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This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.

In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source terms are considered in the fractional differential equations. The classical behaviors are recovered when the fractional order α is equal to 1.

The aim of this paper is to study the fractional model of Lumpy Skin Disease, aiming to enhance our understanding of this disease. Specifically, we employ the recently introduced Caputo–Fabrizio fractional (CFF) derivative to analyze the Lumpy Skin Disease model in detail. To comprehensively study the model’s solutions, we utilize the Picard-Lindelof approach to assess their existence and uniqueness. Furthermore, we employ numerical techniques, specifically the CFF derivative combined with the fundamental theorem of fractional calculus and fixed point theorem, to obtain the solutions of Lumpy Skin Disease in near form using fractional order. This innovative approach offers novel insights into the dynamics of the disease model that were previously unexplored. In addition, numerical simulations are conducted to explore the change in effects of control parameters on specific compartments within the model. The geometric representation of the model provides valuable insights into its complexity and reliability. By simulating each model compartment at various fractional orders and comparing them with integer-order simulations, we highlight the effectiveness of modern derivatives. Overall, our fractional analysis emphasizes the enhanced accuracy of non-integer order derivatives in capturing the dynamics of the Lumpy Skin Disease model. These findings contribute to advancing our understanding of the disease and may have implications for its control and management strategies.

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years. A lot of new fractional operators were used to handle various practical problems. In this article, we mainly study four new fractional operators, namely the Caputo-Fabrizio operator, the Atangana-Baleanu operator, the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k -Prabhakar fractional integral operator. Usually, the theory of the k -Prabhakar fractional integral is regarded as a much broader than classical fractional operator. Here, we firstly give a series expansion of the k -Prabhakar fractional integral by means of the k -Riemann-Liouville integral. Then, a connection between the k -Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown, respectively. In terms of the above analysis, we can obtain this a basic fact that it only needs to consider the k -Prabhakar fractional integral to cover these results from the four new fractional operators.

Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed.

This paper addresses the mathematical modelling of transient processes in a steel membrane oscillating in a nonlinear viscoelastic isotropic medium. For this purpose, three types of mathematical models of the object under study with different degrees of adequacy are presented. The first type of model is presented as a system with distributed mechanical parameters, the equations of state of which are derived from the modified Hamilton‒Ostrogradsky principle and represent a mixed problem. The second is presented as a system with concentrated parameters, which is a Cauchy problem. The third type of model is presented as a modification of the second type via the fractional derivative and integral theory using the Caputo Fabrizio operator. The results of computer simulations for all types of models are presented in the form of analysed figures. Comparative analysis was also carried out for all the models on a model steel membrane with a fixed tension force, which demonstrated that the application of the Caputo–Fabrizio operator to a simplified membrane model improved the degree of adequacy of the latter.

This study solves the coupled fractional differential equations defining the massive Thirring model and the Kundu Eckhaus equation using the Natural transform decomposition method. The massive Thirring model is a dynamic component of quantum field theory, consisting of a coupled nonlinear complex differential equations. Initially, we study the suggested equations under the fractional derivative of Caputo-Fabrizio. The Atangana-Baleanu derivative is then used to evaluate the comparable equations. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approach and the fractional operators in this situation to develop satisfactory approximations to the offered problems. The proposed approach combines the natural transform technique with the efficient Adomian decomposition scheme. Obtaining numerical findings in the form of a fast-converge series significantly improves the scheme’s accuracy. Some graphical plot distributions are presented to show that the present approach is very simple and straightforward. We performed a fractional order analysis of assumed phenomena to demonstrate and validate the effectiveness of the future technique. The behaviour of the approximate series solution for several fractional orders is shown visually. Additionally, the nature of the derived outcome has been observed for various fractional orders. The derived results demonstrate how simple and efficient the proposed method is to apply for analysing the behaviour of fractionally-order complex nonlinear differential equations that arise in related fields of engineering and science.

Malaria continues to pose a significant global health challenge, with its persistent transmission creating major difficulties for healthcare systems worldwide. Tackling this problem calls for innovative and effective methods to enhance understanding and control of the disease. In this work, we proposed a fractional-order mathematical model to study the dynamics of malaria transmission, integrating essential control measures such as treatment of humans and management of mosquito populations. The model employed three different types of non-integer order differential operators: the Caputo operator, the Caputo–Fabrizio operator with exponential decay, and the Atangana–Baleanu operator with an extended Mittag–Leffler kernel. Using fixed-point theory, we proved the existence and uniqueness of solutions for the proposed model. Numerical simulations are carried out to assess the impact of varying fractional orders on the progression of the disease. The results revealed that increasing the fractional order slows down the spread of malaria, reduces the peak number of infections, and prolongs the duration of outbreaks highlighting the memory-dependent nature of fractional systems. Our findings demonstrated that fractional-order models offer a more accurate and flexible approach to capturing the complex dynamics of malaria transmission. The study underscores the importance of integrating both therapeutic interventions and vector control strategies in reducing disease burden. Based on the findings of this study, we recommended the integration of fractional order modeling into malaria control strategies, as it captures the memory effects and long-term dynamics of disease transmission more accurately than classical models. Public health programs should adopt combined intervention approaches incorporating both effective treatment and vector control measures to significantly reduce infection rates. Furthermore, control efforts should be sustained over time, as fractional models reveal that short-term interventions may not be sufficient in curbing prolonged outbreaks. Policymakers are encouraged to use insights from these models to design adaptive, data-driven strategies that enhance the efficiency and sustainability of malaria control programs.

In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.

Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations.