Browsing by Author "Rohul Amin"

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    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.sa
    Fractional 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).
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    Haar Collocations Method for Nonlinear Variable Order Fractional Integro-Differential Equations
    (Natural Sciences Publishing, 2023) Rohul Amin; Thanin Sitthiwirattham; Muhammad Bilal Hafeez; Wojciech Sumelka; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand; email: thanin_sit@dusit.ac.th
    Variable order integrations and differentiations are the natural extensions of the corresponding usual operators. The idea was first introduced by Samko and his coauthors. Due to the importance of the said area, we consider a class of fractional integro-differential equations(FIDEs) under the variable order (VO) differentiation. Our investigation is related to numerical solution. For the said results, we utilize Haar collocation method (HCM). The concerned method has a convergence rate of order two and itself based on BroydenÕs technique. Various examples are testified by using the said techniques. Numerical interpretations are done by using Matlab. © 2023 NSP Natural Sciences Publishing Cor.
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    Two-dimensional Haar Wavelet Method for Numerical Solution of Delay Partial Differential Equations
    (Hindawi Limited, 2022) Rohul Amin; Nichaphat Patanarapeelert; Muhammad Awais Barkat; Ibrahim Mahariq; Thanin Sitthiwirattham; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand; email: thanin-sit@dusit.ac.th
    In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient. © 2022 Rohul Amin et al.