02- Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 Analyzing the Stability of Caputo Fractional Difference Equations with Variable Orders Abstract : This research aims to explore the complex stability of Fractional Variable Order Discrete Time Systems by introducing new stability criteria. To achieve this, we utilize the properties of Volterra convolution-type systems and the Z-transform methodology. We validate these criteria through practical numerical experiments, showing their usefulness in real-world engineering and scientific applications.

 Mathematical modeling of Cholera dynamics and Analysis using Caputo fractional Operator with optimal control Abstract : In this study, we developed a cholera model using the Caputo fractional operator with optimal control strategies to address the dynamic nature of cholera transmission, employing bifurcation analysis. Initially, by applying fixed point theory, we analyzed the existence and uniqueness of the solutions of fractional order derivatives. Furthermore, utilizing the next-generation matrix, we computed the basic reproduction number, crucial for assessing disease dynamics. If this number falls below unity, the equilibrium point remains disease-free, both locally and globally stable, as verified through the Jacobian matrix, Metzler matrix, and Lyapunov function. Otherwise, cholera persistence equilibrium occurs. In addition, we extended the model to include optimal control by integrating three controls: a prevention effort to protect susceptible individuals from contracting the disease and Vibrio cholera, a treatment for those infected with cholera through quarantine, and water sanitation strategies to reduce infectious transmission. These controls were determined using the Pontryagin minimum principle. The model validation was done using numerical experiments. Based on the numerical simulation of fractional order, we observed that the order of derivatives has an impact on controlling disease transmission. The study of this work underscores the benefit of integrating fractional order derivatives with optimal control strategies to mitigate cholera outbreaks effectively. The proposed optimal control framework provides a systematic approach to evaluate the impact of various interventions and inform public health policies. Future research directions include model extensions to incorporate spatial heterogeneity and real-world data integration to further enhance the applicability and robustness of the control strategies.

 Computational Solutions for Fractional Atangana- Baleanu PDEs: An Exploration of Sawi Transform and Homotopy Perturbation Method Abstract : This article introduces an efficient and innovative method for solving fractional Atangana-Baleanu partial differential equations (PDEs) by combining the Sawi transform and the homotopy perturbation method. The proposed approach stands out for its computational efficiency, reducing the time and resources required for solving complex and nonlinear equations. The article’s novelty lies in its interdisciplinary approach, bridging the gap between theoretical and applied aspects of fractional calculus. By offering a robust and versatile solution framework, this method is applicable to a wide array of problems in fields such as fluid mechanics, biological systems, and nonlinear wave equations. To show the efficiency of the presented method, some numerical applications have been solved and discussed. The article sets the stage for future research by providing a method that can be adapted and extended to other types of fractional differential equations and operators. It serves as a significant contribution to the field, opening new avenues for research and application.

 A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations Abstract : Due to the vast range of applications in many scientific c domains, researchers have recently become interested in quadratic Riccati differential equations of fractional order and their solutions. In this research, we propose a new method for solving particular classes of quadratic Riccati fractional differential equations that combine the Ramadan group transform (RGT) and a variant of the accelerated Adomian decomposition method (AADM). It is worth noting that RGT is a generalization for both Laplace and Sumudu transforms. El-kalla proposed the AADM, where the main advantages of AADM are that the polynomials generated are recursive and do not have derivative terms, so the formula is easy to programme and saves much time on the same processor as the traditional Adomian polynomials formula, and thus the solution obtained using this proposed hybrid method accelerated Ramadan group Adomian decomposition method (RGAADM), converges faster than the traditional Adomian decomposition method According to the findings of this work, the solutions obtained by solving a class of quadratic Riccati differential equations of fractional order are extremely compatible with those found via exact solutions. We obtained good performance in all applied cases, which may lead to a promising strategy for many applications.

 Certain traveling wave solutions for fractional extended nonlinear Schrö- 2 dinger equations Abstract : In this study, we investigate a fractional extended nonlinear Schrödinger equation, which is an extension to the linear Schrödinger. We convert the governing model into a nonlinear ordinary differential equation by using certain sophisticated traveling wave transformation. In addition, we generate traveling wave solutions by using the Ansatz approach. Also, we provide 2D and 3D graphs at special parameter values to acquire deeper knowledge on the physical properties of the resulting traveling wave solutions. Further, we describe the way that the characteristics of the solu- tions be affected by the fractional derivative and the findings offer major additions to understand the non-linear dynamics and demonstrate the presence of the bright and kink wave solutions within the fractional extended nonlinear Schrödinger equation. Moreover, we improve the understanding of the fractional derivatives and their effects on wave propagation in the nonlinear media. Over and above, we provide explanations for some characteristics of the ruling model, which have been studied by considering a new classical derivative by using a fractional derivative which makes the study important and novel.

 Exploring the new analytical wave solutions for M-fractional stochastic Nizhnik-Novikov-Veselov system by an efficient analytical technique Abstract : In this paper, we succeed to obtain the new analytical wave solutions to the truncated M-fractional stochastic Nizhnik-Novikov-Veselov (SNNV) system by utilizing the modified simplest equation technique along the multiplicative noise effect. This system is an extension of KdV equation and have many applications in plasma, crystal network, shallow water waves and others. Achieved solutions are verified with the use of Mathematica software. Some of the gained solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots. The gained solutions are helpful in the further research of concerned model. Finally, this technique is simple, fruitful and reliable to handle the nonlinear PDEs.

 Analysis of Burger Equation Using HPM with General Fractional Derivative 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.

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

 On fractional inequalities for general fractional operators Abstract : he Leibniz and the power law rules of differentiations do not hold for fractional derivatives with non-local kernels. Instead, infinite series representations were derived for the Leibniz and the power law rules of fractional derivatives. These rules produce heavy calculations, and they are not practical in implementations. In this paper, we derive certain inequalities of the general fractional operators of Caputo type. These inequalities are similar to the Leibniz and the power law rules for integer derivatives, where we replace the equalitys by inequalities. Because the general fractional operators involve many fractional operators as particular cases, the current study will involve several types of fractional differential and integral operators and include some recent studies in the literature. We present several new inequalities to illustrate the applicability of the obtained results.

 Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials Abstract : In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian system $$\label{eq1} \left\{ \begin{array}{l} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\u\in H^{\alpha}(\mathbb{R}), \end{array}\right.$$ 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.

 Approximating functions by Fractional Lagrange Polynomials Abstract : Approximating functions by matching their values at given points is very important in mathematics and has many applications. In this paper we approximate a function by a new function which can be considered as a generalization of Lagrange and Hermite polynomials. The interpolation error is derived based on the comformable fractional derivative.

 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 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 Hybrid Technique for Space-Time Fractional Parabolic Differential Equations Abstract : The main objective of the present article is to present an accurate efficient technique for approximating the solutions of the diffusion equation of fractional space-time Levy-Feller type. The suggested method depends on a spectral collocation algorithm and implicit non-standard finite difference method. The fractional space-time Levy-Feller diffusion equations are acquired by updating the classical diffusion equations such that the time derivative of the first order will be the fractional Caputo operator and the second-order space derivative will modify to be the Riesz-Feller derivative. The utilized spectral method uses the well known Legendre orthogonal polynomials and the Gauss-Lobatto Chebyshev collocation points. The method depends basically on the conversion of these kinds of fractional differential equations into a system of algebraic equations which may be solved easily using appropriate techniques. The numerical outcomes are presented in the form of tables and graphs to emphasize the reliability of the introduced technique to approximate the solutions of the fractional spacetime-Levy-Feller diffusion equations.

 A Coinfection Model of Malaria and COVID-19 in the context of Conformable-order derivative 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.

 Tow numerical approaches for solving fractional model of Chemical Kinetics problem via Chebyshev polynomials Abstract : We give two new approaches for solving the fractional model of the Chemical Kinetics (CK) problem. The approaches (Chebyshev collocation (CC) method and the Chebyshev Galerkin (CG) method) are constituted of the Chebyshev polynomials, Galerkin method, and the collocation method where these techniques are to convert the system of differential equations into a system of algebraic equations which can be solved easily. Also, we study the error analysis for the methods. This work is expected to contribute to the vast advantage of Haar wavelets in chemical science. A complete agreement is achieved between our new methods. Moreover, We also checked the stability of the proposed methods.

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

 Solution of Conformable Fractional Heat Equation Using Fractional Bessel Functions 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 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 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 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 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 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 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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