indicates that for different cases for convection-dominant and diffusion-dominant problems, our proposed approach gives highly accurate and stable solution. The conventional finite volume method and other approaches result solution with non-physical oscillations. Our obtained numerical results are consistent and support our theoretical approach.

The local as well as the semi-local convergence analysis is provided for two compositions for solving Banach space valued operator nonlinear equations. These compositions are defined on the real line. They were shown to be efficient and of convergence order six. But, the convergence in the local convergence case utilized assumptions reaching the seventh derivative not on the composition. Moreover, no computable error estimates on the distances or uniqueness of the solution regions were provided, limiting the applicability of these compositions. The new convergence analysis is using conditions only on the operators on these compositions. Moreover, computable error estimates and uniqueness results are developed based on $w$-continuity and in the more general setting of Banach space valued operators. Numerical applications are presented to validate the theoretical aspects.

We extend the applicability of a novel seventh-order method for solving nonlinear equations in the setting of Banach spaces. This is done by using assumptions only on the first derivative that does appear on the method, whereas in earlier works up to the eighth derivatives (not on the scheme) were used to establish the convergence. Our technique is so general that it can be used to extend the usage of other schemes along the same lines.