Conformable and Caputo’s Derivatives in Generalized Viscoelastic Models

We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

Non-Noether symmetries of Hamiltonian systems with conformable fractional derivatives

In this paper, we present the fractional Hamilton＇s canonical equations and the fractional non-Noether symmetry of Hamilton systems by the conformable fractional derivative. First/y, the exchanging relationship between isochronous variation and fractional derivatives, and the fractional Hamilton principle of the system under this fractional derivative are proposed. Secondly, the fractional Hamilton＇s canonical equations of Hamilton systems based on the Hamilton principle are established. Thirdly, the fractional non-Noether symmetries, non-Noether theorem and non-Noether conserved quantities for the Hamilton systems with the conformable fractional derivatives are obtained. Finally, an example is given to illustrate the results.

On Conformable Delta Fractional Derivative on Time Scales

In this paper, we introduce and investigate the concept of conformable delta fractional derivative on time scales. By using the theory of time scales, we obtain some basic properties of the conformable delta fractional derivative. Our results extend and improve both the results in [9] and the usual delta derivative.

APPLICATION OF MULTI-DIMENSIONAL OF CONFORMABLE SUMUDU DECOMPOSITION METHOD FOR SOLVING CONFORMABLE SINGULAR FRACTIONAL COUPLED BURGER'S EQUATION

In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger's equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.

On the Nonlinear Neutral Conformable Fractional Integral-Differential Equation

In this paper, we investigate the nonlinear neutral fractional integral-differential equation involving conformable fractional derivative and integral. First of all, we give the form of the solution by lemma. Furthermore, existence results for the solution and sufficient conditions for uniqueness solution are given by the Leray-Schauder nonlinear alternative and Banach contraction mapping principle. Finally, an example is provided to show the application of results.

A different approach for conformable fractional biochemical reaction–diffusion models

This paper attempts to shed light on three biochemical reaction-diffusion models:conformable fractional Brusselator,conformable fractional Schnakenberg,and conformable fractional Gray-Scott.This is done using conformable residual power series(hence-form,CRPS)technique which has indeed,proved to be a useful tool for generating the solution.Interestingly,CRPS is an effective method of solving nonlinear fractional differential equations with greater accuracy and ease.

Exact solutions of conformable time fractional Zoomeron equation via IBSEFM

The nonlinear conformable time-fractional Zoomeron equation is an important mod-el to describe the evolution of a single scalar field.In this paper,new exact solutions of con-formable time-fractional Zoomeron equation are constructed using the Improved Bernoulli Sub-Equation Function Method(IBSEFM).According to the parameters,3D and 2D figures of the solutions are plotted by the aid of Mathematics software.The results show that IBSEFM is an efficient mathematical tool to solve nonlinear conformable time-fractional equations arising in mathematical physics and nonlinear optics.

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.

Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives:A comparative study

This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The SMCH equation is used to model the unidirectional propagation of shallowwater waves.The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived.We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system.Furthermore,some 2D and 3D graphical illustrations of the obtained solutions have been presented.The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted.The derived solutions may contribute to comprehending wave propagation in shallow water.So,the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management,which are critical topics in the ocean and coastal engineering.

Investigation on the new exact solutions of generalized Rosenau-Kawahara-RLW equation with p-th order nonlinearity occurring in ocean engineering models

The main objective of this study is to nd novel wave solutions for the time-fractional generalized Rosenau-Kawahara-RLW equation,which occurs in unidirectional water wave prop-agation.The generalized Rosenau-Kawahara-RLW equation comprises three equations Rosenau equation,Kawahara equation,RLW equation and also p-th order nonlinear term.All these equations describe the wave phenomena especially the wave-wave and wave-wall interactions in shallow and narrow channel waters.The auxiliary equation method is employed to get the analytical results.

On the Viability of Solutions to Conformable Stochastic Differential Equations

The viability of the conformable stochastic differential equations is studied.Some necessary and sufficient conditions in terms of the distance function to K are given.In addition,when the boundary of K is sufficiently smooth,our necessary and sufficient conditions can reduce to two relations just on the boundary of K.Lastly,an example is given to illustrate our main results.

Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic method.The results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.This article fairly clarifies the behaviors of surface waves in shallow waters.In the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great importance.Thus,this study will serve as a reference to guide scientists working in this field.

New Analytical and Numerical Results For Fractional Bogoyavlensky-Konopelchenko Equation Arising in Fluid Dynamics

In this article,(2+1)-dimensional time fractional Bogoyavlensky-Konopelchenko(BK)equation is studied,which describes the interaction of wave propagating along the x axis and y axis.To acquire the exact solutions of BK equation we employed sub equation method that is predicated on Riccati equation,and for numerical solutions the residual power series method is implemented.Some graphical results that compares the numerical and analytical solutions are given for di erent values of.Also comparative table for the obtained solutions is presented.

A popular reaction-difusion model fractional Fitzhugh-Nagumo equation:analytical and numerical treatment

The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population genetics.For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool.The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule.By the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential equation.The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method.This comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.

Multiple-solitons for generalized(2+1)-dimensional conformable Korteweg-de Vries-Kadomtsev-Petviashvili equation

This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation.The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational solutions.The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations.Furthermore,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

SITEM for the conformable space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations

In the present paper,new analytical solutions for the space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations are obtained by using the simplified tan(φ(ξ)2)-expansion method.Here,fractional derivatives are defined in the conformable sense.To show the correctness of the obtained traveling wave solutions,residual error function is defined.It is observed that the new solutions are very close to the exact solutions.The solutions obtained by the presented method have not been reported in former literature.

Multiple soliton solutions for the (3+1) conformable space-time fractional modified Korteweg-de-Vries equations

In the present paper,multiple exact soliton solutions for the old and the newly introduced(3+1)-dimensional modified Korteweg-de-Vries equation(mKdV)will be sought.The mKdV equations considered feature fractional derivative orders in both the space and time variables.A variety of soliton solutions ranging from hyperbolic to periodic function solutions will be constructed using simple ansatze for the equations.Finally,the algebraic equations to be obtained along the way and graphical representations will be carried out by utilizing the Mathematica software.

Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method

In this paper,we used the generalized(G’/G)-expansion method to construct exact solutions for conformable fractional nonlinear partial differential equations.This method is applied to obtain exact solutions for conformable fractional equal width wave equation(EW equation)and conformable fractional modified equal width wave equation(MEW equation).Based on the proposed method,several new exact solutions have been obtained.The proposed method is powerful and easily applicable for solving different types of conformable fractional partial differential equations.