Algorithms and applications of the new modified decomposition method to solve initial-boundary value problems for fractional partial differential equations

This study aims to efficiently solve the class of initial-boundary problems for fractional partial differential equations by expanding the modified Adomian decomposition method. Three new algorithms are proposed, two for handling various fractional initial-boundary problems via the $x$-differential operator, and one for using the $t$-differential operator. The proposed methods are designed to increase computational efficiency and ensure greater accuracy in the solutions. The effectiveness of the techniques is reviewed by applying them to a variety of cases, demonstrating their ability to address a broad range of fractional calculus problems. The results emphasize the flexibility and adaptability of the developed algorithms as reliable methods.