Efficient Numerical Schemes for Fractional Models and Their Applications to Heat Transfer and Nanofluid Flows

Abstract

This thesis presents the development and applications of efficient numerical frameworks for solving fractional partial differential equations (PDEs) and their extension to realistic heat-transfer problems governed by fractional dynamics. The research integrates advanced wavelets based formulations with fast algorithm to overcome the challenges associated with the non-local and memory dependent behavior of fractional operators. In the mathematical component, two computationally efficient solvers are pro- posed, the fast Shifted Legendre Wavelets (SLWs) method and the fast Green–CAS wavelets method. The fast SLWs approach combines the orthogonality of Shifted Legendre Wavelets with the efficient sum-of-exponential (SOE) base approximation of Caputo fractional derivative to accurately and effi- ciently solve two-dimensional, multi-term, and nonlinear fractional PDEs. Convergence analyses and extensive numerical experiments confirm its robustness, demonstrating comparable accuracy but signif- icantly reduced computational cost relative to traditional L1-based and conventional operational matrix based wavelets schemes. The Green–CAS wavelets framework is further enhanced through the introduction of a Green’s- function formulation that completely eliminates the need for operational integration matrices, simpli- fying implementation for boundary value problems. The resulting fast Green–CAS method, designed for time fractional orders α ∈ (0,1), exhibits superior stability and precision. Nonlinear equations are linearized using iterative schemes, and convergence studies verify the reliability of the technique. Nu- merical results reveal that the fast Green–CAS method consistently outperforms classical operational wavelets based and other existing approaches in terms of both efficiency and accuracy. Building upon these methodological advances, the thesis extends fractional modeling to the physical domain of nanofluid convection and radiative heat transfer. A fractional formulation of the Tiwari-Das nanofluid model is developed, incorporating Caputo time fractional derivatives and solved via a ADI based schemes. Parametric analyses demonstrate that the fractional order (γ), Rayleigh number (Ra), nanoparticles volume fraction (φ ), and radiation parameter (Rd) collectively govern the evolution of flow and temperature fields. Lower fractional orders enhance nanofluid flow and improve heat transfer rates, while Multiple Linear Regression (MLR) analyses highlight the dominant influence of nanoparticles concentration in fractional regimes. Comparative results show that fractional models achieve higher thermal efficiency than their integer-order counterparts. Overall, the work establishes a unified, high-accuracy, and computationally efficient framework for fractional modeling, from fundamental numerical formulations to applied thermofluid systems. In future work, these methods will be extended to variable and distributed order fractional systems and adapted to complex, multi-dimensional geometries. Further studies will focus on integrating data driven approaches such as physics-informed and graph neural networks with fractional solvers to enhance predictive capa- bility and parameter estimation in buoyancy-driven nanofluid convection systems.