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(searched for: doi:10.53391/mmnsa.2021.01.001)
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Vijith Raghavendra,
An International Journal of Optimization and Control: Theories & Applications (ijocta), Volume 13, pp 104-115; https://doi.org/10.11121/ijocta.2023.1306

Abstract:
Technology has revolutionized the way transactions are carried out in economies across the world. India too has witnessed the introduction of numerous modes of electronic payment in the past couple of decades, including e-banking services, National Electronic Fund Transfer (NEFT), Real Time Gross Settlement (RTGS) and most recently the Unified Payments Interface (UPI). While other payment mechanisms have witnessed a gradual and consistent increase in the volume of transactions, UPI has witnessed an exponential increase in usage and is almost on par with pre-existing technologies in the volume of transactions. This study aims to employ a modified Lotka-Volterra (LV) equations (also known as the Predator-Prey Model) to study the competition among different payment mechanisms. The market share of each platform is estimated using the LV equations and combined with the estimates of the total market size obtained using the Auto-Regressive Integrated Moving Average (ARIMA) technique. The result of the model predicts that UPI will eventually overtake the conventional digital payment mechanism in terms of market share as well as volume. Thus, the model indicates a scenario where both payment mechanisms would coexist with UPI being the dominant (or more preferred) mode of payment.
An International Journal of Optimization and Control: Theories & Applications (ijocta), Volume 13, pp 46-58; https://doi.org/10.11121/ijocta.2023.1265

Abstract:
In this article, we demonstrated the study of the time-fractional nonlinear Sharma-Tasso-Olever (STO) equation with different initial conditions. The novel technique, which is the mixture of the q-homotopy analysis method and the new integral transform known as Elzaki transform called, q-homotopy analysis Elzaki transform method (q-HAETM) implemented to find the adequate approximated solution of the considered problems. The wave solutions of the STO equation play a vital role in the nonlinear wave model for coastal and harbor designs. The demonstration of the considered scheme is done by carrying out some examples of time-fractional STO equations with different initial approximations. q-HAETM offers us to modulate the range of convergence of the series solution using , called the auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations, the effectiveness and reliability of the considered technique are validated. The implementation of the new integral transform called the Elzaki transform along with the reliable analytical technique called the q-homotopy analysis method to examine the time-fractional nonlinear STO equation displays the novelty of the presented work. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.
Yassine Sabbar, , Nadia Gul, Driss Kiouach, S. P. Rajasekar, Nasim Ullah, Alsharef Mohammad
Aims Mathematics, Volume 8, pp 1329-1344; https://doi.org/10.3934/math.2023066

Abstract:
Exhaustive surveys have been previously done on the long-time behavior of illness systems with Lévy motion. All of these works have considered a Lévy–Itô decomposition associated with independent white noises and a specific Lévy measure. This setting is very particular and ignores an important class of dependent Lévy noises with a general infinite measure (finite or infinite). In this paper, we adopt this general framework and we treat a novel correlated stochastic $ SIR_p $ system. By presuming some assumptions, we demonstrate the ergodic characteristic of our system. To numerically probe the advantage of our proposed framework, we implement Rosinski's algorithm for tempered stable distributions. We conclude that tempered tails have a strong effect on the long-term dynamics of the system and abruptly alter its behavior.
, Nichaphat Patanarapeelert, Muhammad Awais Barkat, ,
Published: 8 June 2022
Journal of Function Spaces, Volume 2022, pp 1-9; https://doi.org/10.1155/2022/7519002

Abstract:
In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient.
Muhammad Imran Asjad, Pongsakorn Sunthrayuth, Muhammad Danish Ikram, Taseer Muhammad, Ali Saleh Alshomrani
Published: 1 June 2022
Chaos, Solitons, and Fractals, Volume 159; https://doi.org/10.1016/j.chaos.2022.112090

Published: 1 April 2022
by MDPI
Journal: Mathematics
Mathematics, Volume 10; https://doi.org/10.3390/math10071125

Abstract:
This work proposes a qualitative study for the fractional second-grade fluid described by a fractional operator. The classical Caputo fractional operator is used in the investigations. The exact analytical solutions of the constructed problems for the proposed model are determined by using the Laplace transform method, which particularly includes the Laplace transform of the Caputo derivative. The impact of the used fractional operator is presented; especially, the acceleration effect is noticed in the paper. The parameters’ influences are focused on the dynamics such as the Prandtl number (Pr), the Grashof numbers (Gr), and the parameter η when the fractional-order derivative is used in modeling the second-grade fluid model. Their impacts are also analyzed from a physical point of view besides mathematical calculations. The impact of the fractional parameter α is also provided. Finally, it is concluded that the graphical representations support the theoretical observations of the paper.
, Ali Akgül, , , M. Mossa Al-Sawalha,
Published: 16 March 2022
Journal of Function Spaces, Volume 2022, pp 1-14; https://doi.org/10.1155/2022/3341754

Abstract:
In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.
Halil Anaç
Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi; https://doi.org/10.25092/baunfbed.984440

Abstract:
Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations. Bazı doğrusal olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemler, homotopi pertürbasyon Elzaki dönüşümü yöntemi ile çözülmüştür. Kesirli türevler Caputo anlamında tanımlanmıştır. Uygulamalar homotopi pertürbasyon Elzaki dönüşümü yöntemi ile incelenmiştir. Bunun yanında, çözümlerin grafikleri MAPLE yazılımında çizdirilmiştir. Ayrıca homotopi pertürbasyon Elzaki dönüşümü yöntemi ve homotopi pertürbasyon Sumudu dönüşümü yöntemi çözümlerinin, lineer olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemlerin tam çözümü ile mutlak hata karşılaştırması sunulmaktadır. Ek olarak, bu mutlak hata karşılaştırması tablolarda belirtilmiştir. Bu makalenin yeniliği, hem Caputo kesir dereceli gaz dinamiği denkleminin hem de Caputo kesir dereceli Klein-Gordon denkleminin bu yöntemle ilk analizidir. Bu nedenle, homotopi pertürbasyon Elzaki dönüşümü yöntemi, zaman-kesirli mertebeden kısmi diferansiyel denklemlerin analitik çözümlerinin elde edilmesinde hızlı ve etkilidir.
Mehmet Yavuz, Müzeyyen Akman, Fuat Usta, Necati Özdemir
Published: 1 January 2022
Abstract:
Tuberculosis (TB) is an infectious disease with a high death rate compared to many infectious diseases. Therefore, many prominent studies have been done on the mathematical modeling and analysis of TB. In this study, an illustrative mathematical model is developed by considering the awareness parameter. In this context, two different treatment strategies that is applied as protective treatment and main treatment that is considered on the infected individuals are taken into account. In the provided model, a six-dimensional compartment system of fractional-order is constructed that includes the susceptible, latent, infected, and recovered population, as well as including the mentioned two treatment strategies. Also positivity and biologically feasible region of the model are provided. In the numerical simulations, Adams–Bashforth which is a well-known numerical scheme is applied to obtain the results.
Mehmet Yavuz, Fatma Özlem Coşar, Fuat Usta
Published: 1 January 2022
Abstract:
Recently, many illustrative studies have been performed on the mathematical modeling and analysis of COVID-19. Due to the uncertainty in the process of vaccination and its efficiency on the disease, there have not been taken enough studies into account yet. In this context, a mathematical model is developed to reveal the effects of vaccine treatment, which has been developed recently by several companies, on COVID-19 in.this study. In the suggested model, as well as the vaccinated individuals, a five-dimensional ordinary differential equation system including the susceptible, infected, exposed and recovered population is constructed. This mentioned system is considered in the fractional order to investigate and point out more detailed analysis in the disease and its future prediction. Moreover, besides the positivity, existence and uniqueness of the solution, biologically feasible region are provided. The basic reproduction number, known as expected secondary infection which means that expected infection among the susceptible populations caused by this infection, is computed. In the numerical simulations, the parameter values taken from the literature and estimated are used to perform the solutions of the proposed model. In the numerical simulations, Adams-Bashforth algorithm which is a well-known numerical scheme is used to obtain the results.
Hassan Khan, Hajira, Qasim Khan, Poom Kumam, Fairouz Tchier, Gurpreet Singh, Kanokwan Sitthithakerngkiet, Ferdous Mohammed Tawfiq
Published: 1 January 2022
Journal: Open Physics
Open Physics, Volume 20, pp 764-777; https://doi.org/10.1515/phys-2022-0072

Abstract:
Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.
Khalid Khan, Amir Ali, Manuel De la Sen, Muhammad Irfan
Aims Mathematics, Volume 7, pp 1580-1602; https://doi.org/10.3934/math.2022092

Abstract:
In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 < \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.
Muhammad Imran Asjad, Abdul Basit, , Sameh Askar,
Published: 28 October 2021
Case Studies in Thermal Engineering, Volume 28; https://doi.org/10.1016/j.csite.2021.101585

The publisher has not yet granted permission to display this abstract.
An International Journal of Optimization and Control: Theories & Applications (ijocta), Volume 11, pp 52-67; https://doi.org/10.11121/ijocta.2021.1177

Abstract:
The Korteweg–De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface critical flow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional operators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physical nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order.
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