Browsing by Author "Shah, Kamal"

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Citation - WoS: 9
Citation - Scopus: 21
Study of Implicit Type Coupled System of Non-Integer Order Differential Equations With Antiperiodic Boundary Conditions
(Wiley, 2019) Shah, Kamal; Khan, Rahmat Ali; Baleanu, Dumitru; Samina
In this paper, the first purpose is to study existence and uniqueness of solutions to a system of implicit fractional differential equations (IFDEs) equipped with antiperiodic boundary conditions (BCs). To obtain the mentioned results, we use Schauder's and Banach fixed point theorem. The second purpose is discussing the Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) stabilities for the respective solutions. An example is provided to illustrate the established results.
Citation - WoS: 143
Citation - Scopus: 157
Semi-Analytical Study of Pine Wilt Disease Model With Convex Rate Under Caputo-Febrizio Fractional Order Derivative
(Pergamon-elsevier Science Ltd, 2020) Jarad, Fahd; Abdeljawad, Thabet; Shah, Kamal; Alqudah, Manar A.
In this paper, we present semi-analytical solution to Pine Wilt disease (PWD) model under the CaputoFabrizio fractional derivative (CFFD). For the proposed solution, we utilize Laplace transform coupled with Adomian decomposition method abbreviated as (LADM). The concerned method is a powerful tool to obtain semi-analytical solution for such type of nonlinear differential equations of fractional order (FODEs) involving non-singular kernel. Furthermore, we give some results for the existence of solution to the proposed model and present numerical results to verify the established analysis. (C) 2020 Elsevier Ltd. All rights reserved.
Citation - WoS: 34
Citation - Scopus: 42
Study of Global Dynamics of Covid-19 Via a New Mathematical Model
(Elsevier, 2020) Seadawy, Aly R.; Shah, Kamal; Ullah, Aman; Baleanu, Dumitru; Din, Rahim Ud
The theme of this paper focuses on the mathematical modeling and transmission mechanism of the new Coronavirus shortly noted as (COVID-19), endangering the lives of people and causing a great menace to the world recently. We used a new type epidemic model composed on four compartments that is susceptible, exposed, infected and recovered (SEIR), which describes the dynamics of COVID-19 under convex incidence rate. We simulate the results by using nonstandard finite difference method (NSFDS) which is a powerful numerical tool. We describe the new model on some random data and then by the available data of a particular regions of Subcontinents.
Citation - WoS: 11
Citation - Scopus: 12
On Stable Iterative Solutions for a Class of Boundary Value Problem of Nonlinear Fractional Order Differential Equations
(Wiley, 2019) Shah, Kamal; Jarad, Fahd; Ali, Sajjad
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 - WoS: 8
Citation - Scopus: 15
Computation of Iterative Solutions Along With Stability Analysis To a Coupled System of Fractional Order Differential Equations
(Springeropen, 2019) Abdeljawad, Thabet; Shah, Kamal; Jarad, Fahd; Arif, Muhammad; Ali, Sajjad
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 - WoS: 30
Citation - Scopus: 30
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) Shah, Kamal; Al-Mdallal, Qasem M.; Jarad, Fahd; Haq, Fazal
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 - WoS: 37
Citation - Scopus: 42
On Nonlinear Conformable Fractional Order Dynamical System Via Differential Transform Method
(Tech Science Press, 2023) Jarad, Fahd; Al-Mdallal, Qasem; Shah, Kamal; Abdeljawad, Thabet
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 - WoS: 2
Citation - Scopus: 2
A New Numerical Treatment for Fractional Differential Equations Based on Non-Discretization of Data Using Laguerre Polynomials
(World Scientific Publ Co Pte Ltd, 2020) Arfan, Muhammad; Abdeljawad, Thabet; Jarad, Fahd; Khan, Adnan; Shah, Kamal
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 - WoS: 66
Citation - Scopus: 80
Efficient Sustainable Algorithm for Numerical Solutions of Systems of Fractional Order Differential Equations by Haar Wavelet Collocation Method
(Elsevier, 2020) Shah, Kamal; Al-Mdallal, Qasem; Jarad, Fahd; Abdeljawad, Thabet; Amin, Rohul
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 - WoS: 6
Citation - Scopus: 9
Computation of Semi-Analytical Solutions of Fuzzy Nonlinear Integral Equations
(Springer, 2020) Ullah, Aman; Shah, Kamal; Baleanu, Dumitru; Ullah, Zia
In this article, we use a fuzzy number in its parametric form to solve a fuzzy nonlinear integral equation of the second kind in the crisp case. The main theme of this article is to find a semi-analytical solution of fuzzy nonlinear integral equations. A hybrid method of Laplace transform coupled with Adomian decomposition method is used to find the solution of the fuzzy nonlinear integral equations including fuzzy nonlinear Fredholm integral equation, fuzzy nonlinear Volterra integral equation, and fuzzy nonlinear singular integral equation of Abel type kernel. We also provide some suitable examples to better understand the proposed method.
Citation - WoS: 3
Citation - Scopus: 4
Investigation of Covid-19 Mathematical Model Under Fractional Order Derivative
(Edp Sciences S A, 2021) Arfan, Muhammad; Deebani, Wejdan; Shutaywi, Meshal; Baleanu, Dumitru; Shah, Kamal
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.
Citation - WoS: 38
Citation - Scopus: 55
Existence and Stability Analysis To a Coupled System of Implicit Type Impulsive Boundary Value Problems of Fractional-Order Differential Equations
(Springer, 2019) Jarad, Fahd; Gupta, Vidushi; Abdeljawad, Thabet; Ali, Arshad; Shah, Kamal
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 - WoS: 7
Citation - Scopus: 9
Mathematical Study of Sir Epidemic Model Under Convex Incidence Rate
(Amer inst Mathematical Sciences-aims, 2020) Alqudah, Manar A.; Abdeljawad, Thabet; Jarad, Fahd; Din, Rahim Ud; Shah, Kamal
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 R-0: 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.
Citation - WoS: 21
Citation - Scopus: 17
An Analytical Study of Fractional Delay Impulsive Implicit Systems With Mittag-Leffler Law
(Ministry Communications & High Technologies Republic Azerbaijan, 2023) Abdo, Mohammed S.; Jarad, Fahd; Abdeljawad, Thabet; Shah, Kamal
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 - WoS: 40
Citation - Scopus: 45
Existence of Positive Solutions for Weighted Fractional Order Differential Equations
(Pergamon-elsevier Science Ltd, 2020) Ali, Saeed M.; Shah, Kamal; Jarad, Fahd; Abdo, Mohammed S.; Abdeljawad, Thabet
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. (C) 2020 Elsevier Ltd. All rights reserved.
Citation - WoS: 86
Qualitative Analysis of a Mathematical Model in the Time of Covid-19
(Hindawi Ltd, 2020) Mahariq, Ibrahim; Jarad, Fahd; Shah, Kamal; Abdeljawad, Thabet
In this article, a qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered. The concerned model is composed of two compartments, namely, healthy and infected. Under the new nonsingular derivative, we, first of all, establish some sufficient conditions for existence and uniqueness of solution to the model under consideration. Because of the dynamics of the phenomenon when described by a mathematical model, its existence must be guaranteed. Therefore, via using the classical fixed point theory, we establish the required results. Also, we present the results of stability of Ulam's type by using the tools of nonlinear analysis. For the semianalytical results, we extend the usual Laplace transform coupled with Adomian decomposition method to obtain the approximate solutions for the corresponding compartments of the considered model. Finally, in order to support our study, graphical interpretations are provided to illustrate the results by using some numerical values for the corresponding parameters of the model.
Citation - WoS: 79
Citation - Scopus: 89
Existence Theory and Numerical Solutions To Smoking Model Under Caputo-Fabrizio Fractional Derivative
(Amer inst Physics, 2019) Shah, Kamal; Zaman, Gul; Jarad, Fahd; Khan, Sajjad Ali
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 - WoS: 8
Citation - Scopus: 8
Qualitative Analysis of a Fuzzy Volterra-Fredholm Integrodifferential Equation With an Atangana-Baleanu Fractional Derivative
(Amer inst Mathematical Sciences-aims, 2022) Shah, Kamal; Jarad, Fahd; Abdo, Mohammed S.; Abdeljawad, Thabet; Almalahi, Mohammed A.; Panchal, Satish K.
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.
Citation - WoS: 25
Citation - Scopus: 29
Note on the Solution of Random Differential Equations Via Ψ-Hilfer Fractional Derivative
(Springer, 2018) Shah, Kamal; Baleanu, Dumitru; Kanagarajan, K.; Harikrishnan, S.
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 - WoS: 12
Citation - Scopus: 12
On a Class of Boundary Value Problems Under Abc Fractional Derivative
(Springer, 2021) Jarad, Fahd; Gul, Rozi; Shah, Kamal; Khan, Zareen A.
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.