Research Article Natural Transform along with HPM Technique for Solving Fractional ADE N. Pareek , 1 A. Gupta , 2 G. Agarwal , 3 and D. L. Suthar 4 1 Department of Mathematics, Bhagat Singh Govt. P.G. College, Jaora, India 2 Department of Mathematics, Govt. M.V.M, Bhopal, India 3 Department of Mathematics and Statistics, Manipal University Jaipur, Rajasthan, India 4 Department of Mathematics, Wollo University, P.O. Box: 1145, Dessie, Ethiopia Correspondence should be addressed to D. L. Suthar; dlsuthar@gmail.com Received 16 April 2021; Accepted 8 July 2021; Published 28 July 2021 Academic Editor: Younes Menni Copyright © 2021 N. Pareek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ ½0, 1. The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases. 1. Introduction and Preliminaries Fractional calculus generalizes the integration and differenti- ation of integer order to arbitrary order that is being studied for the past 300 years. The growing interest of researchers in this field has led to solving the real-world issues in type of fractional differential equations due to their nonlocal behav- ior, and these equations are well suited to describe various phenomena in the field of engineering and science. Also, fractional derivatives are capable to model various processes mathematically which exhibit the memory and hereditary properties [1–5]. The ADE arises in the study of transport of solute or Brownian motion of particles in a fluid occurring due to the simultaneous occurrence of advection and particle disper- sion. Fractional advection-dispersion equation describes the phenomena of anomalous diffusion of the particles in the transport process in a better way; as in anomalous diffusion, the solute transport is quicker or speedier than time’s inferred square root given by Baeumer et al. [6]. The equation is used to study groundwater pollution, pollution of the atmosphere produced by smoke or dust, the spread of chem- ical solutes and contaminant discharges, etc. [7]. Hence, FADE has attracted the attention of many researchers. Hindawi Advances in Mathematical Physics Volume 2021, Article ID 9915183, 11 pages https://doi.org/10.1155/2021/9915183