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Öğe Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator(AMER INST MATHEMATICAL SCIENCES-AIMS, 2022) Akram, Muhammad; Ihsan, Tayyaba; Allahviranloo, Tofigh; Al-Shamiri, Mohammed M. AliThis study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.Öğe An analytical study of Pythagorean fuzzy fractional wave equation using multivariate Pythagorean fuzzy fourier transform under generalized Hukuhara Caputo fractional differentiability(Springernature, 2024) Akram, Muhammad; Yousuf, Muhammad; Allahviranloo, TofighPythagorean fuzzy fractional calculus provides a strong framework for modeling and analyzing complicated systems with uncertainty and indeterminacy. The primary focus of this article is to investigate the analytical solution of the Pythagorean fuzzy fractional wave equation using multivariate Pythagorean fuzzy Fourier transform under generalized Hukuhara Caputo fractional differentiability. To this end, we first establish generalized Hukuhara Caputo fractional differentiability in the context of multivariate Pythagorean fuzzy-valued functions and then we present some results of multivariate Pythagorean generalized Hukuhara Caputo fractional differentiability and generalized Hukuhara integrability. We present the concept of multivariate Pythagorean fuzzy Fourier transform and give some results for Pythagorean fuzzy Fourier transforms of second-order generalized Hukuhara partial differentiability. Finally, we provide a practical application of the Pythagorean fuzzy fractional wave equation to visco-elastic materials including polymers and biological tissues. Their graphs are analyzed to visualize and support the theoretical findings.Öğe An efficient numerical method for solving m-polar fuzzy initial value problems(Springer Science and Business Media Deutschland GmbH, 2022) Akram, Muhammad; Saqib, Muhammad; Bashir, Shahida; Allahviranloo, TofighSeveral problems in the field of science and technology are modeled with information about the situation that is ambiguous, imprecise, or incomplete. That is, the information about values of parameters, functional relationships, or initial conditions is not given in precise. In these circumstance, existing analytic or numerical methods can be applied only to the selected behavior of the system. For example, by fixing the values of unknown parameters to some credible values. On the basis of partial knowledge, it is impossible to describe the behavior of complete system. Thus, fuzzy differential equations arise in many dynamical models. In modeling of several real-world problems, differential equations frequently involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty rather than a single bit. These type of differential equations cannot be well represented by means of fuzzy differential equations or bipolar fuzzy differential equations. Therefore, the theory of m-polar fuzzy sets can be applied to differential equations to handle the problems which have multi-polar information. The aim of this paper is to study differential equation in m-polar fuzzy environment. A fourth-order Runge–Kutta method to solve m-polar FIVPs is presented. The consistency, stability and convergence of suggested method are discussed to ensure its efficiency and validity. Since it requires no higher order function derivatives, the suggested method is straightforward to implement. Euler and Euler modified methods have global truncations errors of O(h) and O(h2) respectively whereas the suggested Runge–Kutta’s global truncation errors of O(h4). Numerical examples are provided to compare the proposed method with Euler and modified Euler methods in terms of global truncation errors (GTE). The numerical findings suggest that the purposed method has an adequate level of accuracy. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.Öğe Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment(Elsevier Science Inc, 2023) Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, TofighFuzzy 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.Öğe An extended multi-objective transportation model based on Fermatean fuzzy sets(Springer, 2023) Akram, Muhammad; Shahzadi, Sundas; Shah, Syed Muhammad Umer; Allahviranloo, TofighFermatean fuzzy sets are a more efficient, flexible, and general model for dealing with uncertainty as compared to Pythagorean fuzzy sets. The multi-objective transportation problem in a Fermatean fuzzy setting is examined in this study. Due to the volatility of competitive marketplaces, transportation costs, supply and demand factors are not always reliable. These parameters are regarded as triangular Fermatean fuzzy numbers in this article. The multi-objective transportation problem is addressed using a novel compromise approach based on the ordering of these triangular Fermatean fuzzy numbers. Also, the proposed solution procedure is used to solve a real-world problem in order to show how useful it is. Lastly, the outcomes of the Fermatean fuzzy multi-objective transportation problem are used to illustrate the benefits of the proposed method over previous methods.Öğe A fully Fermatean fuzzy multi-objective transportation model using an extended DEA technique(Springernature, 2023) Akram, Muhammad; Shahzadi, Sundas; Shah, Syed Muhammad Umer; Allahviranloo, TofighA mathematical technique called data envelope analysis is used to determine the relative efficiency of decision-making units (DMU) with numerous inputs and outputs. Compared to other DMUs, it determines how efficient the DMU is at delivering a specific level of output based on the amount of input it uses. The transportation problem is a linear programming problem for reducing the net transportation cost or maximizing the net transportation profit of moving goods from a number of sources to a number of destinations. In this manuscript, a multi-objective transportation problem is examined in which all the parameters, such as transportation cost, supply, and demand are uncertain with hesitancy and triangular Fermatean fuzzy numbers are used to represent these uncertain parameters. Using Fermatean fuzzy data envelope analysis, a new technique for determining the common set of weights is presented. The fully Fermatean fuzzy multi-objective transportation problem is then solved using a novel data envelopment analysis-based approach. To this end, two different Fermatean fuzzy efficiency scores are derived, first by considering the sources as targets and changing the destinations, and second by considering the destinations as targets and changing the sources. Next, a unique Fermatean fuzzy relative efficiency is determined for each arc by combining these two different Fermatean fuzzy efficiency scores. As a result, a single-objective Fermatean fuzzy transportation problem is constructed, which can be solved using existing techniques. A numerical illustration is provided to support the suggested methodology, and the performance of the proposed method is compared with an existing techniqueÖğe 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, WitoldThis 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.Öğe An interactive method for the solution of fully Z-number linear programming models(Springernature, 2023) Akram, Muhammad; Ullah, Inayat; Allahviranloo, TofighLinear programming is a technique widely used in decision-making nowadays. Linear programming in a fuzzy environment makes it even more interesting due to the vagueness and uncertainty of the available resources and variables. Since the market price and profit of certain goods are not known exactly, considering fuzzy variables and parameters in the linear programming makes it more closer to the real-life situation; therefore, it becomes more attractive for the decision-makers. In a fuzzy environment, there is only one information and that is the possibility of the variable. In many real-world problems, we need the reliability of the information along with its possibility. Zadeh suggested a Z-number Z = (A; B) with two components, A carrying the information of possibility of the variable, and B carrying the information about reliability of the first component A. Linear programming with its parameters and variables carrying the information in the form of Z number is even more exciting for the decision-makers. Because every decision-maker demands information that is more reliable, linear programming in a Z-number environment with both its components taken as fuzzy numbers is a very attractive problem. In this paper, we present linear programming problems with the parameters and variables taken as Z number having triangular fuzzy numbers as possibility and reliability. We also suggest an interactive method to solve Z number linear programming problems by converting Z-numbers into conventional fuzzy numbers and then using the ranking of fuzzy numbers. We also present applications of the proposed models by solving numerical examples. We also test the authenticity of the proposed method by comparing the results with the existing techniques.Öğe A method for solving bipolar fuzzy complex linear systems with real and complex coefficients(Springer Science and Business Media Deutschland GmbH, 2022) Akram, Muhammad; Ali, Muhammad; Allahviranloo, TofighAn extensive variety of human decision making is based on two-sided or bipolar cognitive assessment of a positive side and a negative side. Zhang’s bipolar fuzzy set is represented as {?(x,y)|(x,y)?[-1,0]×[0,1]}.The multiagent decision analysis is based on bipolar judgment of a positive side and a negative side. In this paper, we consider the issues posed by bipolar fuzzy linear systems (BFLSs) and bipolar fuzzy complex linear (BFCL) systems of equations with real and complex coefficients. Primarily, we find the solution of the BFLS of equations by the resort into two parts: one is negative(?) and other is positive(?) n× n linear fuzzy systems; to this aim, we use n× n bipolar fuzzy center system and 2 n× 2 n bipolar fuzzy width system. We describe the fundamentals of proposed method with the aid of n= 2 , 5 -dimensional numerical examples, and we get the weak and strong solutions of the system. Subsequently, we describe a technique to solve BFCL systems of equations whose coefficients are real and complex by a pair of positive(?) and negative(?) n× n real and complex crisp linear systems; to this aim, we use bipolar fuzzy complex center and width. Ultimately, we use our proposed technique to solve a current flow circuit which is defined in terms of a BFCL system with complex coefficients, and we get the unknown current in the form of bipolar fuzzy complex number. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.Öğe 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, GhadaThe 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.Öğe A New Framework for Numerical Techniques for Fuzzy Nonlinear Equations(Mdpi, 2023) Abbasi, Fazlollah; Allahviranloo, Tofigh; Akram, MuhammadThis paper describes a computational method for solving the nonlinear equations with fuzzy input parameters that we encounter in engineering system analysis. In addition to discussing the existence of solutions, the definition and formalization of numerical solutions is based on a new fuzzy computation operation as a transmission average. Error analysis in numerical solutions is described. Finally, some examples are presented to implement the proposed method and its effectiveness compared to other previous methods.Öğe A new maximal flow algorithm for solving optimization problems with linguistic capacities and flows(ELSEVIER SCIENCE INC, 2022) Akram, Muhammad; Habib, Amna; Allahviranloo, TofighThe maximal flow problems (MFPs) are among the most significant optimization problems in network flow theory with widespread and diverse applications. To represent qualitative aspects of uncertainty in the maximal flow model, which asks for the largest amount of flow transported from one vertex to another, the use of linguistic variables has effective means for experts in expressing their views. In this paper, we first define trapezoidal Pythagorean fuzzy numbers (TrPFNs) along with some new arithmetic operations which cover the gaps in previously defined operations. For defuzzification of TrPFNs, we introduce a ranking procedure based on value and ambiguity indices. This work puts forward a the-oretical framework for a new Pythagorean fuzzy maximal flow algorithm (PFMFA), which helps to solve different optimization problems with PF information by considering linguis-tic capacities and flows. The implementation of the algorithm is elaborated by considering two case studies. Firstly, we examine the maximum flow of a water distribution pipeline network in Pyigyitagon Township, Mandalay, Myanmar. Secondly, we compute maximum PF power flow in a 14-bus electricity network provided by the IEEE working group, con-cerning the example data from the University of Washington. The results illustrate the superiority of the proposed method and give a detailed analysis of flow connected with several practical performances. In addition, the Pythagorean fuzzy optimal flows corre-sponding to each network arc are compared and performance comparison of our method is investigated which shows the increasing and decreasing trends of backward and forward arcs of the network, respectively. Moreover, the runtime analysis of existing well-known maximal flow algorithms is provided. Finally, we present the advantages of our technique to promote its cogency. (c) 2022 Elsevier Inc. All rights reserved.Öğe A new method for the solution of fully fuzzy linear programming models(Springer Science and Business Media Deutschland GmbH, 2022) Akram, Muhammad; Ullah, Inayat; Allahviranloo, TofighIn this study, we first show that the existing arithmetic operations of trapezoidal fuzzy numbers do not satisfy the basic properties. Then, for trapezoidal fuzzy numbers, we define new arithmetic operations. Furthermore, we demonstrate that the existing Simplex method for addressing fully fuzzy linear programming problems has some drawbacks. Finally, we provide a new strategy for solving fully fuzzy linear programming problems and compare our results with the existing methods. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.Öğe A new method to solve linear programming problems in the environment of picture fuzzy sets(Iranian Journal of Fuzzy Systems, 2022) Akram, Muhammad; Ullah, Iftikhar; Allahviranloo, TofighPicture fuzzy set is characterized by neutral membership function along with the membership and non-membership functions, and is, therefore, more general than the intuitionistic fuzzy set which is only characterized by membership and non-membership functions. In this paper, first, we are going to point out a drawback and try to fix it by the existing trapezoidal picture fuzzy number. Furthermore, we define an LR flat picture fuzzy number, which is a generalization of trapezoidal picture fuzzy numbers. We also discuss a linear programming model with LR flat picture fuzzy numbers as parameters and variables and present a method to solve these type of problems using a generalized ranking function.Öğe Solution of the Pythagorean fuzzy wave equation with Pythagorean fuzzy Fourier sine transform(Springernature, 2023) Akram, Muhammad; Yousuf, Muhammad; Allahviranloo, TofighThe main objective of this research article is to study the analytical solution of the Pythagorean fuzzy wave equation under the generalized Hukuhara partial differentiability using the Pythagorean fuzzy Fourier sine transform. Some concepts of multivariate Pythagorean fuzzy-valued functions and their gH-partial differentiability along with integrability are given. The notions of Pythagorean fuzzy Fourier sine transform and Pythagorean fuzzy Fourier inverse sine transform are introduced along with some fundamental properties. Furthermore, a new Pythagorean fuzzy wave equation model is developed under gH-differentiability using the Pythagorean fuzzy Fourier sine transform. Some numerical examples are solved with the proposed method and their solutions are displayed graphically to verify and support theoretical results. A practical application of the Pythagorean fuzzy wave equation to magnetic resonance imaging is also described.Öğe 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, GhadaThe 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.
