In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.
<abstract><p>In this paper, we find the solution of the time-fractional Newell-Whitehead-Segel equation with the help of two different methods. The newell-Whitehead-Segel equation plays an efficient role in nonlinear systems, describing the stripe patterns' appearance in two-dimensional systems. Four case study problems of Newell-Whitehead-Segel are solved by the proposed methods with the aid of the Antagana-Baleanu fractional derivative operator and the Laplace transform. The numerical results obtained by suggested techniques are compared with an exact solution. To show the effectiveness of the proposed methods, we show exact and analytical results compared with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, which confirms that the solution gets closer to the exact solution as the value tends from fractional-order towards integer order. Moreover, proposed methods are interesting, easy and highly accurate in solving various nonlinear fractional-order partial differential equations.</p></abstract>
The present paper is related to the analytical solutions of some heat like equations, using a novel approach with Caputo operator. The work is carried out mainly with the use of an effective and straight procedure of the Iterative Laplace transform method. The proposed method provides the series form solution that has the desired rate of convergence towards the exact solution of the problems. It is observed that the suggested method provides closed-form solutions. The reliability of the method is confirmed with the help of some illustrative examples. The graphical representation has been made for both fractional and integer-order solutions. Numerical solutions that are in close contact with the exact solutions to the problems are investigated. Moreover, the sample implementation of the present method supports the importance of the method to solve other fractional-order problems in sciences and engineering.
This paper aims at the analysis of the VdP heartbeat mathematical model. We have analysed the conditionality of a mathematical model which represents the oscillatory behaviour of the heart. A novel neuroevolutionary approach is chosen to analyse the mathematical model. The characteristics of the cardiac pulse of the heart are examined by considering two major scenarios with sixteen different cases. Artificial neural networks (ANNs) are constructed to obtain the best solutions for the heartbeat model. Unknown weights are finely tuned by a combination of a global search technique the Harris Hawks Optimizer (HHO) and a local search technique the Interior Point Algorithm (IPA). Stable behaviour of solutions obtained by considering different cases demonstrates that the model under consideration is well-conditioned. The accuracy of our novel procedure is established by getting the lowest residual errors in our solution for all cases. Graphical and statistical analysis are added to further elaborate the accuracy of our approach. INDEX TERMSCardiac pulse model, hybridized soft computing, artificial neural networks, non-linear ordinary differential equations, heuristics, interior-point algorithm, Harris Hawks optimizer. Ph.D. degree in software engineering from De Montfort University, in 2015. From 2004 to 2007, he worked in the Software Development Industry, where he implemented several systems and solutions for a National Academic Institution. His research interests include algorithms, semantic web, and optimization techniques. He focuses on enhancing real-world matching systems using machine learning and data analytics in the context of supporting decision-making.
In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.