Research Projects

Function approximation and data interpolation are essential in fields such as machine learning, signal processing, numerical analysis, biomedical imaging, time-series forecasting, and financial modeling. Traditional methods like polynomials, splines, and standard interpolation techniques often fail to handle data with local irregularities, noise, discontinuities, or self-similar structures. As real-world datasets frequently exhibit these complexities, there is a growing need for advanced approximation tools capable of modeling such behaviors effectively. Fractal functions, developed via the Iterated Function System (IFS) framework, provide a powerful approach to model non-differentiable, irregular, and self-similar phenomena. Their ability to represent data with intricate, fine-scale patterns and non-integer fractal dimensions makes them valuable for capturing the complexities found in diverse domains such as geophysics, biomedical science, and finance. Fractal interpolation offers a flexible means of dealing with datasets where classical smooth interpolants fall short. In parallel, Radial Basis Functions (RBFs) are widely used for multivariate approximation, particularly in high-dimensional and scattered data scenarios. RBFs are known for their mesh-free structure, rotational invariance, and their reliance on distances between data points. These properties have made them integral to machine learning, kernel methods, and scientific computation. However, standard RBFs often struggle to accurately approximate data characterized by non-smoothness, discontinuities, or self-affinity. To address these challenges, this project proposes the development of Fractal Radial Basis Functions (FRBFs) by embedding fractal features like self-similarity, scale variance, and local irregularity into classical RBFs. This integration will extend traditional models such as Gaussians, multiquadrics, thin-plate splines, inverse quadratics, Cauchy kernels, Wendland functions, Matern kernels, and polyharmonic splines, creating a new class of basis functions capable of handling both smooth global trends and detailed local structures. The project will establish several families of FRBFs with properties including finite-order smoothness, tunable roughness, and fractal dimensionality. Their convergence, stability, approximation strength, and error bounds will be thoroughly investigated. Applications of FRBFs will focus on developing mesh-free numerical methods to solve complex nonlinear differential and integral equations, such as the Burgers’ equation, Sine-Gordon equation, Black-Scholes model, and Degasperis-Processi equation, frequently encountered in fluid dynamics, finance, and quantum mechanics. Additionally, FRBFs will be applied to 3D surface reconstruction and geometric modeling, especially in medical imaging, cultural heritage conservation, geosciences, and materials science, where point cloud data is often noisy or incomplete. Their shape-preserving and multiscale adaptability make them suitable for reconstructing surfaces with high fidelity. In machine learning and AI, FRBFs will serve as advanced kernels for Support Vector Machines, kernel regression, and RBF neural networks. The incorporation of fractal parameters offers improved adaptability to sparse, nonlinear, or self-similar datasets, enhancing classification, prediction accuracy, and robustness. Overall, this research sits at the intersection of pure mathematics, numerical analysis, differential equation, integral equation, machine learning, and data science, contributing both theoretically and practically to the field. The project is poised to advance the mathematical understanding of fractal-based approximations and extend their application to real-world problems characterized by structural complexity and data irregularity. The outcomes will benefit a wide range of professionals, including applied mathematicians, computational scientists, engineers, and AI researchers.