This book is about the simulation and modeling of novel chaotic systems within the frame of fractal-fractional operators. The methods used, their convergence, stability, and error analysis are given, and this is the first book to offer mathematical modeling and simulations of chaotic problems with a wide range of fractal-fractional operators, to find solutions. Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling provides details for stability, convergence, and analysis along with numerical methods and their solution procedures for fractal-fractional operators. The book offers applications to chaotic problems and simulations using multiple fractal-fractional operators and concentrates on models that display chaos. The book details how these systems can be predictable for a while and then can appear to become random. Practitioners, engineers, researchers, and senior undergraduate and graduate students from mathematics and engineering disciplines will find this book of interest._

The study was conducted on the wheat crop for the management of aphids in the research area of University of Agriculture, Faisalabad during 2009. Fedding efficacy of various larval instars and adult of C. Septempunctata L. was studied under laboratory conditions of Biodiversity and Biosystematics Lab. Department of Agri- Entomology university of Agriculture, Faisalabad. The results revealed that the fourth larval of C. Septempunctata L. consumed more aphids (36.67) followed by third (30.32), second (16.47) and first instars (2.10) respectively. Whereas the adult of C. Septempunctata L. was found very much voracious and consumed 74.07 aphids. All the insecticides were statistically equal, in controlling the population of aphids. However Advantage caused more mortality followed by Acelan, Imidacloprid, Black Warent and Rigorous in field conditions. All the insecticides were also tested against aphid in controlled conditions of labs. The effects of these insecticides were statistically equal. However However Advantage caused more mortality followed by Acelan, Imidacloprid, Black Warent and Rigorous in field conditions.