Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

New Optical Soliton Solutions for Coupled Resonant Davey-Stewartson System with Conformable Operator

This paper investigates the novel soliton solutions of the coupled fractional system of the resonant Davey-Stewartson equations. The fractional derivatives are considered in terms of conformable sense. Accordingly, we utilize a complex traveling wave transformation to reduce the proposed system to an integer-order system of ordinary differential equations. The phase portrait and the equilibria of the obtained integer-order ordinary differential system will be studied. Using suitable mathematical assumptions, the new types of bright, singular, and dark soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing system. To achieve this, illustrative examples of the fractional Davey-Stewartson system are provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of conformable derivatives on behavior of acquired solutions for different fractional orders is also discussed. By comparing the proposed method with the other existing methods, the results show that the execute of this method is concise, simple, and straightforward. The results are useful for obtaining and explaining some new soliton phenomena.

A THEORETICAL STUDY ON FRACTIONAL EBOLA HEMORRHAGIC FEVER MODEL

The Ebola virus infection (EVI), generally known as Ebola hemorrhagic fever, is a major health concern. The occasional outbreaks of virus occur primarily in certain parts of Africa. Many researches have been devoted to the study of the Ebola virus disease. In this paper, we have taken susceptible-infected-recovered-deceased-environment (SIRDP) system to investigate the dynamics of Ebola virus infection. We adopted fractional operators for a better illustration of model dynamics and memory effects. Initially, the Ebola disease model is modified with Caputo–Fabrizio arbitrary operator in Caputo sense (CFC) and we employed the fixed-point results for the existence and uniqueness of the solution of the fractional system. Further, we adopted the arbitrary fractional conformable and [Formula: see text]-conformable derivatives to the alternative representation of the model. For the numerical approximation of the system, we show a numerical technique based on the fundamental theorem of fractional calculus for CFC derivative and a numerical scheme called the Adams–Moulton for conformable derivatives. Finally, for the validation of theoretical results, the numerical simulations are displayed.

Extremal Solutions for Caputo Conformable Differential Equations with p-Laplacian Operator and Integral Boundary Condition

The Caputo conformable derivative is a new Caputo-type fractional differential operator generated by conformable derivatives. In this paper, using Banach fixed point theorem, we obtain the uniqueness of the solution of nonlinear and linear Cauchy problem with the conformable derivatives in the Caputo setting, respectively. We also establish two comparison principles and prove the extremal solutions for nonlinear fractional p -Laplacian differential system with Caputo conformable derivatives by utilizing the monotone iterative technique. An example is given to verify the validity of the results.

On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed, and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

Newton’s Law of Cooling with Generalized Conformable Derivatives

In this communication, using a generalized conformable differential operator, a simulation of the well-known Newton’s law of cooling is made. In particular, we use the conformable t1−α, e(1−α)t and non-conformable t−α kernels. The analytical solution for each kernel is given in terms of the conformable order derivative 0<α≤1. Then, the method for inverse problem solving, using Bayesian estimation with real temperature data to calculate the parameters of interest, is applied. It is shown that these conformable approaches have an advantage with respect to ordinary derivatives.