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Analytical Solutions of the Electrical RLC Circuit via Liouville–Caputo Operators with Local and Non-Local Kernels
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.
Journal Article
A rumor of war
\"In March of 1965, Lieutenant Philip J. Caputo landed at Danang with the first ground combat unit deployed to Vietnam. Sixteen months later, having served on the line in one of modern history's ugliest wars, he returned home-- physically whole but emotionally wasted, his youthful idealism forever gone. 'A Rumor of War' is far more than one soldier's story. Upon its publication in 1977, it shattered America's indifference to the fate of the men sent to fight in the jungles of Vietnam. In the years since then, it has become not only a basic text on the Vietnam War but also renowned classic in the literature of wars throughout history and, as the author writes, of 'the things men do in war and the things war does to them'\"--Back cover.
Book
Solitary wave solutions of the time fractional Benjamin Bona Mahony Burger equation
The present study examines the approximate solutions of the time fractional Benjamin Bona Mahony Burger equation. This equation is critical for characterizing the dynamics of water waves and fluid acoustic gravity waves, as well as explaining the unidirectional propagation of long waves in nonlinear dispersive systems. This equation also describes cold plasma for hydromagnetic and audio waves in harmonic crystals. The natural transform decomposition method is used to obtain the analytical solution to the time fractional Benjamin Bona Mahony Burger equation. The proposed method uses the Caputo, Caputo Fabrizio, and Atangana Baleanu Caputo derivatives to describe the fractional derivative. We utilize a numerical example with appropriate initial conditions to assess the correctness of our findings. The results of the proposed method are compared to those of the exact solution and various existing techniques, such as the fractional homotopy analysis transform method and the homotopy perturbation transform technique. As a result, bell shaped solitons are discovered under the influence of hyperbolic functions. By comparing the outcomes with tables and graphs, the findings demonstrate the efficacy and effectiveness of the suggested approach.
Journal Article
The longest road : overland in search of America, from Key West to the Arctic Ocean
Philip Caputo, who had just turned seventy, his wife, and their two English setters, took off in a truck hauling an Airstream camper from Key West, Florida, en route via back roads and state routes to Deadhorse, Alaska. The journey took four months and covered sixteen thousand miles, during which Caputo interviewed more than eighty Americans from all walks of life to get a picture of what their lives and the life of the nation are really about in the twenty-first century.
Book
Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives
In recent years the world has witnessed the arrival of deadly infectious diseases that have taken many lives across the globe. To fight back these diseases or control their spread, mankind relies on modeling and medicine to control, cure, and predict the behavior of such problems. In the case of Ebola, we observe spread that follows a fading memory process and also shows crossover behavior. Therefore, to capture this kind of spread one needs to use differential operators that posses crossover properties and fading memory. We analyze the Ebola disease model by considering three differential operators, that is the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators. We present brief detail and some mathematical analysis for each operator applied to the Ebola model. We present a numerical approach for the solution of each operator. Further, numerical results for each operator with various values of the fractional order parameter α are presented. A comparison of the suggested operators on the Ebola disease model in the form of graphics is presented. We show that by decreasing the value of the fractional order parameter α , the number of individuals infected by Ebola decreases efficiently and conclude that for disease elimination, the Atangana–Baleanu operator is more useful than the other two.
Journal Article
A DISCRETE GRÖNWALL INEQUALITY WITH APPLICATIONS TO NUMERICAL SCHEMES FOR SUBDIFFUSION PROBLEMS
We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose temporal derivatives are singular at t = 0. The main result is a type of fractional Grönwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar LI approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.
Journal Article
Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential Problems Using Dhage’s Approach
In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve these goals, we utilize the well-known fixed point theorems attributed to Dhage for both BVPs. Moreover, we present two numerical examples to validate our analytical findings.
Journal Article
Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel
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 σ. 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 Entropy 2015, 17 6290 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.
Journal Article
A novel technique to study the solutions of time fractional nonlinear smoking epidemic model
The primary goal of the current work is to use a novel technique known as the natural transform decomposition method to approximate an analytical solution for the fractional smoking epidemic model. In the proposed method, fractional derivatives are considered in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo senses. An epidemic model is proposed to explain the dynamics of drug use among adults. Smoking is a serious issue everywhere in the world. Notwithstanding the overwhelming evidence against smoking, it is nonetheless a harmful habit that is widespread and accepted in society. The considered nonlinear mathematical model has been successfully used to explain how smoking has changed among people and its effects on public health in a community. The two states of being endemic and disease-free, which are when the disease dies out or persists in a population, have been compared using sensitivity analysis. The proposed technique has been used to solve the model, which consists of five compartmental agents representing various smokers identified, such as potential smokers
V
, occasional smokers
G
, smokers
T
, temporarily quitters
O
, and permanently quitters
W
. The results of the suggested method are contrasted with those of existing numerical methods. Finally, some numerical findings that illustrate the tables and figures are shown. The outcomes show that the proposed method is efficient and effective.
Journal Article