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02- Progress in Fractional Differentiation and Applications
An International Journal



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 well-known operators. We use the effective analytical method known as the homotopy perturbation method (HPM) to get generalised Burgers equations solution. A real-world 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

Abstract :
In this paper, the conformable Schr\"odinger equation for hydrogen-like 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 Integro-Differential 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 double-conforming Sumudu-Elzaki transformation (CDSET) and a decomposition method are combined to develop a new method that solves the nonlinear sub-problem considering some specific conditions. This combination can be referred to as the Conformal Double Sumudu-Elzaki 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 Liouville-Weyl 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 environmental-disruptive 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 H-functions, incomplete H-functions, and incomplete I-functions, and demonstrating their relationship with existing results.


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

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 multi-order 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 Korteweg-de Vries-Burgers (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, Ho-Hon 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 fractal-fractional 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 Pig-Human Population by Riemann-Liouville’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 forward-backward sweep scheme using the Adams-type predictor-corrector 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 COVID-19 in the context of Conformable-order 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 (COVID-19) is now common infection in the Sub-Saharan Africa and this study examines both theoretical and numerical dynamics of coinfection of malaria and COVID-19. The existence and uniqueness of solution is studied using the Fixed-point theorem and Picard iterative method. Conformable-order derivative as a mathematical tool is used to investigate the dyanamics of the coinfected malaria and COVID-19 model. It is concluded that Conformable-order 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 number-valued 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 Hyers-Ulam stability of the proposed model. Furthermore, Corrector-Predictor 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 off-treatment 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 ESBL-producing K. pneumoniae Infections for Carbapenems and Piperacillin-Tazobactam

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 piperacillin-tazobactam (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 SIS-type model, while non-ESBL K. pneumoniae infections are seemed to remain susceptible to both antibiotic groups, the opposite is true for ESBL-producing K. pneumoniae infections. For ESBL-producing 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 ESBL-producing 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 ESBL-producing 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 fractional-order breast cancer model with modified parameters. This model formulated in Caputo fractional derivative sense. The non-negative solutions of this fractional-order model (FOM) are proved. We seek to study the equilibrium points and their stability of both the disease-free 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 analytic-approximate solution is called generalized Mittag-Leffler function method (GMLFM) and another method express to the numerical solution is named predictor-corrector method (PCM). The numerical simulations for the proposed model have been supported to verify the theoretical results obtained.


Fractional Approach for Belousov-Zhabotinsky Reactions Model with Unified Technique

Chandrali Baishya, P. Veeresha,
Abstract :
The Belousov-Zhabotinsky reaction model represents chemical oscillators that exhibit periodic vibrations as a result of complex physic-chemical phenomena. The non-linear behaviour exhibited by Belousov-Zhabotinsky 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 Belousov-Zhabotinsky model by a novel numerical approach q-Homotopy 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 well-organized tool to analyze the multifaceted coupled partial differential equation representing Belousov-Zhabotinsky 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 Riemann-Liouville differentiability. Also, the Bagley-Torvik equation involving additive delta function on the interval right-hand side is solved to validate the theoretical results. the Bagley-Torvik 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 Riemann-Liouville, Caputo and Caputo-Fabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by ||D^{1-\alpha}f-f||_p for p=1 and p=\infty and we prove that for both Caputo and Caputo-Fabrizio operators the order of convergence is a positive real r\in(0,1). Finally, we compare the speed of convergence between Caputo and Caputo-Fabrizio operators obtaining that they are related by the Digamma function.


Analytic solutions of a 3-D 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 time-fractional wave dynamical equation, equations of Khokhlov-Zabolotskaya (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 one-dimensional 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 psi-Caputo fractional derivative

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.

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