Yazar "Muhammad, Ghulam" seçeneğine göre listele

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    Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment
    (Elsevier Science Inc, 2023) Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, Tofigh
    Fuzzy fractional models have attracted considerable attention because of their comprehensive and broader understanding of real-world problems. Analytical studies of these models are often complex and difficult. Therefore, it is beneficial to develop a suitable and comprehensive technique to solve these models analytically. In this paper, an explicit analytical technique for solving two-dimensional incommensurate linear fuzzy systems of fractional Caputo differential equations (FLSoCFDEs) considering generalized Hukuhara differentiability (g H-differentiability) is presented and demonstrated. This extracted explicit solution is presented for different classes of such systems, including homogeneous and non-homogeneous cases with commensurate and incommensurate fractional orders. Moreover, the potential solution of FLSoCFDEs in terms of the Mittag-Leffler function involving double series is presented. The originality of the proposed technique is that the fuzzy Cauchy problem is transformed into a system of fuzzy linear Volterra integral equations of second kind and then the solution is extracted using the iterative Picard scheme based on the Banach fixed point theorem. Moreover, several interesting results are derived from FLSoCFDEs in terms of the Mittage-Leffler function for both homogeneous and inhomogeneous cases. To understand the proposed technique, we solve a diffusion process problem (a biological model) and several mass-spring systems as applications. Their graphs are analyzed to illustrate and support the theoretical results.
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    Incommensurate non-homogeneous system of fuzzy linear fractional differential equations using the fuzzy bunch of real functions
    (Elsevier, 2023) Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, Tofigh; Pedrycz, Witold
    This article aims to introduce and investigate the analytical fuzzy solution of the incommensurate non-homogeneous system of fuzzy linear fractional differential equations (INS-FLFDEs) using trivariate Mittag-Leffler functions. Entries of the coefficient matrix of the given system are treated as real numbers, initial-values are triangular fuzzy numbers (TFNs), and the forcing function is a fuzzy set (or a bunch) of real function. We extract the potential solution in the form of a fuzzy bunch of real functions (FBoRFs) rather than the solution of fuzzy-valued functions. We formulate the fuzzy initial value problem as a set of classical initial value problems by taking the forcing function from the class of FBoRFs and the initial value from the collection of TFNs (as a special case). The solution of this system is in the form of a trivariate Mittag-Leffler function. We interpret this solution as an element of the fuzzy solution set and assign the minimum value of membership that takes from the forcing function and the initial value in the fuzzy set. The originality of the proposed technique is that the uncertainty is smaller compared to the uncertainty extracted from other techniques. In addition, generalized derivatives increase the order and dimension of the system. Therefore, the proposed technique is better in terms of complexity because it reduces the order and dimension of the system. Finally, to grasp the proposed technique, we solve the electrical network and multiple mass-spring systems as applications and analyze their graphs to visualize and support theoretical results.
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    New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense
    (AMER INST MATHEMATICAL SCIENCES-AIMS, 2022) Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, Tofigh; Ali, Ghada
    The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.
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    Solution of initial-value problem for linear third-order fuzzy differential equations
    (SPRINGER HEIDELBERG, 2022) Muhammad, Ghulam; Allahviranloo, Tofigh; Pedrycz, Witold
    Every real-world physical problem is inherently based on uncertainty. It is essential to model the uncertainty then solve, analyze and interpret the result one encounters in the world of vagueness. Generally, science and engineering problems are governed by differential equations. But the parameters, variables and initial conditions involved in the system contain uncertainty due to the lack of information in measurement, observations and experiment. However, It is necessary to develop a comprehensive approach for solving differential equations in an uncertain environment. The purpose of this work is to study and investigate the fuzzy solution of linear third-order fuzzy differential equations using the concept of strongly generalized Hukuhara differentiability (SGHD). To make our analysis possible, we apply the first and second differentiability up to the third-order fuzzy derivative of the fuzzy-valued function. Moreover, we develop an important result concerning the relationship between Laplace transform of fuzzy-valued function and third-order derivative. We construct an algorithm to determine a potential solution of linear third-order fuzzy initial-value problem using the Laplace transform technique. All these solutions are represented in terms of the Mittag-Leffler function involving a single series. Furthermore, we discuss the switching points of linear third-order differential equations and their corresponding solutions in fuzzy environments. To enhance the novelty of the proposed technique, some illustrative examples are presented as applications are analyzed to visualize and support theoretical results.
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    A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations
    (American Institute of Mathematical Sciences, 2022) Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, Tofigh; Ali, Ghada
    The purpose of this study is to extend and determine the analytical solution of a twodimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized H-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the MittagLeffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.