THE ANALYTICAL APPROXIMATE SOLUTION OF SYSTEM OF NONHOMOGENEOUS FRACTIONAL ORDERED PARTIAL DIFFERENTIAL EQUATION USING LAPLACE VARIATIONAL ITERATION METHOD (LVIM)

This thesis presents the analytical approximate solution for inhomogeneous fractional-order partial differential equations and their associated nonlinear systems using the Laplace Variational Iteration Method. Specifically, the Laplace Variational Iteration Method was employed to derive approximate anaiytical solutions for the time fractional Navier-Stokes equations in Cartesian coordinates. The Navier-Stokes equations, which describe the motion of fluids are essential in fluid dynamics but often pose significant challenges due to their non-linear characteristics and complexity, especially in fractional form. The numerical solutions of this system was obtained with the help of MATLAB software which provided LVIM algorithm for the given problem. Moreover, the results of the proposed method were compared with the exact solution of the problems which has confirmed that as the terms of the series increases the approximate solutions converges to the exact solution of each problem. This method is a powerful tool to handle nonlinear time fractional Navier-Stokes equation. The main property of the method is its ability to solve nonlinear equations, accurately and easily. Using Laplace variational iteration method, it is possible to find the exact solution or a closed approximate solution of a problem. To illustrate the application of the method, three non-linear 2D and 3D Navier-stokes equations had been considered. The scheme was found to be very reliable, effective, efficient and powerful technique to solve the proposed time fractional Navier-Stokes equation. The solutions of the three numerical problems solved in this thesis had shown that the rate of convergence of the approximate analytical exact solutions using Laplace variational Iteration Method was overlapping with that of Laplace Adomain Decomposition Method and Fractional Reduced Differential Transform Method.