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    A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay
    (Hindawi Ltd, 2022) Liaqat, Muhammad Imran; Khan, Adnan; Akgul, Ali; Ali, Md. Shajib
    Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He's and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.
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    Adaptation on power series method with conformable operator for solving fractional order systems of nonlinear partial differential equations
    (Pergamon-Elsevier Science Ltd, 2022) Liaqat, Muhammad Imran; Khan, Adnan; Akgul, Ali
    The aim of this research work is to modify the power series solution method to fractional order in the sense of conformable derivative to solve a coupled system of nonlinear fractional partial differential equations. We called it the conformable fractional power series method. To evaluate its efficiency and consistency, absolute errors of three problems are considered numerically. Consequences established that our recommended method is unpretentious, accurate, valid, and capable. When solving the nonlinear complications, it has a powerful superiority over the homotopy analysis and Adomian decomposition methods. Additional as in the residual power series method through generating the coefficients for a series, it is compulsory to calculate the fractional derivatives on every occasion, whereas this method only needs the idea of equating coefficients. The convergence and error analyses of the series solutions are also presented.(c) 2022 Elsevier Ltd. All rights reserved.
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    Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform
    (Hindawi Ltd, 2022) Shah, Nehad Ali; El-Zahar, Essam R.; Akgul, Ali; Khan, Adnan; Kafle, Jeevan
    A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.
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    Decision Making Under Pythagorean Fuzzy Soft Environment
    (World Scientific Publ Co Pte Ltd, 2023) Khan, Adnan; Farman, Muhammad; Akgul, Ali
    This research article illustrates the notion of strong and complete Pythagorean fuzzy soft graphs (PFSGs). Different operations on PFSGs including union of two PFSGs, join of two PFSGs, lexicographic product of two PFSGs, strong product of two PFSGs, Cartesian product of two PFSGs, composition of two PFSGs are also analysed here. Some properties related to these products are discussed here. The idea of complememt of a PFSG is also eloborated here. Moreover, we establish the application of PFSG in the decision making (DM) problem.
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    Dynamical Behavior andMathematical Analysis of Fractional Order Smoking Model
    (L and H Scientific Publishing, LLC, 2023) Ahmad, Aqeel; Farman, Muhammad; Akgul, Ali; Khan, Adnan
    In this paper the fractional order smoking model is represent with Caputo and Caputo Fabrizio fractional derivative operator of order ? ? (0,1] for dynamical transmission of smoking. Human beings face dangerous diseases caused by smoking, including arms, lungs, stomach, cervix, breast, pancreatic cancer and many others. Stability and qualitative analysis of model is studied to show the dynamical behaviour of the model in feasible region. It’s important to note that a more powerful approach for computing convergent solutions is applied for mathematical models based on a fractional order differential equation structure. Study of the convergence is often provided to demonstrate the process’s effectiveness. It shows the stability, uniqueness and applicability of the model for the control of smoking in the society. Numerical simulation are established to show the actual behavior of the smoking spread. © 2023 L&H Scientific Publishing, LLC. All rights reserved.
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    Fractional Order Glucose Insulin Model with Generalized Mittag-Leffler Kernel
    (Natural Sciences Publishing, 2023) Akgul, Ali; Farman, Muhammad; Ahmad, Aqeel; Khan, Adnan; Zahran, Siraj; Awad, Wasan Shakir
    TIn this paper, We formulate a fractional-order mathematical model for the populations of diabetic patients consist three-compartment G, X, and I. Diabetes Model is investigated with fractal-fractional operator for normal and type-1 diabetes. Also, the deterministic mathematical model for diabetes mellitus is investigated with the effect of the fractional parameters. Solutions are derived to investigate the influence of fractional operator which shows the impact of the disease for type-1 diabetes. The existence and uniqueness results of the fractional-order model are derived using fixed point theory. Simulation has been made for developed solutions of fractional order diabetes model to check the actual behavior of a normal person as well as a type-1 diabetes patient. © 2023.
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    Some results on generalised Euler-type integrals related to the k-Wright function
    (Inderscience Enterprises Ltd, 2023) Asif, Muhammad; Khan, Adnan; Akgul, Ali; Shimelis, Biniyam
    Special functions such that Zeta, Bessel, Whittaker, Struve, Airy, Weber-Hermite and k-Wright functions are obtained as a solution to complex differential equations in engineering. In this work, generalised Euler-type integrals involving k-Wright function are suggested. Some special cases of this type of generalised integrals that are corresponding to well-known results in the literature are also inferred. We also study extended beta and associated functions (Gauss hypergeometric and confluent hypergeometric functions) connected to k-Wright function. For the newly extended beta, Gauss hypergeometric and confluent hypergeometric functions.
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    The Extended Laguerre Polynomials {Aq,n (a)} (x) Involving qFq, q > 2
    (Hindawi Ltd, 2022) Khan, Adnan; Kalim, Muhammad; Akguel, Ali; Jarad, Fahd
    In this paper, for the proposed extended Laguerre polynomials {A(q,n )((alpha))}, the generalized hypergeometric function of the type (F)(q)(q), q > 2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue's formula are determined. Some corollaries are also discussed at the end.
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    The K Extended Laguerre Polynomials Involving {Ar,n,k (?) (x)}rFr, r > 2
    (Hindawi Ltd, 2022) Khan, Adnan; Mateen, M. Haris; Akguel, Ali; Ali, Md. Shajib
    In this manuscript, we present the generalized hypergeometric function of the type F-r(r), r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials {A(r, n k)((alpha))(x)}. Additionally, we describe the K generating function, K recurrence relations, and KS Rodrigues formula.
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    Two Dimensional Laplace Transform Coupled with the Marichev-Saigo-Maeda Integral Operator and the Generalized Incomplete Hypergeometric Function
    (Mdpi, 2021) Khan, Yasir; Khan, Adnan; Shaeel, Muhammad; Akguel, Ali
    This paper represents the processing of the two-dimensional Laplace transform with the two-dimensional Marichev-Saigo-Maeda integral operators and two-dimensional incomplete hypergeometric function. This article provides an entirely new perspective on the Marichev-Saigo-Maeda operators and incomplete functions. In addition, we have included some interesting results, such as left-sided Saigo-Maeda operators and right-sided Saigo-Maeda operators, making this a good direction for symmetry analysis.