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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    Fully bipolar fuzzy linear systems: bounded and symmetric solutions in dual form
    (Springer nature, 2025) Muhammad, Ghulam; Allahviranloo, Tofigh; Hussain, Nawab; Mrsic, Leo
    The aim of this article is to develop a new and simple technique to determine the approximate fuzzy solution of the fully bipolar fuzzy linear system of equations (BFLSEs) AU = eta, where U and eta are the triangular bipolar fuzzy vectors and A is either the crisp or fuzzy matrix. The solution of the proposed system is extracted in two fold. First, we obtain the crisp solution of the proposed system. To achieve this, we solve the BFLSEs using(-1,1)-cut expansion. Second, we assign unknown symmetric parameters to each row of this crisp system in(-1,1)-cut expansion. Thus, this system will transform into a system of interval equations. The unknown symmetric parameters corresponding to each element of a bipolar fuzzy vector are obtained by solving such an interval system of equations. Additionally, we demonstrate that the bounded and symmetric solutions (B&SSs) of the fully BFLSEs will reside within the tolerable solution set (TSS) and in the controllable solution set (CSS), respectively. To enhance the novelty of the proposed technique, we present several theorems that serve as a formal foundation for our approach. Furthermore, a numerical example is provided to demonstrate the effectiveness and validity of the proposed technique.
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    Fuzzy Langevin fractional delay differential equations under granular derivative
    (Elsevier Inc., 2024) Muhammad, Ghulam; Akram, Muhammad; Hussain, Nawab; Allahviranloo, Tofigh
    Analytical studies of the class of the fuzzy Langevin fractional delay differential equations (FLFDDEs) are frequently complex and challenging. It is necessary to construct an effective technique for the solution of FLFDDEs. This article presents an explicit analytical representation of the solution to the class of FLFDDEs with the general fractional orders under granular differentiability. The closed-form solution to the FLFDDEs is extracted for both the homogeneous and non-homogeneous cases using the Laplace transform technique and presented in terms of the delayed Mittag-Leffler type function with double infinite series. Moreover, the existence and uniqueness of the solutions of the FLFDDEs are investigated using the generalized contraction principle. An illustrative example is provided to support the proposed technique. To add to the originality of the presented work, the FLFDDEs with constant delay are solved by applying vibration theory and visualizing their graphs to support the theoretical results. © 2024 Elsevier Inc.
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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.