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In this paper, we investigate the approximate solutions for the fractional fuzzy acoustic wave equations. We use the Laplace transform and an iterative technique with the fractional Atangana–Baleanu–Caputo operator to archive the series form results for the mains equations, and the expected outcomes of the suggested kinds were calculated. Finally, numerical examples are presented to show the usability of our major results. [ABSTRACT FROM AUTHOR]

Purpose – This paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results. Design/methodology/approach – This paper considered theoretical and numerical methodologies. Findings – This paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures. Originality/value – The novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

This article aims to introduce and analyze a diabetes mellitus model of fractional order, utilizing the ABC derivative. Diabetes mellitus is a prevalent and significant disease worldwide, ranking among the top causes of mortality. It is characterized by chronic metabolic dysfunction, leading to elevated blood glucose levels and subsequent damage to vital organs including the nerves, kidneys, eyes, blood vessels, and heart. The fractional ABC derivative is used in this study to describe and analyze diabetes mellitus mathematically while removing hereditary influences. The investigation begins by exploring the initial points of the diabetes mellitus model. Under the fractional ABC operator, Picard's theorem is used to prove the existence and uniqueness of solutions. For the numerical approximation of solutions in the fractional-order diabetes mellitus model, this study used a specialized technique that combines the principles of fractional calculus and a two-step Lagrange polynomial interpolation. Finally, the obtained results are visually presented through graphical representations, serving as empirical evidence to support our theoretical findings. The numerical experiments showed that the proportion of patients with diabetes mellitus increased as the fractional dimension (θ) reduced. The combination of mathematical modelling, analysis, and numerical simulations provides insights into the dynamics of diabetes mellitus, offering valuable contributions to the understanding and management of this prevalent disease. Additionally, the proposed scheme can be enhanced by incorporating the ABC operator, allowing for the simulation of real-world dynamics and behavior in the coexistence of diabetes mellitus and tuberculosis.

In this article, the Korteweg-de Vries-Caudrey-Dodd-Gibbon (KdV-CDG) equation is explored via a fractional operator. A nonlocal differential operator with a nonsingular kernel is used to study the KdV-CDG equation. Some theoretical features concerned with the existence and uniqueness of the solution, convergence, and Picard-stability of the solution by using the concepts of fixed point theory are discussed. Analytical solutions of the KdV-CDG equation by using the Laplace transformation (LT) associated with the Adomian decomposition method (ADM) are retrieved. The solutions are presented using 3D and surface graphics.

The aim of this paper is to analyze a fractional model of the Ebola virus. This study is important because it contributes to our understanding of the Ebola virus transmission dynamics using the notion of non-local differential operators. We aim to apply the recently implemented Atangana–Baleanu–Caputo (ABC) fractional derivative with the Mittag-Leffler kernel to study the Ebola virus model closely. The Picard–Lindelof approach is used to do a comprehensive study of the existence and uniqueness of the model’s solutions. The approximate solutions of the fractional order Ebola virus model were obtained using a numerical technique with the ABC operator, a combination of the fundamental theorem of fractional calculus and the two-step Lagrange polynomial interpolation. This innovative approach may offer new insights into the Ebola virus model that were not previously explored. Finally, the numerical simulations illustrate how the control parameters impact specific compartments within the model. The geometrical representation gives significant information about the model’s complexity and reliable information about the model. We simulate each model compartment at various fractional orders and compare them with integer-order simulations, highlighting the effectiveness of modern derivatives. The fractional analysis underscores the enhanced accuracy of non-integer order derivatives in capturing the Ebola virus model’s dynamics.

This article aims to introduce and analyze a diabetes mellitus model of fractional order, utilizing the ABC derivative. Diabetes mellitus is a prevalent and significant disease worldwide, ranking among the top causes of mortality. It is characterized by chronic metabolic dysfunction, leading to elevated blood glucose levels and subsequent damage to vital organs including the nerves, kidneys, eyes, blood vessels, and heart. The fractional ABC derivative is used in this study to describe and analyze diabetes mellitus mathematically while removing hereditary influences. The investigation begins by exploring the initial points of the diabetes mellitus model. Under the fractional ABC operator, Picard's theorem is used to prove the existence and uniqueness of solutions. For the numerical approximation of solutions in the fractional-order diabetes mellitus model, this study used a specialized technique that combines the principles of fractional calculus and a two-step Lagrange polynomial interpolation. Finally, the obtained results are visually presented through graphical representations, serving as empirical evidence to support our theoretical findings. The numerical experiments showed that the proportion of patients with diabetes mellitus increased as the fractional dimension (θ) reduced. The combination of mathematical modelling, analysis, and numerical simulations provides insights into the dynamics of diabetes mellitus, offering valuable contributions to the understanding and management of this prevalent disease. Additionally, the proposed scheme can be enhanced by incorporating the ABC operator, allowing for the simulation of real-world dynamics and behavior in the coexistence of diabetes mellitus and tuberculosis. [ABSTRACT FROM AUTHOR]

In this article, the Korteweg-de Vries-Caudrey-Dodd-Gibbon (KdV-CDG) equation is explored via a fractional operator. A nonlocal differential operator with a nonsingular kernel is used to study the KdV-CDG equation. Some theoretical features concerned with the existence and uniqueness of the solution, convergence, and Picard-stability of the solution by using the concepts of fixed point theory are discussed. Analytical solutions of the KdV-CDG equation by using the Laplace transformation (LT) associated with the Adomian decomposition method (ADM) are retrieved. The solutions are presented using 3D and surface graphics. [ABSTRACT FROM AUTHOR]

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.

We present analytical solutions of the Fornberg–Whitham equations with the aid of two well-known methods: Adomian decomposition transform and variational iteration transform involving fractional-order derivatives with the Atangana–Baleanu–Caputo derivative. The Elzaki transformation is used in the Atangana–Baleanu–Caputo derivative to find the solution to the Fornberg–Whitham equations. Using certain exemplary situations, the proposed method's viability is assessed. Comparative analysis for both integer and fractional-order results is established. For validation, the solutions of the suggested methods are compared with the actual results available in the literature. Two examples are considered to check the accuracy and effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vriesmodified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems. [ABSTRACT FROM AUTHOR]

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models. [ABSTRACT FROM AUTHOR]

We present analytical solutions of the Fornberg–Whitham equations with the aid of two well-known methods: Adomian decomposition transform and variational iteration transform involving fractional-order derivatives with the Atangana–Baleanu–Caputo derivative. The Elzaki transformation is used in the Atangana–Baleanu–Caputo derivative to find the solution to the Fornberg–Whitham equations. Using certain exemplary situations, the proposed method’s viability is assessed. Comparative analysis for both integer and fractional-order results is established. For validation, the solutions of the suggested methods are compared with the actual results available in the literature. Two examples are considered to check the accuracy and effectiveness of the proposed techniques.

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.