Browsing by Author "Kamal Shah"
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Item Application of asymptotic homotopy perturbation method to fractional order partial differential equation(MDPI, 2021) Haji Gul; Sajjad Ali; Kamal Shah; Shakoor Muhammad; Thanin Sitthiwirattham; Saowaluck Chasreechai; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.thIn this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item EXTENSION of HAAR WAVELET TECHNIQUES for MITTAG-LEFFLER TYPE FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS(World Scientific, 2023) Jiraporn Reunsumrit; Kamal Shah; Aziz Khan; Rohul Amin; Israr Ahmad; Thanin Sitthiwirattham; A. Khan; Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia; email: akhan@psu.edu.saFractional order integro-differential equation (FOIDE) of Fredholm type is considered in this paper. The mentioned equations have many applications in mathematical modeling of real world phenomenon like image and signal processing. Keeping the aforementioned importance, we study the considered problem from two different aspects which include the existence theory and computation of numerical approximate solution. FOIDEs have been investigated very well by using Caputo-Type derivative for the existence theory and numerical solutions. But the mentioned problems have very rarely considered under the Mittage-Leffler-Type derivative. Also, for FOIDE of Fredholm type under Mittage-Leffler-Type derivative has not yet treated by using Haar wavelet (HW) method. The aforementioned derivative is non-singular and nonlocal in nature as compared to classical Caputo derivative of fractional order. In many cases, the nonsingular nature is helpful in numerical computation. Therefore, we develop the existence theory for the considered problem by using fixed point theory. Sufficient conditions are established which demonstrate the existence and uniqueness of solution to the proposed problem. Further on utilizing HW method, a numerical scheme is developed to compute the approximate solution. Various numerical examples are given to demonstrate the applicability of our results. Also, comparison between exact and numerical solution for various fractional orders in the considered examples is given. Numerical results are displayed graphically. © 2023 The Author(s).Item Hyers-Ulam Stability, Exponential Stability, and Relative Controllability of Non-Singular Delay Difference Equations(Hindawi Limited, 2022) Sawitree Moonsuwan; Gul Rahmat; Atta Ullah; Muhammad Yasin Khan; ÊKamran Ê; Kamal Shah; K. Shah; Department of Mathematics and Sciences, Prince Sultan University, Riyadh, P. O. Box 66833, 11586, Saudi Arabia; email: kamalshah408@gmail.comIn this paper, we study the uniqueness and existence of the solutions of four types of non-singular delay difference equations by using the Banach contraction principles, fixed point theory, and Gronwall's inequality. Furthermore, we discussed the Hyers-Ulam stability of all the given systems over bounded and unbounded discrete intervals. The exponential stability and controllability of some of the given systems are also characterized in terms of spectrum of a matrix concerning the system. The spectrum of a matrix can be easily obtained and can help us to characterize different types of stabilities of the given systems. At the end, few examples are provided to illustrate the theoretical results. © 2022 Sawitree Moonsuwan et al.Item Semi-analytical solutions for fuzzy caputoÐfabrizio fractional-order two-dimensional heat equation(MDPI, 2021) Thanin Sitthiwirattham; Muhammad Arfan; Kamal Shah; Anwar Zeb; Salih Djilali; Saowaluck Chasreechai; A. Zeb; Department of Mathematics, Abbotabad Campus, COMSATS University of Islamabad, Khyber Pakhtunkhwa, 22060, Pakistan; email: anwar@cuiatd.edu.pkIn the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition tech-niques and the Adomian polynomial under the CaputoÐFabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the approximate solution of the proposed equation. For the validation of our scheme, three different examples have been provided, and the solutions were calculated in fuzzy form. All the three illustrations simulated two different fractional orders between 0 and 1 for the upper and lower portions of the fuzzy solution. The said fractional operator is nonsingular and global due to the presence of the exponential function. It globalizes the dynamical behavior of the said equation, which is guaranteed for all types of fuzzy solution lying between 0 and 1 at any fractional order. The fuzziness is also included in the unknown quantity due to the fuzzy number providing the solution in fuzzy form, having upper and lower branches. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item STUDY OF A COUPLED SYSTEM WITH ANTI-PERIODIC BOUNDARY CONDITIONS UNDER PIECEWISE CAPUTO-FABRIZIO DERIVATIVE(Serbian Society of Heat Transfer Engineers, 2023) Nichaphat Patanarapeelert; Asma Asma; Arshad Ali; Kamal Shah; Thabet Abdeljawad; Thanin Sitthiwirattham; T. Abdeljawad; Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia; email: tabdeljawad@psu.edu.saA coupled system under Caputo-Fabrizio fractional order derivative (CFFOD) with antiperiodic boundary condition is considered. We use piecewise version of CFFOD. Sufficient conditions for the existence and uniqueness of solution by applying the Banach, KrasnoselskiiÕs fixed point theorems. Also some appropriate results for Hyers-Ulam (H-U) stability analysis is established. Proper example is given to verify the results. © 2023 Society of Thermal Engineers of Serbia Published by the Vin_a Institute of Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditionsItem Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative(American Institute of Mathematical Sciences, 2022) Thanin Sitthiwirattham; Rozi Gul; Kamal Shah; Ibrahim Mahariq; Jarunee Soontharanon; Khursheed J. Ansari; K. Shah; Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan; email: kamalshah408@gmail.comThis article is devoted to investigate a class of non-local initial value problem of implicit-impulsive fractional differential equations (IFDEs) with the participation of the Caputo-Fabrizio fractional derivative (CFFD). By means of KrasnoselskiiÕs fixed-point theorem and BanachÕs contraction principle, the results of existence and uniqueness are obtained. Furthermore, we establish some results of Hyers-Ulam (H-U) and generalized Hyers-Ulam (g-H-U) stability. Finally, an example is provided to demonstrate our results. © 2022 the Author(s),.Item STUDY OF INTEGER AND FRACTIONAL ORDER COVID-19 MATHEMATICAL MODEL(World Scientific, 2023) Rujira Ouncharoen; Kamal Shah; Rahim Ud Din; Thabet Abdeljawad; A.L.I. Ahmadian; Soheil Salahshour; Thanin Sitthiwirattham; T. Abdeljawad; Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia; email: tabdeljawad@psu.edu.saIn this paper, we study a nonlinear mathematical model which addresses the transmission dynamics of COVID-19. The considered model consists of susceptible (S), exposed (E), infected (I), and recovered (R) individuals. For simplicity, the model is abbreviated as SEIR. Immigration rates of two kinds are involved in susceptible and infected individuals. First of all, the model is formulated. Then via classical analysis, we investigate its local and global stability by using the Jacobian matrix and Lyapunov function method. Further, the fundamental reproduction number R0 is computed for the said model. Then, we simulate the model through the Runge-Kutta method of order two abbreviated as RK2. Finally, we switch over to the fractional order model and investigate its numerical simulations corresponding to different fractional orders by using the fractional order version of the aforementioned numerical method. Finally, graphical presentations are given for the approximate solution of various compartments of the proposed model. Also, a comparison with real data has been shown. © 2023 The Author(s).