Modified approach for linear and non-linear IBVPs with fractional dynamics

Abstract: Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical techniques for FPDEs with boundary conditions, although some methods do exist to address such problems. In this paper, a modified technique based on the Adomian decomposition method with Laplace transformation is presented, which effectively treats initial-boundary value problems. The non-linear term has been controlled by Daftardar-Jafari polynomials. Our proposed technique is applied to several initial and boundary value problems and the obtained results are presented through graphs. The differing behavior of the solutions for the suggested problems is observed by using various fractional orders. It is found that our proposed technique has a high rate of convergence towards the exact solutions of the problems. Moreover, while implementing this modification, higher accuracy is achieved with a small number of calculations, which is the main novelty of the proposed technique. The present method requires a new approximate solution in each iteration that adds further accuracy to the solution. It demonstrates that our suggested technique can be used effectively to solve initial-boundary value problems of FPDEs.