Executive Summary : | The concept of fractional calculus has dragged considerable demand and popularity because of its high application in the broader field of technology and sciences. The main advantage of fractional differential equations (FDEs) is that it provides a powerful tool for depicting the systems with memory, long-range interactions and hereditary properties of several materials. A wide range of relevant physical phenomena are characterized by time-fractional wave equations (TFWEs). TFWEs are derived by replacing the first-order time derivative of the standard diffusion equations with α-order (0≤α≤1), fractional derivative and TFWEs are derived by replacing the second-order time derivative of the classical diffusion or wave equations with α-order 1 to 2. In this project our goal is to find the adaptive computational approach for Riesz aractional advection dispersion wave equations with α-order 1 to 2. Previous studies at International level and National level have shown that the numerical approximations of FDs with non-uniform mesh have better accuracy than the uniform mesh and there are only very few numerical methods available with non-uniform mesh as compared to uniform mesh for the case of order 1 to 2. Therefore, an efficient numerical techniques are required for their treatment. So, some new wavelets will be constructed and based on these wavelets, numerical techniques will be established and implemented on concerned problem. Most of the existing numerical schemes for the time-fractional Caputo derivative with order αϵ(1,2) have convergence rate (3-α) and based on uniform mesh. There are very less study on numerical methods which have convergence rate more than (3-α) , and especially for the case of non-uniform mesh as compared to the uniform mesh till now. The development of some high convergence rate schemes based on the non-uniform mesh is really a challenging and interesting task. This investigation motivated the P.I. to propose a new rapid convergent approximation for the Caputo fractional derivative with convergence rate (5 - α) on a non-uniform mesh. Using new approximation of Caputo derivative in the time direction and second-order central difference discretization in the spatial direction, expected outcome will be a high order adaptive numerical algorithm to solve the TFDWEs numerically. Set up of the designed algorithm will be such that the algorithm changes its behavior automatically according to the value of α. Further, an adaptive high order stable implicit difference scheme will also be designed for the time-fractional wave equations (TFWEs) by using estimation of high order for the Caputo derivative in the time domain on non-uniform mesh. The proposed algorithm will allow one to build adaptive nature where the scheme is adjusted according to the behavior of α in order to keep the numerical errors very small and converge to the solution very fast as compared to the previously investigated schemes. |