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The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected.

Smoking has become one of the major causes of health problems around the globe. It harms almost every organ of the body. It causes lung cancer and damage of different muscles. It also produces vascular deterioration, pulmonary disease, and ulcer. There is no advantage to smoking except the monetary one to the tobacco producers, manufacturers, and advertisers. Due to these facts, a passive verse active exposure of mathematical smoking model has been analyzed subject to the dynamical aspects for giving up smoking. In this context, mathematical modelling and qualitative analysis have been traced out for smoking model having five classes. Mathematical forms of smoke absent and smoke present points of equilibrium have been calculated for knowing optimal and dynamical control. By making use of the Lyapunov function theory, we have shown the global asymptotic behavior of smoke-free equilibrium for threshold parameter R0

In this article, we study existence, uniqueness, and stability analysis in Hyer–Ulam settings of delay fractional differential equations with nonlinear singular p‐Laplacian operator ϕp. The fractional operators are taken in the Caputo sense. The fractional differential equation is converted into an integral form with the Green's theorem, also a fixed point approach is studied for this article. We include an example with specific parameter and assumptions to show the results of the proposal. Our results generalize some of the works in the literature. [ABSTRACT FROM AUTHOR]

Our main aim in this paper is to propose and investigate new models of fractional blood ethanol and fractional two‐cell cubic autocatalysis reaction models. In particular, we evaluate the numerical solutions of these models by means of the power law and the Mittag–Leffler kernel. The numerical solutions are based mma7188 the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. We compare the numerical solutions for the blood ethanol with exact solutions while for the two‐cell cubic autocatalysis reaction model, we compare with numerical solutions that derived by using the finite‐difference method. In the case of the fractional order of the second model, the comparison is made with the spectral collocation method. In all cases, we found an excellent agreement. The novelty of this work is to study these models with the proposed algorithm for first time. Finally, we illustrate the behaviors of the numerical solutions for different values of order of the models and their parameters. [ABSTRACT FROM AUTHOR]

We consider the equations of motion of a bar, with given density, infinite in both directions, subjected to longitudinal vibrations under the action of an external load, and a stress–strain relation represented by a fractional‐order operator. Using three types of fractional operators, the initial‐boundary value problems associated with the described phenomenon are posed and solved. Through the bivariate Mittag–Leffer function, which has been recently introduced, we find the fundamental solution of these problems and calculate their moments. [ABSTRACT FROM AUTHOR]

The present research endeavor contains formulation of a new ψ‐Hilfer differential equation equipped with integral‐type subsidiary conditions. Utilizing Picard operator method, Banach contraction principle, and Gronwall inequality, we explore solution's properties of the underlying problem. We provide some assumptions to set up results related to uniqueness of solution for the underlying model. Furthermore, stability analysis is studied in the sense of Ulam Hyers Mittag Leffler's definition. We furnish illustrative examples for the vindication of our obtained analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new numerical method based on two‐step Lagrange polynomial interpolation to get numerical simulations and adaptive anti‐synchronization schemes for two fractional conformable attractors of variable order. It was considered the fractional conformable derivative in Liouville‐Caputo sense. The novel numerical method was applied to derive new results from the anti‐synchronization of the identical uncertain Wang‐Sun attractors and three‐dimensional chaotic system using fractional conformable sliding mode control. Numerical examples show the effectiveness of the adaptive fractional conformable anti‐synchronization schemes for the uncertain chaotic systems considered in this paper. [ABSTRACT FROM AUTHOR]

In this article, an efficient variable‐order Chebyshev collocation method which is based on shifted fifth‐kind Chebyshev polynomials is applied to solve a nonlinear variable‐order fractional reaction–diffusion equation with Mittag–Leffler kernel. The operational matrix of shifted fifth‐kind Chebyshev polynomials is derived for variable‐order ABC derivatives. The Chebyshev operational matrix together with the collocation method are applied to concerned nonlinear physical model with Mittag–Leffler kernel which is converted into a system of nonlinear algebraic equations, this system can be solved by using Newton method. The main focus of this paper is finding the convergence analysis of the approximation and high convergence order for small grid approximation. Few test examples with a comparison of maximum absolute error between the obtained numerical solution and existing known solution are being reported to show the accuracy and stability of the scheme. The physical presentation of the absolute errors for considered nonlinear variable‐order reaction–diffusion equations involving the Mittag–Leffler kernel with their exact solutions shows that the method is good for finding the solution of these kind of problems. [ABSTRACT FROM AUTHOR]

This research work is related to investigate sufficient conditions for existence, uniquely of solution to a coupled system of nonlinear fractional order hybrid differential equations (NFOHDEs). With the help of Burton's fixed point theorem, we develop the required conditions for existence of at least one solution to the considered problem of NFOHDEs. [ABSTRACT FROM AUTHOR]

This paper presents the solution for a fractional Bergman's minimal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo fractional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homotopy polynomial we obtain analytical approximate solutions for both derivatives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homotopy analysis transform method are simple and straightforward. [ABSTRACT FROM AUTHOR]

In this work we present a numerical method based on the Adams-Bashforth-Moulton scheme to solve numerically fractional delay differential equations. We focus in the fractional derivative with Mittag-Leffler kernel of type Liouville-Caputo with variable-order and the Liouville-Caputo fractional derivative with variable-order. Numerical examples are presented to show the applicability and efficiency of this novel method. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich-Fabrikant, Thomas' cyclically symmetric attractor and Newton-Leipnik. Fractional conformable and β-conformable derivatives of Liouville-Caputo type are considered to solve the proposed systems. A numerical method based on the Adams-Moulton algorithm is employed to approximate the numerical simulations of the fractional-order conformable attractors. The results of the new type of fractional conformable and β-conformable attractors are provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional cancer model was considered using the Caputo-Fabrizio-Caputo and the new fractional derivative with Mittag-Leffler kernel in Liouville-Caputo sense. Special solutions using an iterative scheme via Laplace transform, Sumudu-Picard integration method and Adams-Moulton rule were obtained. We studied the uniqueness and existence of the solutions. Novel chaotic attractors with total order less than three are obtained. [ABSTRACT FROM AUTHOR]

In this work, the study of the fractional behavior of the Bateman-Feshbach-Tikochinsky and Caldirola-Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler-Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville-Caputo, Caputo-Fabrizio-Caputo and the new fractional derivative based on the Mittag-Leffler kernel with arbitrary order a. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when a is equal to 1. [ABSTRACT FROM AUTHOR]

In this work, a control strategy is presented to produce hydrogen on demand to feed an internal combustion (IC) engine. For this purpose, the modeling of the IC engine fueled by gasoline blended with 10% v/v of anhydrous ethanol (E10) and hydrogen as an additive is developed. It is considered that the hydrogen gas is produced according to the IC engine demand, and that the hydrogen gas is obtained by an alkaline electrolyzer. The gasoline-ethanol blend added into the combustion chamber is determined according to the stoichiometric ratio and the production of hydrogen gas is regulated by a proportional and integral controller (P.I.). The controller reference is varying according to the mass flow air induced into the cylinder, in order to ensure an adequate production of hydrogen gas for any operating condition of the IC engine. The main contribution of this work is the control scheme developed, through simulation, in order to produce hydrogen on demand for any operating point of an internal combustion engine fueled by an E10 blend. The simulation results showed that the use of hydrogen gas as an additive in an E10 blend decreases the E10 fuel consumption 23% on average, and the thermal efficiency is increased approximately 2.13%, without brake power loss in the IC engine. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo-Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter . The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag-Leffler function; for the Caputo-Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one. [ABSTRACT FROM AUTHOR]

An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as 0 < β, γ = 1 for the space and time domain, respectively. he fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters σx and σt are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leler function depending on the parameters β and γ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior. [ABSTRACT FROM AUTHOR]