




Analysis of Burger Equation Using HPM with General Fractional Derivative


Sachin Kumar,
Manvendra Narayan Mishra,
Ravi Shanker Dubey,


Abstract
: 

To solve the generalised Burgers equation, the most recent operator in fractional calculus, is introduced in this research work. A more manageable form of problem can be obtained by reducing general fractional derivative into three wellknown operators. We use the effective analytical method known as the homotopy perturbation method (HPM) to get generalised Burgers equations solution. A realworld example is used to demonstrate the findings, and we also analyzed all three reduced operators. A graphical analysis is also supplied to demonstrate how the solution functions. By demonstrating how to solve generalised Burgers problem using this approach and general fractional derivative, this study makes a contribution in field of nonlinear differential equations. 






Analytical solution of conformable Schrodinger wave equation with Coulomb potential


Mohamed.AlMasaeed,


Abstract
: 

In this paper, the conformable Schr\"odinger equation for hydrogenlike systems with conformable Coulombs potential is constructed. Then, the conformable eigenfunctions and energy eigenvalues are obtained. The analytic solutions are expressed in terms of conformable spherical harmonics and conformable Laguerre functions that are appeared and defined in this work. Some aspects of the results are discussed. For instance, the probability density for the first three levels and different values of $\alpha$ is plotted, and it is observed that the probability density gradually converts to $\alpha = 1$ for all levels. The traditional version for this problem is recovered when the fractional parameter $\alpha = 1$. The conformable eigenfunctions could be useful as a basis for approximation methods developed for the conformable counterparts that appeared in conformable quantum mechanics. 






The Fuzzy conformable IntegroDifferential Equations


Atimad Harir,


Abstract
: 

The fuzzy generalized conformable fractional derivative is a novel fuzzy fractional derivative based on the basic limit definition of the derivative in cite{cfd}. We introduce the convolution product of fuzzy mapping and a crisp function in
this paper. The conformable Laplace convolution formula is proved under the generalized conformable fractional derivatives concept and used to solve fuzzy integrodifferential equations with a kernel of convolution type. The method is demonstrated by solving two examples, and the related theorems and properties are proved in detail. 






Applying Conformable Double Sumudu – Elzaki Approach to Solve Nonlinear Fractional Problems


Shams A. Ahmed,
Rania Saadeh,
Ahmad Qazza,
Tarig M. Elzaki,


Abstract
: 

In this study, a doubleconforming SumuduElzaki transformation (CDSET) and a decomposition method are combined to develop a new method that solves the nonlinear subproblem considering some specific conditions. This combination can be referred to as the Conformal Double SumuduElzaki Decomposition Method (CDSEDM). Furthermore, we explained and discussed the main features and main results of the presented method. The CDSEDM presents analytic series solutions with high convergence to the exact solution in closed form. The benefit of utilizing the proposed technique is that it presents analytic series solutions to the objective equations without the requirement for any constrained assumptions, transformation or discretization. In addition, various numerical experiments were presented to prove the efficiency of the obtained. The results show the power and effectiveness of the proposed approach in handling a range of physical and engineering problems. 






Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials


Wafa Selmi,
Mohsen Timoumi,


Abstract
: 

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian system
\begin{equation}
\label{eq1}
\left\{
\begin{array}{l}
_{t}D_{\infty}^{\alpha}(_{\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\u\in H^{\alpha}(\mathbb{R}),
\end{array}\right.
\end{equation}
where $_{\infty}D_{t}^{\alpha}$ and $_{t}D^{\alpha}_{\infty}$ are left and right LiouvilleWeyl fractional derivatives of order $\frac{1}{2}<\alpha<1$ on the whole axis respectively, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix valued function unnecessary coercive and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below and unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that (\ref{eq1}) possesses infinitely many solutions via a variant Symmetric Mountain Pass Theorem. 






Analytical Methods for the Solution of Linear Fractional Order Systems


Djamel Boucherma,


Abstract
: 

This paper presents the systems of fractional order described by differential equations of fractional order whose orders of their derivatives are real numbers. Differential equations do not have exact analytical solutions, so numerical and approximation techniques are widely used in the derivation of their analytical and numerical solutions. The main objective of this work is to present the analytical methods and to obtain an explicit expression for the solution of the equation of state of fractional linear systems of commensurate order. 






Fractional Order Mathematical Modelling for Studying the Impact on the Emergence of Pollution and Biodiversity Pertaining to Incomplete Aleph functions


Shaher Momani,
Rahul Sharma,
Jagdev Singh,
Yudhveer Singh,
Samir Hadid,


Abstract
: 

Civilizations are always known for utilization, destruction and ignoring the environment. Different opinions have been put at the root of our environmentaldisruptive tendencies, religion, social and economic structures and the adoption of technology. It is conventional to blame environmental conditions but we do not utilize practically our knowledge and understanding of nature. It seems that we continuously blame pollution and its destruction instead of our efforts. With the use of a fractional order mathematical model involving incomplete Aleph (א) functions (IAFs) and the Caputo fractional derivative, we explore the impact of the emergence of pollution and study the effect of industrial operations on the environment of the region. In this work, we evaluate the Caputo fractional derivative of incomplete Aleph (א) functions. Here, we define the concentration C(x,t) in terms of incomplete Aleph (א) functions and obtain some particular cases by giving specific values to the parameters of our primary results and expressing them in terms of special functions, notably Hfunctions, incomplete Hfunctions, and incomplete Ifunctions, and demonstrating their relationship with existing results.







Analytical Solution of Cancer Cells Interaction with Virotherapy in the Framework of Fractional Derivative Operator


MANISH SHARMA,


Abstract
: 

In the present paper, we introduce a mathematical model of virotherapy for cancer treatment by using fractional calculus. This model contains four differential equations which describes interactions among uninfected tumor cells, infected tumor cells, virions and effector T cells. We discuss the existence and uniqueness of the given model and study about local stability of equilibrium points. We also find the numerical solutions to investigate the effect of fractional order derivative and different variables. We also plot some graphs to illustrated the results. 






Fractional spectral approaches for some fractional partial differential equations


S. Z. Rida,
H. S. Hussien,
A. H. Noreldeen,
M. M. Farag,


Abstract
: 

The multiorder linear and nonlinear partial differential equations of fractional order have been solved numerically. A computational strategy based on fractional spectral operational matrices (OM) and generalized fractional Lageurre (GFL), generalized fractional shifted Legendre (GFSL), generalized fractional modified Bernstein (GFMB) are provided. Our fractional spectral methods have been used to solve nonlinear fractional partial differential equations and fractional Kortewegde VriesBurgers (KdVB) equation.
Making use of a variety of test and application examples, the effectiveness of the numerical solution have been satisfied. 






A Computational Numerical study of Burger Equation with Fractal Fractional Derivative


Sonal Jain,
HoHon Leung,
Firuz Kamalov,


Abstract
: 

In the several real world problem we can see the concept of fractal medium. In this paper we show that the concept of the fractal derivative is not only represent the fractal sharps but also describe the movement of the fluid within these media. In this paper we develop the solution of viscous Burger equation with different kernel of fractalfractional derivative. We solve the numerical technique using Newton’s Polynomial method for solving fractional Burgers equation. To show the applicability of the present model in fractal media we present simulations for different values of fractal dimensions.







Exploring the solvability for coupled system of nonlinear fractional Langevin equations


Ahmed S. Ghanem,


Abstract
: 

In this article, we investigate the existence and uniqueness of the solution to a nonlinear fractional coupled system of Langevin equations with boundary conditions. The required results are established based on fixed point theorem and fractional calculus. Finally, an example is given to illustrate the obtained results. 






Mathematical Analysis of Streptococcus Suis Infection in PigHuman Population by RiemannLiouville’s Fractional Operator


Awatif Muflih Alqahtani,
Manvendra Narayan Mishra,


Abstract
: 

In the current article, we have discussed a mathematical model for a rare disease known as Streptococcus Suis which is a contaminated and infectious disease. The disease Streptococcus Suis is spread by the infection found in pigs that is transmitted in humans, later. In human transmissions, this disease takes very severe form that can result death as well with serious illness. In this paper, we have taken seven population groups (some are from human populations and some are from pig’s populations as well) under consideration. We have applied the Riemann Liouville’s fractional derivative with Laplace transformation to analysis and study of the presented mathematical model of the disease. We have also validated the presence and oneness of solution with graphical presentation of the solutions. 






An Optimal Control Problem for Dengue Fever Model using Caputo Fractional Derivatives


M. Vellappandi,
V. Govindaraj,
Pushpendra Kumar,
Kottakkaran Sooppy Nisar,


Abstract
: 

In this paper, we study the optimal control of the dengue disease model with the vertical transmission in terms of the Caputo fractional derivatives. We apply the control parameters like larvicide, fogging, vaccination, and isolation to stop the spread of the dengue epidemic and explore the influence of the fractional order α (0.6 ≤ α ≤ 1) on the dengue transmission model. We apply a forwardbackward sweep scheme using the Adamstype predictorcorrector approach for solving the proposed control problem. Finally, the effects of optimal controls considering three different cases in the given model are discussed.







A Coinfection Model of Malaria and COVID19 in the context of Conformableorder derivative


Francis Ohene Boateng,


Abstract
: 

Confection infection diseases are increasing everyday on the globe which could be as a result of climate issues. Malaria and Coronavirus (COVID19) is now common infection in the SubSaharan Africa and this study examines both theoretical and numerical dynamics of coinfection of malaria and COVID19. The existence and uniqueness of solution is studied using the Fixedpoint theorem and Picard iterative method. Conformableorder derivative as a mathematical tool is used to investigate the dyanamics of the coinfected malaria and COVID19 model. It is concluded that Conformableorder derivative has a great impart on the spread of the disease.







Intuitionistic fuzzy generalized conformable fractional derivative


Atimad Harir,


Abstract
: 

The innovative idea of Atanassovs intuitionistic fuzzy sets (IFSs) is to get a more comprehensive and detailed description of the ambiguity and uncertainty by introducing a membership function and a nonmembership function. Each element in an IFS is expressed by an ordered pair, which is called an intuitionistic fuzzy number (IFN). In this paper, We introduced a new definition of the generalized conformable fractional derivative of the intuitionistic fuzzy numbervalued functions, Using this definition, we prove some results and with the help of $\alpha$cut set, the Hukuhara difference between intuitionistic fuzzy numbers are defined and proved. An intuitionistic fuzzy conformable nuclear decay equation with the initial condition given to show the new theorems and is solved under a new generalized conformable fractional derivative concept. 






Numerical approximation of fractional order transmission of worms in wireless sensor network in sense of Caputo operator


Aziz Khan,


Abstract
: 

In this manuscript, we study the uncertain attacking of some worms in wireless sensor network
(WSNs). We established a mathematical formulation for the WSNs model in the sense of Caputo
fractional operator. Applying the fixed point theory, certain theoretic solutions of existence and
uniqueness are considered for the fractional model. In addition, to investigate reproduction number,
local stability and HyersUlam stability of the proposed model. Furthermore, CorrectorPredictor
algorithm is utilized for fractional dynamics and numerical solutions. 






Existence of Solution for a New Class of Fractional Differential Equations


Meryeme El Harrak,
Hajar Sbai,
Meryeme El Harrak,
Ahmed Hajji,


Abstract
: 

In this paper, we investigate the global existence and uniqueness of a solution to a specific class of $\Phi$fractional differential equations with nonlocal condition in Banach spaces. Our problem is solved by constructing a special closed subset by using Banach fixed point theorem. Moreover, we give some illustrative examples which exhibited the applicability of the founded hypothesis. 






An Exploration of Discrete Fractional Calculus with Applications to Intermittent Oncological Modeling


Raegan Higgins,
Casey J Mills,


Abstract
: 

In this work, we use and unify time scale calculus and discrete fractional calculus to develop a new approach to modeling intermittent androgen deprivation therapy, a standard prostate cancer treatment. The novel time scale model previously developed assumes a constant length of time for on and offtreatment intervals. By creating a time scale that more accurately represents time data, we explore the use of fractional calculus to model treatment. Current fractional calculus theory only allows for strictly continuous or discrete domains. We create a strictly discrete time scale and construct a dynamic equation on this time scale. We then develop theory that allows us to calculate the fractional difference of this dynamic equation. Finally, we model intermittent androgen deprivation therapy using this fractional difference and find that an improved fit is achieved for most of the patients tested. 






Construction by the Tikhonov method of a nonzero solution of the homogeneous Cauchy problem for one equation with a fractional derivative


Irgashev Bakhrom,


Abstract
: 

The article constructs a nonzero solution to the homogeneous Cauchy problem for a homogeneous equation of even order with a fractional Caputo derivative. 






Fractional Model of Susceptibility Characteristics of ESBLproducing K. pneumoniae Infections for Carbapenems and PiperacillinTazobactam


CEMILE BAGKUR,


Abstract
: 

In this paper, we provide a fractional dynamical model of extended spectrum beta lactamase (ESBL)producing Klebsiella pneumoniae to visualise future perspective of susceptibility characteristics of K. pneumoniae infections for antibiotics; carbapenems and piperacillintazobactam (PTZ). Our approach is based on a system of fractional differential equations, where the fractional derivative was obtained in Caputo type. We demonstrate that the stability of the disease free equilibrium is locally stable and for the endemic equilibrium point the stability analysis is shown by Hurwitz criteria. Additionally, it is shown that the system has a unique solution due to the Lipschitz condition being provided. According to the SIStype model, while nonESBL K. pneumoniae infections are seemed to remain susceptible to both antibiotic groups, the opposite is true for ESBLproducing K. pneumoniae infections. For ESBLproducing K. pneumoniae infections, resistance to both carbapenems and PTZ will occur approximately 7.5 years later in line with the mathematical interpretation of the data. In addition, resistance to PTZ in ESBLproducing infections is observed to be less than carbapenems. As a result, it is seen that the formation of resistance against agents called last line antibiotics, which are preferred in the treatment of ESBLproducing infections, continues in the future, and measures to prevent this issue should be improved. 






Solution of Conformable Fractional Heat Equation Using Fractional Bessel Functions


Al. Naamneh,
Sh. Alsharif,
E. A. E.Rawashdeh,


Abstract
: 

With the use of the fractional Bessel function of the first kind of complex
order and the Wronskian matrix, a second order linearly independent solution
of the fractional Bessel equation is defined. Moreover as an application an exact
solution of a reformulated fractional type heat equation in a circular plate in
one and two dimension is obtained. 






Dynamical Behavior of Fractional Order Breast Cancer Model: an Analytical and Numerical Study


Hegagi Mohammed Ali,


Abstract
: 

In this paper, we investigate the dynamical behavior of the fractionalorder breast cancer model with modified parameters. This model formulated in Caputo fractional derivative sense. The nonnegative solutions of this fractionalorder model (FOM) are proved. We seek to study the equilibrium points and their stability of both the diseasefree and endemic cases for the FOM. Moreover, the basic reproduction number R0 is calculated and sensitivity analysis with respect to the parameters for the FOM is achieved. We solved this FOM by two methods one of them gave an analyticapproximate solution is called generalized MittagLeffler function method (GMLFM) and another method express to the numerical solution is named predictorcorrector method (PCM). The numerical simulations for the proposed model have been supported to verify the theoretical results obtained. 






Fractional Approach for BelousovZhabotinsky Reactions Model with Unified Technique


Chandrali Baishya,
P. Veeresha,


Abstract
: 

The BelousovZhabotinsky reaction model represents chemical oscillators that exhibit periodic vibrations as a result of complex physicchemical phenomena. The nonlinear behaviour exhibited by BelousovZhabotinsky model is the cause of Turing patterns, birth of spiral waves, rise of limit cycle attractors, and deterministic chaos in many chemical reaction processes. Due to these noteworthy characteristics, in this paper, we have analyzed mathematical BelousovZhabotinsky model by a novel numerical approach qHomotopy analysis transformation method. To interpret new observations, we have incorporated Caputo fractional derivative in the model. The numerical result are presented graphically and concerning the absolute error of solutions. With the help of the homotopy parameter curve, we have projected the convergence region with reference to diverse values of fractional derivative. This work establishes that the projected numerical algorithm is a wellorganized tool to analyze the multifaceted coupled partial differential equation representing BelousovZhabotinsky type reactions. 






Impulsive fractional differential equations under uncertainty: Application in fluid mechanics


Soheil Salahshour,
Morteza Pakdaman,
Ali Ahmadian,


Abstract
: 

In this research, we study impulsive fractional differential equations (IFDEs) under interval uncertainty using Laplace transforms. For this purpose, the solution of IFDEs is obtained under RiemannLiouville differentiability. Also, the BagleyTorvik equation involving additive delta function on the interval righthand side is solved to validate the theoretical results. the BagleyTorvik equation arises in fluid mechanics.







About Convergence and Order of Convergence of some Fractional Derivatives


Sabrina Roscani,
Lucas Venturato,


Abstract
: 

In this paper we establish some convergence results for RiemannLiouville, Caputo and CaputoFabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by D^{1\alpha}ff_p for p=1 and p=\infty and we prove that for both
Caputo and CaputoFabrizio operators the order of convergence is a positive real r\in(0,1). Finally, we compare the speed of convergence between Caputo and CaputoFabrizio operators obtaining that they are related by the Digamma function. 






Analytic solutions of a 3D propagated wave dynamical equation formulated by conformable calculus


Shaher Mohammed Momani,


Abstract
: 

Researchers show that there is a fundamental association between the symmetric and
traveling wave solutions. They have shown that all symmetric waves are traveling waves. In this paper,
we establish new analytic solution collections of nonlinear conformable timefractional wave dynamical
equation, equations of KhokhlovZabolotskaya (KZ) type in a complex domain. For this purpose,
we build a new definition of a symmetric conformable differential operator (SCDO). The operator
has a symmetric illustration in the open unit disk. By using SCDO, we propagate a class of special
wave dynamical equation type KZ equation. The consequences show that the obtainable methods are
powerful, dependable and formulate to apply to all classes of complex differential equations. 






A Convenience Approximate Method for Solving an Inverse Heat Conduction Problem


Rahmat Darzi,
Bahram Agheli,


Abstract
: 

In this research, fractional type onedimensional inverse heat conduction problem (FIHCP) have been studied. This problem is devoted to calculating the temperature distribution in its range and the thermal flux on the bound while the temperature is clear at some of the domain points. The new approach of homotopic perturbation method (NHPM) is employed to recovering unknown functions and obtaining a solution for the problem. At the end, some appropriate examples are given for introducing and implementing the proposed approach in solving FIHCP. 






Theory of stochastic pantograph differential equations with psiCaputo fractional derivative


D. VIVEK,
SABRI T. M. THABET,
K. KANAGARAJAN,


Abstract
: 

In this paper, we mainly study the existence of analytical solution of stochastic pantograph differential equations. The standard Picard’s iteration method is used to obtain the theory. 




