Natural Transform along with HPM Technique for Solving Fractional ADE
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...
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doaj-fc976b270b594d86b929ea522ad4499f2021-08-09T00:00:45ZengHindawi LimitedAdvances in Mathematical Physics1687-91392021-01-01202110.1155/2021/9915183Natural Transform along with HPM Technique for Solving Fractional ADEN. Pareek0A. Gupta1G. Agarwal2D. L. Suthar3Department of MathematicsDepartment of MathematicsDepartment of Mathematics and StatisticsDepartment of MathematicsThe 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∈0,1. 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.http://dx.doi.org/10.1155/2021/9915183 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
N. Pareek A. Gupta G. Agarwal D. L. Suthar |
spellingShingle |
N. Pareek A. Gupta G. Agarwal D. L. Suthar Natural Transform along with HPM Technique for Solving Fractional ADE Advances in Mathematical Physics |
author_facet |
N. Pareek A. Gupta G. Agarwal D. L. Suthar |
author_sort |
N. Pareek |
title |
Natural Transform along with HPM Technique for Solving Fractional ADE |
title_short |
Natural Transform along with HPM Technique for Solving Fractional ADE |
title_full |
Natural Transform along with HPM Technique for Solving Fractional ADE |
title_fullStr |
Natural Transform along with HPM Technique for Solving Fractional ADE |
title_full_unstemmed |
Natural Transform along with HPM Technique for Solving Fractional ADE |
title_sort |
natural transform along with hpm technique for solving fractional ade |
publisher |
Hindawi Limited |
series |
Advances in Mathematical Physics |
issn |
1687-9139 |
publishDate |
2021-01-01 |
description |
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∈0,1. 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. |
url |
http://dx.doi.org/10.1155/2021/9915183 |
work_keys_str_mv |
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