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Item Generalization of fractional calculus operators with applications(UMT Lahore, 2018-07-31) Muhammad Khurshid AzamGeneralized forms of fractional calculus operators (integrals and derivatives) are introduced. Caputo k-fractional derivative and Hadamard Caputo type k-fractional derivative are discussed and their results with some applications are presented. Extensions of Weyl k-fractional integral and Hadamard k-fractional integral are also introduced. Boundedness of the extended Hadamard k-fractional integral in spaces is determined. The generalized k-fractional derivative and generalized Caputo type k-fractional derivative are introduced and their properties and results are found. Finally, the generalized type k-fractional integral (unifying eleven existing fractional integrals) is introduced and its boundedness in spaces is also determined. Further, integral transforms of k-fractional and extended k-fractional operators are found. Proofs of properties including semigroup, commutative and some other results for k-Weyl fractional integral are given. Moreover, some inequalities for k-Weyl fractional integral are discussed and examples are also given to illustrate the results. Relationship between these new generalized forms of fractional calculus operators with the existing fractional operators are discussed by substituting the different values of involved parameters. Integral transforms of new fractional operators and their applications are also given.Item Multi-criteria decision making techniques based on some extensions of fuzzy set(UMT Lahore, 2018-07) Shahzad FaiziMulti-criteria decision making (MCDM) is a common activity in everyday life and the objective is to select the most feasible alternative from a set of given alternatives with the highest level of satisfaction in the presence of multiple, usually conflicting, criteria. Similarly, there are many real-life complex problems, where we need involve wide domain of knowledge which are beyond a single expert. Therefore, it is usually necessary to allocate more than one expert to decision process from different fields, including the education backgrounds, work experience and knowledge structure. Consequently, the multi-criteria group decision making (MCGDM) is also an important tool to deal with human activities and their problems of daily lives. In this thesis, different MCDM/MCGDM techniques are discussed based on some important extensions of fuzzy set. This thesis is comprised of three stages. In the first stage, the MCDM method called the Characteristic Objects Method (COMET) is extended to solve problems for MCGDM with hesitant fuzzy sets (HFSs) and intuitionistic fuzzy sets (IFSs). It is a completely new idea for solving problems of group decision making (GDM) under uncertainty. In this approach, we use L-R-type generalized fuzzy numbers (GFNs) and triangular intuitionistic fuzzy numbers (TIFNs) to get the degree of hesitancy in the form of hesitant fuzzy elements (HFEs) and the degree of membership and non-membership in the form of intuitionistic fuzzy numbers (IFNs) respectively for an alternative under a certain criterion. Therefore, the classical COMET method was adapted to work with GFNs and TIFNs in GDM problems. The proposed extensions are presented in detail, along with the necessary background information. The second stage of the thesis is comprised of three parts. In the first part, an outranking method is constructed using hesitant intuitionitic fuzzy linguistic term sets (HIFLTSs) for ranking alternatives in MCGDM problems based on intuitionistic fuzzy support function (IFSF), intuitionistic fuzzy risk function (IFRF), intuitionistic fuzzy credibility function (IFCF) and the net outranking flow index (NOFI). In the second part, the notion of directional Hausdorff distance between two HIFLTSs has been proposed and used it to formulate ELECTRE-based outranking method in hesitant intuitionistic fuzzy linguistic (HIFL) environment.Item Analytical solutions for different motions of differential and rate type fluids with fractional derivatives.(UMT Lahore, 2018-10-24) Muhammad Bilal RiazIn this dissertation, we present the analytical studies of some fluid flow models. We analyze the fractional models for the flow of non-Newtonian fluids via classical computational techniques to obtain analytical solutions. This study includes the investigation of the unsteady natural convection flow of Maxwell fluid with fractional derivative over an exponentially accelerated infinite vertical plate. Slip condition, chemical reaction, transverse magnetic field and Newtonian heating effects are also considered using a modern definition of fractional derivative. Moreover, the unsteady flow of Maxwell fluid with non-integer order derivatives through a circular cylinder of infinite length in a rotating frame is studied. The motion of Maxwell fluid is generated by a time dependent torsion applied to the surface of the cylinder. As novelty, the dimensionless governing equation related to the non-trivial shear stress is used and the first exact solution analogous to a ramped shear stress on the surface is obtained. The rotational flow of an Oldroyd-B fluid with fractional derivative induced by an infinite circular cylinder that applies a constant couple stress to the fluid is investigated. It is worth mentioning that the considered problem of Oldroyd-B fluid in the settings of fractional derivatives has not been found in the literature. Some unsteady Couette flows of an Oldroyd-B fluid with non-integer derivative in an annular region of two infinite co-axial circular cylinders are investigated. Flows are due to the motion of the outer cylinder, that rotates about its axis with an arbitrary time dependent velocity while the inner cylinder is held fixed. Finally, the analysis of the second grade fluid with fractional derivative is made. The fluid fills the annulus region between two coaxial cylinders in which one cylinder is at rest while the other experiences time dependent shear stress. In all the flow models, we obtained the exact or semi analytical solutions for the motions with technical relevance. These solutions correspond to some flows in which either velocity or the shear stress is given on the boundary are established for different kinds of rate and differential type fluids. The obtained solutions presented in all the fluid flow models satisfy the imposed initial and boundary conditions. Further, the flow properties and comparison of models with respect to derivative (fractional or ordinary) are highlighted by graphical illustrations.Item Aggregation operators based on some extension of fuzzy sets.(UMT Lahore, 2018-03-22) Raja Noshad JamilBonferroni mean (BM) and heronian mean (HM) operators are useful tools for group decision making problems, when arguments are interrelated to each other. In this thesis, we developed some BM and HM based aggregation operators. We defined some aggregation operators for dual hesitant fuzzy (DHF) sets, for instance, we defined dual hesitant fuzzy geometric bonferroni mean (DHFGBM) and different properties of DHFGBM are discussed. Some special cases are also studied in detail for DHFGBM. In addition, dual hesitant fuzzy weighted geometric bonferroni mean (DHFWGBM) and dual hesitant fuzzy chouqet geometric bonferroni mean (DHFCGBM) proposed. We also model a system of fuzzy soft differential equations (FSDEs) to analyze the behavior over the time of an individual depending on their companion’s actions under any particular situation against some decision by the help of BM. Using the ability of BM to capture the interrelationship of arguments, we defined bonferroni fuzzy soft matrix (BFSM) and weighted bonferroni fuzzy soft matrix (WBFSM) for data representation. WBFSM is a decision matrix and provide optimum fuzzy soft constant (OFSC), which is the key element of FSDEs. By utilizing the OFSC, we developed a system of FSDEs to study a dynamical process with nonlinear uncertain and vague data. We presented a novel efficient technique for analyzing the future attitude of people due to their present decisions. To illustrate the practicality and feasibility of proposed technique, an example is also discussed with the help of phase portrait and line graphs. With respect to multiple attribute group decision making problems, in which the value of the attributes are taken in the form of hesitant 2-tuple (H2T) or intuitionistic 2-tuple (I2T) linguistic information are called CW. H2T linguistic arguments are used to evaluate the group decision making problems which have inter-dependent or interactive attributes. Some operational laws are developed for H2T linguistic elements and based on these operational laws hesitant 2-tuple weighted averaging (H2TWA) operator and generalize hesitant 2-tuple averaging (GH2TA) operator are proposed. Combining choquet integral (CI) with H2T linguistic information, defined hesitant 2-tuple correlated averaging (H2TCA) and generalize hesitant 2-tuple correlated averaging (GH2TCA) operators. In the existing literature review, we observed that during aggregation procedure for H2T, more hesitation produces in the resultant element. We targeted this issue and developed a diminishing hesitant 2-tuple averaging operator (DH2TA) operator for H2T linguistic arguments. DH2TA operator work in the way that it’s aggregate all H2T linguistic elements and during the aggregation process it also controls the hesitation in the translation of the resulting aggregated xv xvi linguistic term. We developed a scalar product for H2T linguistic elements and based on the scalar product, a diminishing weighted hesitant 2-tuple averaging operator (DWH2TA) is introduced. Moreover, combining CI with H2T linguistic information, the diminishing choquet hesitant 2-tuple average operator (DCH2TA) operator is defined. Most of existing operational laws in literature for handling the process for CW are not bounded and hence a logical problem comes. We targeted this issue and developed closed operational laws based on Archimedean t-norm and t-conorm. Some aggregation operators intuitionistic 2-tuple linguistic heronian mean (I2THM) and intuitionistic 2-tuple linguistic chouqet heronian mean (I2TCHM) based on these closed operational laws developed and discussed desired properties of the proposed operators. Linkages between industry and university are the significant parts in the entire advancement of any country. To assess university’s reputation for industry, we proposed a fusion approach by using heronian intuitionistic fuzzy analytic hierarchy process (HIF-AHP), fuzzy geometric bonferroni mean (FGHM) operator and 2-tuple fuzzy linguistic elements. In each chapter, we developed some techniques based on proposed operators and demonstrated the validity and feasibility of these techniques by some examples. Educational note: Sentence case formatting prioritizes readability by limiting capitalization to only the first word of each sentence, proper nouns/adjectives (e.g., "Archimedean," derived from the mathematician Archimedes), and acronyms (e.g., BM, DHF). This consistency avoids visual clutter, which is particularly useful for academic texts like the thesis excerpt provided—readers can quickly parse sentence boundaries without distraction from overcapitalized terms. Acronyms are retained in uppercase to preserve their symbolic meaning and avoid confusion with generic terms (e.g., "BM" remains distinct from the common noun "bm").Item The dynamical study of compact objects in General Relativity(UMT Lahore, 2018-03-16) Syed Ali Mardan AzmiIn this thesis, we discuss the dynamical stability of charged compact objects with the help of some mathematical models. For this purpose, we have selected three different models of charged compact objects to discuss the possible occurrence of cracking under different conditions. In first selected model, we discuss charged anisotropic compact objects with a linear equation of state. In second model, we study anisotropic charged compact object PSR J1614-2230 in quadratic regime, while in third model, we study charged compact stars corresponding to embedded class one metric with perfect inner fluid distribution. We investigate the impact of electromagnetic field on the stability regions of charged self-gravitating compact objects by using the concept of cracking. For this, we have applied local density perturbation scheme to the hydrostatic equilibrium equation as well as on physical parameters involved in the model. In particular, we have examined the cracking of charged compact objects (a) PSR J1614-2230, PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3 with linear equation of state (b) PSR J1614-2230 with quadratic equation of state (c) Her X-1, PSR 1937+21, PSR J1614-2230, PSR J0348+0432 and RX J1856-37 corresponding to embedded class one metric. We conclude that these objects exhibit cracking and stability regions decreases with the increase of charge. We also extend two conventional polytropic equations of state to generalized polytropic equations of state for spherical and cylindrical symmetries in the context of general relativity. For this purpose, we formulate the general framework to discuss the physical properties of spherical and cylindrical polytropes with charged anisotropic inner fluid distribution under conformally flat condition. We investigate the stability of generalized polytropic models through Tolman-mass and Whittaker formula for spherical and cylindrical symmetries respectively. We also discuss the possible occurrence of cracking in charged anisotropic polytropes developed under the assumption of generalized polytropic equation of state in two different ways (i) by carrying out local density perturbation under conformally flat condition (ii) by parametric perturbations. We conclude that one of the generalized polytropic equations of state results into a physically viable model and cracking appears for a specific range of density and model parameters.