NUMERICAL AND MACHINE LEARNING APPROACH FOR SINGULARLY PERTURBED PROBLEMS WITH APPLICATION TO CYCLONE MODELING

Abstract

This thesis contributes to numerical methods and Machine learning approach for singularly perturbed differential and its application in cyclone modeling. The pur- pose of this research is to approximate singularly perturbed differential equations using approximation methods. We investigate, develop, and analyze numerical methods, as well as their implementations, for such challenging problems. The chapter-by-chapter structure of the thesis is as follows. Chapter 1 is introductory in nature and gives a brief review of Singularly Per- turbed Problems (SPPs) and a survey on numerical analysis of Singularly Per- turbed Differential equations (SPDEs) and numerical techniques. Objectives, lit- erature survey, and a brief summary of the present work are also included in this chapter. Chapter 2 is about the numerical collocation methods based on Bernstein collo- cation method using equispaced nodes and Chebyshev-Gauss-Lobatto Nodes for solving linear and non linear singularly perturbed differntial equations, and results are then compared with traditional methods and error analysis is carried out. In chapter 3, is devoted to the Szasz Mirakyan operator . In this chapter we solved both linear and non linear singularly perturbed different equations and compared with traditional Methods Stability Analysis and error analysis is car- ried out. In chapter 4, This chapter presents a novel framework combining the Bernstein- Chlodowsky operator with neural networks termed the Bernstein-Chlodowsky Neu- ral Network (BNN) and the Bernstein-Chlodowsky collocation method (CCM) for solving singularly perturbed differential equations (SPDEs) in the context of at- mospheric science and cyclone modeling. We solve linear and nonlinear SPDEs using the Bernstein-Chlodowsky collocation method and BNN, emphasizing how well these methods capture the convection dominated processes observed in cy- clonic systems. The superior convergence of the operator and the resolution of the boundary layer make it an effective tool for cyclone modeling. Chapter 5, This chapter focuses on predicting the Accumulated Cyclone En- ergy (ACE) in the North Indian Ocean (NIO) during monsoon using an optimized Artificial Neural Network (ANN) model. The permutation feature is essential in finding the most influential features to improve model performance Chapter 6, This chapter aims to forecast, using yearly cyclone data, the Accu- mulated Cyclone Energy (ACE) values for the North Indian Ocean (NIO) area. We study a predictive framework for estimating Accumulated Cyclone Energy (ACE) in the North Indian Ocean (NIO) using historical cyclone data from 1982 to 2023, encompassing the Arabian Sea (AS) and Bay of Bengal (BOB) basins. Interpo- lation was performed using the Szász-Mirakyan operator, which preserves the statistical properties of the underlying distribution while reconstructing missing and zero values more effectively than conventional methods. XGBoost and neu- ral networks are two machine learning methods compared in the study. Chapter 7 Finally, this Chapter is devoted to conclusion of the study and discus- sion on some future directions of the current research work. Finally, the bibliography and list of author’s publications have been given at the end of the thesis.