Published Paper Details:

This research paper focuses mainly on the analysis, development and investigation of numerical methods for solving the further generalized time- fractional diffusion equation, a mathematical model with diverse applications in the fields of physics and engineering. A hybrid method combining finite difference and spectral techniques for representing the spatial and temporal domains, addressing the challenges presented by fractional -order derivatives and nonlinearity is suggested. Analysis of convergence, stability, and asymptotic behavior of the proposed numerical approach is the main target. By the first theorem, the convergence order of the proposed numerical method which shows that the numerical solution converges to the exact solution with a specified order as the spatial and temporal discretization parameters approach zero is established. The stability , boundedness, and error analysis of the method, providing insights into the method's behavior across a range of parameters is directed by the second theorem. The third theorem analyses the asymptotic convergence, stability in time, and uniform convergence in time, shedding light on the long-term behaviour and accuracy of the numerical solution. This exploration of asymptotic convergence, stability in time and uniform convergence in time has further strengthened the credibility of the proposed numerical approach. The proposed numerical method provides stability and converges accurately to the exact solution, providing a robust framework for solving further generalized time- fractional diffusion equations. The finding results supply to the knowledge and development of effective computational methods for a broach class of fractional partial differential equations which will cover the way for applications in different scientific fields. Thus this my critical analysis work gives foundation for advancing the understanding and application of numerical techniques in the context of time- fractional diffusion equations