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02- Progress in Fractional Differentiation and Applications
An International Journal



Certain traveling wave solutions for fractional extended nonlinear Schro ̈ dinger equations

Rania Momani, Asad Frihat, Mohammed Alabedalhadi, Shrideh Al-Omari,
Abstract :
The fractional extended nonlinear Schro ̈dinger equation is investigated in this study. We effectively convert the governing model into a nonlinear ordinary differential equation using a sophisticated traveling wave transformation. By employing the Ansatz approach, we generate and study traveling wave solutions, specifically bright and kink wave solutions that satisfy certain parameter requirements. To gain deeper insights into the physical properties of the resulting traveling wave solutions, we employ 3D and 2D graphs at specific parameter values. This visualization provides valuable information about the characteristics and behavior of the solutions. Furthermore, we explore the influence of the fractional derivative on the obtained solution characteristics. Our research findings contribute to the understanding of nonlinear dynamics and demonstrate the existence and characteristics of bright and kink wave solutions in the fractional extended nonlinear Schro ̈dinger equation. Moreover, we enhance our understanding of fractional derivatives and their effects on wave propagation in nonlinear media.


Computational Solutions for Fractional Atangana- Baleanu PDEs: An Exploration of Sawi Transform and Homotopy Perturbation Method

Rania Saadeh, Abdelilah Kamal Sedeeg, Ahmad Qazza,
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

Ahmed Abdelaziz Elsayed,
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

Rania Momani, Asad Frihat, Mohammed Alabedalhadi, Shrideh Al-Omari,
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

Ahmet Bekir,
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

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.


From Calculus to α- Calculus

Mohammed Shehata,
Abstract :
In the previous definitions of fractional (α−) calculus there were a mismatch in some properties to classical calculus. This is because that definitions were built in an unusual way, they were built from the definition of integral to derivative. For example, in the Riemann-Liouvellie definition of derivative,the derivative of a constant may be not zero. In this paper we will overcome incompatibility, by accurately construct α− derivative and α− integral by usual way, so it coincides with the classical ones. We also generalized some basic formulas and theorems.


A Fractional-Order Modeling and Sensitivity Analysis in the Investigation of Colorectal Cancer

Prof. Dr. Evren Hincal,
Abstract :
This research paper focuses on studying colorectal cancer using a sensitivity analysis via the fractional differential equations (FDE) model. The study aims to develop an accurate model for predicting the progression of the disease and its response to treatments, by capturing all the important cells and factors involved. The existence and uniqueness of solutions are proven using the Banach contraction principle, and global stability is shown using the Lyapunov function. Results show that the Epithelial cell growth rate ($\lambda_E$), TGF-$\beta$-induced growth rate of epithelial cells ($\gamma_{TE}$), and competition rate for nutrients with normal cells ($\lambda_{KE}$) are the most sensitive parameters, with the concentration of the epithelial cells ($E(t)$) and APC genes ($A(t)$) as most sensitive compartments. The research concludes that the developed model can be used as a powerful tool for predicting the diseases behavior and assessing the efficacy of different treatment strategies. Overall, this study provides valuable insights into the treatment of colorectal cancer.\


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.


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 Newton Explicit Group Method for Time-Fractional Nonlinear Porous Medium Equations

Jackel Vui Lung Chew, Andang Sunarto, Jumat Sulaiman, Mohammad Fadhli Asli‬‬‬,
Abstract :
This paper presents a fractional Newton explicit group method to solve time-fractional nonlinear porous medium equations. The method utilizes implicit finite difference schemes with the Caputo time-fractional derivative operator. The paper aims to evaluate the accuracy and efficiency of the proposed method in solving initial boundary value problems of porous medium equations at different orders of time-fractional derivatives. The method is experimented with repeatedly using several large systems of equations to illustrate the consistency of the method’s performance. The method is also experimented with by solving some physics problems, which can show the method’s efficacy in solving realistic phenomena.


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.


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.


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.


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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