Mathematical Modelling and Numerical Simulation with Applications (MMNSA) <p>Mathematical Modelling and Numerical Simulation with Applications (MMNSA),,</p> Mehmet Yavuz en-US Mathematical Modelling and Numerical Simulation with Applications (MMNSA) 2791-8564 <p>Articles published in <em>MMNSA</em> are made freely available online immediately upon publication, without subscription barriers to access. All articles published in this journal are licensed under the Creative Commons Attribution 4.0 International License (<a href="">click here</a> to read the full-text legal code). This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.</p> <p>Under the Creative Commons Attribution 4.0 International License, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in <em>MMNSA</em>, so long as the original authors and source are credited.</p> <p><strong>The readers are free to:</strong></p> <ul> <li><strong>Share</strong> — copy and redistribute the material in any medium or format</li> <li><strong>Adapt</strong> — remix, transform, and build upon the material for any purpose, even commercially.</li> <li>The licensor cannot revoke these freedoms as long as you follow the license terms.</li> </ul> <p><strong>under the following terms:</strong></p> <ul> <li><strong>Attribution</strong> — You must give <strong>appropriate credit</strong>, provide a link to the license, and <strong>indicate if changes were made</strong>. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.</li> </ul> <ul> <li><strong>No additional restrictions</strong> — You may not apply legal terms or <strong>technological measures</strong> that legally restrict others from doing anything the license permits.</li> </ul> Stability analysis of an incommensurate fractional-order SIR model <p>In this paper, a fractional-order generalization of the susceptible-infected-recovered (SIR) epidemic model for predicting the spread of an infectious disease is presented. Also, an incommensurate fractional-order differential equations system involving the Caputo meaning fractional derivative is used. The equilibria are calculated and their stability conditions are investigated. Finally, numerical simulations are presented to illustrate the obtained theoretical results.</p> Bahatdin Daşbaşı Copyright (c) 2021 Bahatdin Daşbaşı 2021-09-30 2021-09-30 1 1 44 55 10.53391/mmnsa.2021.01.005 The Hermite-Hadamard type inequality and its estimations via generalized convex functions of Raina type <p>The theory of convexity plays an important role in various branches of science and engineering. The main objective of this work is to introduce the idea of a generalized convex function by unifying s-type m-convex function and Raina type function. In addition, some beautiful algebraic properties and examples are discussed. Applying this new definition, we explore a new sort of Hermite-Hadamard inequality. Furthermore, to enhance the paper we investigate several new estimations of Hermite-Hadamard type inequality. The concepts and tools of this paper may invigorate and revitalize for additional research in this mesmerizing and absorbing field of mathematics.</p> Muhammad Tariq Hijaz Ahmad Soubhagya Kumar Sahoo Copyright (c) 2021 Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo 2021-09-04 2021-09-04 1 1 32 43 10.53391/mmnsa.2021.01.004 Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation <p>In this study, an alternative method has been applied to obtain the new wave solution of mathematical equations used in physics, engineering, and many applied sciences. We argue that this method can be used for some special nonlinear partial differential equations (NPDEs) in which the balancing methods are not integer. A number of new complex hyperbolic trigonometric traveling wave solutions have been successfully generated in the Eckhaus equation (EE) and nonlinear Klein-Gordon (nKG) equation models associated with the Schrödinger equation. The graphs representing the stationary wave are presented by giving specific values to the parameters contained in these solutions. Finally, some discussions about new complex solutions are given. It is discussed by giving physical meaning to the constants in traveling wave solutions, which are physically important as well as mathematically. These discussions are supported by three-dimensional simulation. In order to eliminate the complexity of the process and to save time, computer package programs have been utilized.</p> Asıf Yokuş Copyright (c) 2021 Asıf Yokuş 2021-08-24 2021-08-24 1 1 24 31 10.53391/mmnsa.2021.01.003 Numerical solutions and synchronization of a variable-order fractional chaotic system <p>In the present paper, we implement a novel numerical method for solving differential equations with fractional variable-order in the Caputo sense to research the dynamics of a circulant Halvorsen system. Control laws are derived analytically to make synchronization of two identical commensurate Halvorsen systems with fractional variable-order time derivatives. The chaotic dynamics of the Halvorsen system with variable-order fractional derivatives are investigated and the identical synchronization between two systems is achieved. Moreover, graph simulations are provided to validate the theoretical analysis.</p> Zakia Hammouch Mehmet Yavuz Necati Özdemir Copyright (c) 2021 Zakia Hammouch, Mehmet Yavuz, Necati Özdemir 2021-08-20 2021-08-20 1 1 11 23 10.53391/mmnsa.2021.01.002 A numerical approach to the coupled atmospheric ocean model using a fractional operator <p>In the present framework, the coupled mathematical model of the atmosphere-ocean system called El Nino-Southern Oscillation (ENSO) is analyzed with the aid Adams-Bashforth numerical scheme. The fundamental aim of the present work is to demonstrate the chaotic behaviour of the coupled fractional-order system. The existence and uniqueness are demonstrated within the frame of the fixed-point hypothesis with the Caputo--Fabrizio fractional operator. Moreover, we captured the chaotic behaviour for the attained results with diverse order. The effect of the perturbation parameter and others associated with the model is captured. The obtained results elucidate that, the present study helps to understand the importance of fractional order and also initial conditions for the nonlinear models to analyze and capture the corresponding consequence of the fractional-order dynamical systems.</p> Pundikala Veeresha Copyright (c) 2021 Pundikala Veeresha 2021-08-15 2021-08-15 1 1 1 10 10.53391/mmnsa.2021.01.001