Department of Mathematics
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Item Annihilation of Bumps in Two-Population Neural Field Model Under the Influence of Gaussian External Input(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Hafiz Shahzad Ali; FA18-RMT-019; LHR TP 6456; Dr. Muhammad Yousaf BhattiIn this work, we have examined the impact of Gaussian-type spatio-temporal external input on annihilation of self-managed activity states (bumps) in a Wilson-Cowan type two- population neural field model. These activity states plays the role of the intellectual framework with a restricted limit that can hold information known as working memory in the prefrontal cortex. The effect of various parameters engaged with the external input on annihilation of these persistent activity states is researched in detail. This very choice of Gaussian-type spatio-temporal external input is quite near the nature as seen by Roth et al. [4]. In this work eraser mechanism is used to annihilate the spatially symmetric solutions. If there is an activity in the network, inhibitory external input with no excitatory part in the external input can kill the activity. In the whole of our investigation, the prominent thing is Gaussian-type function which yields quite better findings in spite of three sided (triangular function) used by Yousaf et al. [5]. It is additionally discovered that the relative inhibition time constant serves as a pivotal role in annihilation of the activity. Runge-Kutta fourth order method has been used for mathematical simulations of this work.Item PT-Symmetric and W-Shaped Solitons for Nonlinear Schrödinger Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ujala Safdar; FA18-RMT-035; LHR TP 6548; Dr. Syed Tahir Raza RizviSolitons are the special types of solitary waves who does not change their speed and shape during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical Solitons. Optical solitons are produced due to interaction between group velocity dispersion and self-phase modulation. In this thesis, we studied the different forms of optical solitons in non-Kerr law media, parabolic, power, quadratic-cubic and anti-cubic laws with parity-time (PT) symmetric nonlinear and linear lattices. We obtain various types of soliton solutions with the as sistance of the inverse engineering scheme (IES). Under diverse physical conditions, we produce bright, dark and singular solitons. We will also obtained the dynamical behaviour of W-shaped soliton in cigar-shaped Bose Einstein condensate (BEC) in various nonlinear itiesItem Lump and Interaction Solution for the Nonlinear Schrödinger Equation(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Iram Javed; FA18-RMT-075; LHR TP 6550; Dr. Syed Tahir Raza RizviSolitons are the special types of solitary waves which do not vary their shape and frequency during propagation speed and is also an attractive area of current research in nonlinear physics and mathematics as well as in fiber optics and communication technologies. There are many forms of solitons such as singular soliton, bright soliton, dark soliton and lump soliton. In this thesis, we acquire lumpsolitonsolutionsfor paraxial nonlinear Schr¨odinger equation (NLSE) with the aid of Hirota bilinear method (HBM). In space, lump soliton solutions are rationally positioned in every direction. We also get optical dromions for paraxial NLSE by using modified extended Tanh method. We get domain walls, singular dromions and combined dark-singular dromions. We also list the constraint conditionsItem Edge Reflexive Irregularity Strength of -graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Muhammad Amir Asif; FA18-RMT-036; LHR TP 6547; Dr. Hafiz Muhammad AfzalThe labeling of a simple finite graph J is allocation of positive numbers (called labels) to the elements (vertices and/or edges) of a graph. The weight of the graph element is the sum of all labels correlated with a vertex or an edge. The labeling is irregular if the weights of edges and/or vertices are different. For a graph J we define a total labeling, such that the edge labels and vertex labels are positive integers and even integers respectively and for different edges we have different weights, then the labeling is referred as edge reflexive irregular total labeling and the edge reflexive irregularity strength of graph J is res(J). In this thesis, we give the exact values of edge reflexive irregularity strength of 𝑓 graphs generated by different families of graphs such as cycle, path and star like graphs etc.Item Local Fractional Metric Dimension of Different Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) MUHAMMAD ZULQARNAIN; FA18-RMT-078; LHR TP 7012; Dr. Muhammad HussainIn this thesis we find out different chemical structures and then convert them into planar network graphs. Consequently, to calculate the local fractional metric dimension of these network graphs. We are also interested to extend the proposal to the rooted product of different families of graphs. We also find local metric dimension and local fractional metric dimension on the line graph of subdivided honeycomb network.Item Certain Results on m-polar Fuzzy Groups(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Musawwar Hussain; FA18-RMT-015; LHR TP 6455; Dr. Adeel FarooqIn this thesis we have developed some fundamental results from group theory to fuzzy and m-polar fuzzy groups. We have defined some basic characteristics of fuzzy groups and have provided the m-polar fuzzy analogues of fuzzy subsets, groups, level sets, and cosets. We have investigated an equivalence relation to distinguish whether a m-polar fuzzy subgroup of a group G is equivalent to any other m-polar fuzzy subgroup of that same group G, or not. We determined the maximal chains and a general formula for quaternion and dihedral groups to give the cardinality of their fuzzy and m-polar fuzzy subgroups. We intro duced aninequality which gives quantitative relationship between maximal chains, level subgroups and fuzzy subgroups. Moreover, we mainly constructed the m-polar fuzzy analogues of isomorphism theorems and butterfly lemma.Item Degree Base Topological Indices of Metallic Organic Compound(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Abdul Qadeer; FA18-RMT-064; LHR TP 6463; Dr. Muhammad Faisal NadeemA topological index describes the chemical information as numerical number. Topological index is a graph invarient and describe the topology of that graph. In development of QSARs(quantitative structure-activity relationships) topological indices are very useful. QSARs relates biological activity of a molecule with its chemical structure. QSAR are useful in toxicity prediction and in drug discovery. There are some types of topological indices like degree-based topological indicex, distance-based topological indicex, eigen value toplogical indicex and matching and mixed topological index. There are some famous toplogical indices like geometric-arithmetic index, Randić index which have many applications in chemistry and useful in pharmacology, ABC index(achivements of babies and children). In this thesis we discuss the metallic non-metallic behaviour of TM-TCNB graph. We compute topoligical indices ABC index, geometric arithmetic index and Randić index for variation of alpha. We also calculated sum connectivity index and symmetric division index for different results. In this thesis there are six chapters. In first chapter we will define some funddamental definitions. In 2nd chapter we will discuss different types of topological indices like Hosoya index, Winner index Estrada index etc. 3rd chapter is related to literature review of TCNB graph. In 4th chaper there is basic structure of TCNB and results of TCNB graph. Conlcusion and relevant refrences are given in 5th and 6th chapters respectively.Item Algorithm Development for the Bayesian Optimization of Complex Data Sets(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Muhammad Javed Zahoor; FA18-RMT-057; LHR TP 6461; Dr. Ayesha SohailOptimization is a subfield of applied mathematics and is used in the fields of nano-medicine, applied sciences and computational biology extensively. The recent trends of optimizations are focused on the algorithm development of nonlinear programs for complex data sets. During this research, an algorithm is developed for the application of “Bayesian Optimization”, for the selection of the most accurate hyper-parameter, using the Python framework as well as the MatlabTM 2020 packages. The developed model is then implemented on the experimental data sets, extracted from the online repositories. To ensure the optimalilty of the forecasting, the algorithms are exploited in detailed manner during this research.Item Growth of Matter Index and Cosmic Acceleration(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ahmad Hassan; FA18-RMT-005; LHR TP 6452; Dr. Yousaf HabibThe aim of this research is to construct G-symplectic general linear methods for solution of ordinary differential equation in which parasitism can be controlled. Due to multi-value nature of general linear methods, there exist a parasitic component which disturb the actual solution and destroy the overall accuracy. We want to construct parasitic free G-symplectic general linear methods.Item Metric Dimension of Line Graphh of Theta Graphh(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Rabia Liaquat; Fa18 RMT 048; LHR TP 6026; Dr. Muhammas Hussain