The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multi dimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator. KEYWORDS HPTM and RPSM, multidimensional heat equations, new fractional derivative, new YAC operator, Prabhakar or generalized Mittag-Leffler function MSC CLASSIFICATION 26A33; 34A08; 34A34; 60G22 Math Meth Appl Sci. 2020;43:6062-6080. wileyonlinelibrary.com/journal/mma

We introduce an alternative to the method of matched asymptotic expansions. In the "traditional" implementation, approximate solutions, valid in different (but overlapping) regions are matched by using "intermediate" variables. Here we propose to match at the level of the equations involved, via a "uniform expansion" whose equations enfold those of the approximations to be matched. This has the advantage that one does not need to explicitly solve the asymptotic equations to do the matching, which can be quite impossible for some problems. In addition, it allows matching to proceed in certain wave situations where the traditional approach fails because the time behaviors differ (e.g., one of the expansions does not include dissipation). On the other hand, this approach does not provide the fairly explicit approximations resulting from standard matching. In fact, this is not even its aim, which to produce the "simplest" set of equations that capture the behavior.

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For a given West African country, we constructed a model describing the spread of the deathly disease called Ebola hemorrhagic fever. The model was first constructed using the classical derivative and then converted to the generalized version using the beta-derivative. We studied in detail the endemic equilibrium points and provided the Eigen values associated using the Jacobian method. We furthered our investigation by solving the model numerically using an iteration method. The simulations were done in terms of time and beta. The study showed that, for small portion of infected individuals, the whole country could die out in a very short period of time in case there is not good prevention.

In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels to the concepts of fractional differentiation with no singularity. We make use of the newly developed Caputo-Fabrizio fractional derivative with no singular kernel to establish the model. For existence and uniqueness of the continuous solution to the model, conditions on the perturbation parameters ν, µ and the derivative order α are provided. Numerical approximations are performed for some values of the perturbation parameters. This shows similar behaviors of the solution for close values of the fractional order α.

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Kermack-McKendrick epidemic model is considered as the basis from which many other compartmental models were developed. But the development of fractional calculus applied to mathematical epidemiology is still ongoing and relatively recent. We provide, in this article, some interesting and useful properties of the Kermack-McKendrick epidemic model with nonlinear incidence and fractional derivative order in the sense of Caputo. In the process, we used the generalized mean value theorem (Odibat and Shawagfeh in Appl. Math. Comput. 186:286-293, 2007) extended to fractional calculus to conclude some of the properties. A model of the Kermack-McKendrick with zero immunity is also investigated, where we study the existence of equilibrium points in terms of the nonlinear incidence function. We also establish the condition for the disease free equilibrium to be asymptotically stable and provide the expression of the basic reproduction number.

After having the issues of singularity and locality addressed recently in mathematical modelling, another question regarding the description of natural phenomena was raised: How influent is the second parameter β of the two-parameter Mittag-Leffler function Eα,β(z), z∈ℂ? To answer this question, we generalize the newly introduced one-parameter derivative with non-singular and non-local kernel [A. Atangana and I. Koca, Chaos, Solitons Fractals 89, 447 (2016); A. Atangana and D. Bealeanu (e-print)] by developing a similar two-parameter derivative with non-singular and non-local kernel based on Eα , β(z). We exploit the Agarwal/Erdelyi higher transcendental functions together with their Laplace transforms to explicitly establish the Laplace transform's expressions of the two-parameter derivatives, necessary for solving related fractional differential equations. Explicit expression of the associated two-parameter fractional integral is also established. Concrete applications are done on atmospheric convection process by using Lorenz non-linear simple system. Existence result for the model is provided and a numerical scheme established. As expected, solutions exhibit chaotic behaviors for α less than 0.55, and this chaos is not interrupted by the impact of β. Rather, this second parameter seems to indirectly squeeze and rotate the solutions, giving an impression of twisting. The whole graphics seem to have completely changed its orientation to a particular direction. This is a great observation that clearly shows the substantial impact of the second parameter of Eα , β(z), certainly opening new doors to modeling with two-parameter derivatives.