Browsing by Author "Shah, Kamal"

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A NEW NUMERICAL TREATMENT FOR FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON NON-DISCRETIZATION OF DATA USING LAGUERRE POLYNOMIALS
(2020) Khan, Adnan; Shah, Kamal; Arfan, Muhammad; Abdeljawad, Thabet; Jarad, Fahd; 234808
In this research work, we discuss an approximation techniques for boundary value problems (BVPs) of differential equations having fractional order (FODE). We avoid the method from discretization of data by applying polynomials of Laguerre and developed some matrices of operational types for the obtained numerical solution. By applying the operational matrices, the given problem is converted to some algebraic equation which on evaluation gives the required numerical results. These equations are of Sylvester types and can be solved by using matlab. We present some testing examples to ensure the correctness of the considered techniques.
Citation Count: Thabet, Abdeljawad...et.al. (2023). "An Analytical Study Of Fractional Delay Impulsive Implicit Systems With Mittag-Leffler Law", Applied and Computational Mathematics, Vol.22, No.1, pp.34-44.
An Analytical Study Of Fractional Delay Impulsive Implicit Systems With Mittag-Leffler Law
(2023) Abdeljawad, Thabet; Shah, Kamal; Abdo, Mohammed S.; Jarad, Fahd; 234808
The Atangana-Baleanu-Caputo fractional derivative is a novel operator with a non-singular Mittag-Leffler kernel that we use to solve a class of Cauchy problems for delay impulsive implicit fractional differential equations. We also show the existence and uniqueness of the solution to the proposed problem. Our study makes use of the Gro center dot nwall inequality in the context of the Atangana-Baleanu fractional integral. Additionally, by the use of fixed point theorems due to Banach, Schaefer, and nonlinear functional analysis, necessary and sufficient conditions are developed under which the considered problem has at least one solution. By providing a relevant example, the results are demonstrated.
Citation Count: Haq, Fazal...et al. (2019). "Application of a hybrid method for systems of fractional order partial differential equations arising in the model of the one-dimensional Keller-Segel equation", European Physical Journal Plus, Vol. 134, No. 9.
Application of a hybrid method for systems of fractional order partial differential equations arising in the model of the one-dimensional Keller-Segel equation
(Springer Heidelberg, 2019) Haq, Fazal; Shah, Kamal; Al-Mdallal, Qasem M.; Jarad, Fahd; 234808
In this paper, we apply a hybrid method due to coupling the Laplace transform with the Adomian decomposition method (LADM) for solving nonlinear fractional differential equations that appear in the model of Keller-Segel equations with one dimension. We explain the adopted method is with several examples. It turns out that the reliability of LADM and the reductions in computations show that LADM is widely applicable. We also compare our results with the results of homotopy decomposition method (HDM).
Citation Count: Ali, S...et al. (2019). "Computation of Iterative Solutions Along With Stability Analysis to A Coupled System of Fractional Order Differential Equations",Advances in Difference Equations, Vol. 2019, No. 1.
Computation of Iterative Solutions Along With Stability Analysis to A Coupled System of Fractional Order Differential Equations
(Springer International Publishing, 2019) Ali, Sajjad; Abdeljawad, Thabet; Shah, Kamal; Fahd Jarad; Arif, Muhammad; 234808
In this research article, we investigate sufficient results for the existence, uniqueness and stability analysis of iterative solutions to a coupled system of the nonlinear fractional differential equations (FDEs) with highier order boundary conditions. The foundation of these sufficient techniques is a combination of the scheme of lower and upper solutions together with the method of monotone iterative technique. With the help of the proposed procedure, the convergence criteria for extremal solutions are smoothly achieved. Furthermore, a major aspect is devoted to the investigation of Ulam–Hyers type stability analysis which is also established. For the verification of our work, we provide some suitable examples along with their graphical represntation and errors estimates.
Citation Count: Abdeljawad, Thabet...et al. (2020). "Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method", Alexandria Engineering Journal, Vol. 59, No. 4, pp. 2391-2400.
Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method
(2020) Abdeljawad, Thabet; Amin, Rohul; Shah, Kamal; Al-Mdallal, Qasem; Jarad, Fahd; 234808
This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). Based on these techniques, we find the numerical solution to var-ious systems of FODEs. We compare the obtain solution with the exact solution of the considered problems at integer orders. Also, we compute the maximum absolute error to demonstrate the effi-ciency and accuracy of the proposed method. For the illustration of our results we provide four test examples. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is defined in the Caputo sense. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).
Citation Count: Ali, Arshad...et al. (2019). "Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations", Advances in Difference Equations.
Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations
(Springer Open, 2018) Ali, Arshad; Shah, Kamal; Jarad, Fahd; Gupta, Vidushi; Abdeljawad, Thabet; 234808
In this paper, we study a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs). We use the Schaefer fixed point and Banach contraction theorems to obtain conditions for the existence and uniqueness of positive solutions. We discuss Hyers-Ulam (HU) type stability of the concerned solutions and provide an example for illustration of the obtained results.
Citation Count: Jarad, Fahd...et al. (2020). "Existence and stability results to a class of fractional random implicit differential equations involving a generalized hilfer fractional derivative", Discrete and Continuous Dynamical Systems - Series S, Vol. 13, No. 3, pp. 723-739.
Existence and stability results to a class of fractional random implicit differential equations involving a generalized hilfer fractional derivative
(2020) Jarad, Fahd; Harikrishnan, Sugumaran; Shah, Kamal; Kanagarajan, Kuppusamy; 234808
In this paper, the existence, uniqueness and stability of random implicit fractional differential equations (RIFDs) with nonlocal condition and impulsive effect involving a generalized Hilfer fractional derivative (HFD) are discussed. The arguments are discussed via Krasnoselskii's fixed point theorems, Schaefer's fixed point theorems, Banach contraction principle and Ulam type stability. Some examples are included to ensure the abstract results. © 2020 American Institute of Mathematical Sciences. All rights reserved.
Citation Count: Abdo, Mohammed S...et al. (2020). "Existence of positive solutions for weighted fractional order differential equations", Chaos, Solitons and Fractals, Vol. 141.
Existence of positive solutions for weighted fractional order differential equations
(2020) Abdo, Mohammed S.; Abdeljawad, Thabet; Ali, Saeed M.; Shah, Kamal; Jarad, Fahd; 234808
In this paper, we deliberate two classes of initial value problems for nonlinear fractional differential equations under a version weighted generalized of Caputo fractional derivative given by Jarad et al. (2020a) [25]. We get a formula for the solution through the equivalent fractional integral equations to the proposed problems. The existence and uniqueness of positive solutions have been obtained by using lower and upper solutions. The acquired results are demonstrated by building the upper and lower control functions of the nonlinear term with the aid of Banach and Schauder fixed point theorems. The acquired results are demonstrated by pertinent numerical examples along with the Bashforth Moulton prediction correction scheme and Matlab. © 2020 Elsevier Ltd
Citation Count: Alqudah, Manar A...et al. (2020). "Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative", Advances in Difference Equations, Vol. 2020, No. 1.
Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative
(2020) Alqudah, Manar A.; Abdeljawad, Thabet; Eiman; Shah, Kamal; Jarad, Fahd; Al-Mdallal, Qasem; 234808
This manuscript considers a nonlinear coupled system under nonsingular kernel type derivative. The considered problem is investigated from two aspects including existence theory and approximate analytical solution. For the concerned qualitative theory, some fixed point results are used. While for approximate solution, the Laplace transform coupled with Adomian method is applied. Finally, by a pertinent example of prey–predator system, we support our results. Some graphical presentations are given using Matlab. © 2020, The Author(s).
Citation Count: Khan, Sajjad Ali...et al. (2019). "Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative", Chaos, Vol. 29, No. 1.
Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative
(Amer Inst Physics, 2019) Khan, Sajjad Ali; Shah, Kamal; Zaman, Gül; Jarad, Fahd; 234808
In this paper, taking fractional derivative due to Caputo and Fabrizo, we have investigated a biological model of smoking type. By using Sumudu transform and Picard successive iterative technique, we develop the iterative solutions for the considered model. Furthermore, some results related to uniqueness of the equilibrium solution and its stability are discussed utilizing the techniques of nonlinear functional analysis. The dynamics of iterative solutions for various compartments of the model are plotted with the help of Matlab. Published under license by AIP Publishing.
Citation Count: Shah, Kamal...et al. (2021). "Investigation of COVID-19 mathematical model under fractional order derivative", Mathematical Modelling of Natural Phenomena, Vol. 16.
Investigation of COVID-19 mathematical model under fractional order derivative
(2021) Shah, Kamal; Arfan, Muhammad; Deebani, Wejdan; Shutaywi, Meshal; Baleanu, Dumitru; 56389
The given article is devoted to presentation of some results regarding existence and uniqueness of solution to a fractional order model that addressing the effect of immigration on the transmission dynamics of a population model. Further, in view of this investigation the effect of immigration have been checked on transmission of recent pandemic known as Corona virus COVID-19. The concerned results have been established by using fixed point theory approach. After investigation qualitative analysis of the considered model, by applying Laplace transform along with decomposition method, we have calculated some series type results for the concerned model. The unknown quantities of each equation have been decomposed into small quantities to calculate each small quantity very easily for the series solution by adding first few terms of the said quantities. Approximate results of some testing data with different cases are given to illustrate the results. © The authors. Published by EDP Sciences, 2021.
Citation Count: Ali, Arshad...et al. (2020). "Mathematical analysis of nonlocal implicit impulsive problem under caputo fractional boundary conditions", Mathematical Problems in Engineering, Vol. 2020.
Mathematical analysis of nonlocal implicit impulsive problem under caputo fractional boundary conditions
(2020) Ali, Arshad; Gupta, Vidushi; Abdeljawad, Thabet; Shah, Kamal; Jarad, Fahd; 234808
This paper is related to frame a mathematical analysis of impulsive fractional order differential equations (IFODEs) under nonlocal Caputo fractional boundary conditions (NCFBCs). By using fixed point theorems of Schaefer and Banach, we analyze the existence and uniqueness results for the considered problem. Furthermore, we utilize the theory of stability for presenting HyersUlam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam-Rassias stability results of the proposed scheme. Finally, some applications are offered to demonstrate the concept and results. The whole analysis is carried out by using Caputo fractional derivatives (CFDs). Copyright © 2020 Arshad Ali et al.
Citation Count: Din, Rahim Ud...et al. (2020). "Mathematical study of sir epidemic model under convex incidence rate", AIMS Mathematics, Vol. 5, No. 6, pp. 7548-7561.
Mathematical study of sir epidemic model under convex incidence rate
(2020) Din, Rahim Ud; Shah, Kamal; Alqudah, Manar A.; Abdeljawad, Thabet; Jarad, Fahd; 234808
In this manuscript, we examine the SIR model under convex incidence rate. We first formulate the famous SIR model under the aforesaid incidence rate. Further, we develop some sufficient analysis to examine the dynamical behavior of the model under consideration. We compute the basic reproductive number R0. Also we study the global attractivity results via using Dulac function theory. Further, we also provide some information about the stability of the endemic and disease free equilibria for the considered model. In addition, we use nonstandard finite difference scheme to perform numerical simulation of the considered model via using Matlab. We provide different numerical plots for two different values of contact rate and taking various initial values for compartments involved in the considered model. © 2020 the Author(s), licensee AIMS Press.
Citation Count: Harikrishnan, S...et al. (2018). Note on the solution of random differential equations via psi-Hilfer fractional derivative, Advances in Difference Equations.
Note on the solution of random differential equations via psi-Hilfer fractional derivative
(Springer International Publishing AG, 2018) Harikrishnan, S.; Baleanu, Dumitru; Shah, Kamal; Kanagarajan, K.; 56389
This manuscript is devoted to an investigation of the existence, uniqueness and stability of random differential equations with psi-Hilfer fractional derivative. The concerned investigation of existence and uniqueness is obtained via the Schauder fixed point theorem and Banach contraction principle, respectively. Furthermore, for the respective solutions, some results related to different kinds of Ulam type stability including Hyers-Ulam, and generalized Hyers-Ulam, Hyers-Ulam-Rassias are obtained.
Citation Count: Gul, Rozi...et al. (2021). "On a class of boundary value problems under ABC fractional derivative", Advances in Difference Equations, Vol. 2021, No. 1.
On a class of boundary value problems under ABC fractional derivative
(2021) Gul, Rozi; Shah, Kamal; Khan, Zareen A.; Jarad, Fahd; 234808
In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results. © 2021, The Author(s).
Citation Count: Din, Anwarud...et al. (2020). "On a new conceptual mathematical model dealing the current novel coronavirus-19 infectious disease", Results in Physics, Vol. 19.
On a new conceptual mathematical model dealing the current novel coronavirus-19 infectious disease
(2020) Din, Anwarud; Shah, Kamal; Seadawy, Aly; Alrabaiah, Hussam; Baleanu, Dumitru; 56389
The present paper describes a three compartment mathematical model to study the transmission of the current infection due to the novel coronavirus (2019-nCoV or COVID-19). We investigate the aforesaid dynamical model by using Atangana, Baleanu and Caputo (ABC) derivative with arbitrary order. We derive some existence results together with stability of Hyers-Ulam type. Further for numerical simulations, we use Adams–Bashforth (AB) method with fractional differentiation. The mentioned method is a powerful tool to investigate nonlinear problems for their respective simulation. Some discussion and future remarks are also given. © 2020 The Author(s)
Citation Count: Shah, Kamal; Jarad, Fahd; Abdeljawad, Thabet (2020). "On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative", Alexandria Engineering Journal, Vol. 59, No. 4, pp. 2305-2313.
On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative
(2020) Shah, Kamal; Jarad, Fahd; Abdeljawad, Thabet; 234808
In this manuscript, we investigate epidemic model of dengue fever disease under Caputo and Fabrizio fractional derivative abbreviated as (CFFD). The respective investigation is devoted to qualitative theory of existence of solution for the model under consideration by using fixed point theory. After the establishing the qualitative aspect, we apply Laplace transform coupled with Adomian decomposition method to develop an algorithm for semi analytical solution under CFFD. In same line, we also develop the semi analytical solution for the considered model under usual Caputo fractional derivative (CFD). By using Matlab, we present both type of solutions via graphs and hence give some comparative remarks about the nature of the solutions of both derivatives.
Citation Count: Shah, Kamal;...et.al. (2023). "On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method", CMES - Computer Modeling in Engineering and Sciences, Vol.136, No.2, pp.1457-1472.
On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method
(2023) Shah, Kamal; Abdeljawad, Thabet; Jarad, Fahd; Al-Mdallal, Qasem; 234808
This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system. For the proposed study, we consider the model under the conformable fractional order derivative (CFOD). We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution. Indeed, Schauder and Banach fixed point theorems are utilized to prove our claim. Further, an algorithm for the approximate analytical solution to the proposed problem has been established. In this regard, the conformable fractional differential transform (CFDT) technique is used to compute the required results in the form of a series. Using Matlab-16, we simulate the series solution to illustrate our results graphically. Finally, a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.
Citation Count: Ali, Sajjad; Shah, Kamal; Jarad, Fahd, "On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations", Mathematical Methods in the Applied Sciences, Vol. 42, No. 3, pp. 969-981, (2019).
On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations
(Wiley, 2019) Ali, Sajjad; Shah, Kamal; Jarad, Fahd; 234808
In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results.
Citation Count: Almalahi, Mohammed A.;...et.al. (2022). "Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative", AIMS Mathematics, Vol.7, No.9, pp.15994-16016.
Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative
(2022) Almalahi, Mohammed A.; Panchal, Satish K.; Jarad, Fahd; Abdo, Mohammed S.; Shah, Kamal; Abdeljawad, Thabet; 234808
The point of this work was to analyze and investigate the sufficient conditions of the existence and uniqueness of solutions for the nonlinear fuzzy fractional Volterra Fredholm integrodifferential equation in the frame of the Atangana-Baleanu-Caputo fractional derivative methodology. To begin with, we give the parametric interval form of the Atangana-Baleanu-Caputo fractional derivative on fuzzy set-valued functions. Then, by employing Schauder’s and Banach’s fixed point procedures, we examine the existence and uniqueness of solutions for fuzzy fractional Volterra Fredholm integro-differential equation with the Atangana-Baleanu-Caputo fractional operator. It turns out that the last interval model is a combined arrangement of nonlinear equations. In addition, we consider results by applying the Adams Bashforth fractional technique and present two examples that have been numerically solved using graphs.