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We introduce a new family of iterative methods for solving mathematical models whose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.

In this paper we study the geometry of relative superior Mandelbrot sets through S-iteration scheme. Our results are quit significant from other Mandelbrot sets existing in the literature. Besides this, we also observe that S-iteration scheme converges faster than Ishikawa iteration scheme. We believe that the results of this paper can be inspired those who are interested in creating automatically aesthetic patterns.

Let be a nonempty closed convex subset of a real Banach space , let : → be nonexpansive, and let : → be Lipschitz strongly pseudocontractive mappings such that ∈ ( ) ∩ ( ) = { ∈ : = = } and − ≤ − and − ≤ − for all , ∈ . Let { } be a sequence in [0, 1] satisfying (i) ∑ ∞ =1 = ∞; (ii) lim → ∞ = 0. For arbitrary 0 ∈ , let { } be a sequence iteratively defined by = , = (1 − ) −1 + , ≥ 1. Then the sequence { } converges strongly to a common fixed point p of S and T.

In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method. We study the convergence analysis and also present various numerical examples to demonstrate the validity and efficiency of our methods.