Mathematics

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https://hdl.handle.net/20.500.14540/131

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Advection-Dispersion Equation For Pollutant Concentration
(Department of Mathematics, 2022) Paudel, Keshav
The advection-dispersion-reaction equation is used to describe the dispersion process. Here, we solve one-dimensional steady advection-dispersion equation numerically by using nite di erence method. Also, we formulate the model to minimize the cost of wastewater treatment. Analytical solution to unsteady advection-dispersion equation using Laplace transformation technique is derived to describe the pollutant concentration C(x; t). We have obtained analytic unsteady solution by taking the water velocity u in the x-direction as a linear function of x and dispersion coe cient D as zero in case of concentration of pollutant in one region. Numerical studies show variation of C with time t. If the added pollutant rate along the river q is very small amount, the variation of C along the river at di erent times coincide to each other. In case of concentration of pollutant in two regions, analytical solutions are determined by taking dispersion coe cient D as non-zero. A coupled system of advection-dispersion equations based water pollution model is presented that incorporates di erent parameters. We have proposed analytical solution for mathematical model. One dimensional model is used to observe the concentrations by taking dimension along the length of river. By considering the removal of pollutant by aeration, event of steady states is investigated. In this model, coupled advection-dispersion equations are solved by taking dispersion coe cient as zero and non-zero, respectively. Keywords: Pollutant, Concentration, Laplace transformation, Dispersion, Analytical solution, Dissolved oxygen.
Bottleneck Just-in-Time Sequencing for Mixed-Model Production Systems
(Department of Mathematics, 2008) Poudyal, Chudamani
Due to today’s competitive automotive industrial challenges of providing a variety of products at a very low cost by smoothing productions on a flexible transfer line, one of the most important and fertile research topic in industrial mathematics is to penalize jobs both for being early and for being tardy. A problem is to determine a production sequence for producing different types of products on the line. Just-in-Time (JIT) mixedmodel production system is used to address this problem, which involves producing only the right products of different models of a common base product in evenly balanced sequences in the exact quantities, at the right times, at the right place. Sequencing JIT production system can be formulated as a challenging nonlinear integer programming problem. The goal of such system is to balance the rate of production of products. Minimization of the variation in demand rates for outputs of supplying processes is the output rate variation problem (ORVP) and minimization of the variation in the rate at which different products are produced on the line is the product rate variation problem (PRVP). The problem for minimizing of deviations between actual and desired production for PRVP can be solved efficiently in pseudo-polynomial time complexity. However, the ORVPs for two or more levels are strongly NP-hard. Heuristic algorithms and dynamic programming to solve such NP-hard problems are summarized. But ORVPs with pegging assumption are solvable by reducing them to the corresponding weighted PRVPs. The cyclic sequences are optimal for both sum and max deviation PRVPs. For the bottleneck PRVP, a binary search technique is used to test the existence of a perfect matching and thereby to get optimal sequence. A feasible sequence always exists such that, at all times, the deviation of actual production from the desired level of production for every product is never more than one unit for the max-absolute and maxsquared PRVPs. An elegant algebraic concept of balanced words is used to deal the bottleneck PRVP. The max-absolute PRVP is shown to be Co-NP with leaving its general complexity open. In this thesis, we study several interesting algebraic structures, properties, existence of cyclic solutions and two applications of bottleneck PRVP. An optimal sequence for an instance of max-absolute PRVP is obtained. With considering two min-sum and maxabsolute objectives, a bicriterion objective for balancing the sequence is analyzed. A comparative study of different objectives is also summarized. Moreover, several directions for further research are also explored including some emerged conjectures.
Dynamic Network Contraflow Evacuation Planning Problem
(Department of Mathematics, 2020) Bhandari, Phanindra Prasad
Evacuation planning problem gives effcient way-out on existing road network that attempts to shift evacuees from risk zone to safer in minimum time with minimum casualty during disasters. Its domain based on network ow problems has been ourished with models and solutions with various network attributes. A common feature on almost all of these models is that the ow function obeys conservation constraints at each intermediate vertex. In particular, maximum dynamic ow (MDF) problem, earliest arrival ow (EAF) problem and quickest ow (QF) problem have great applicability in evacuation planning problems. Contra ow approach recon gures the network identifying ideal direction and reallocating available capacity for each arc to improve ow egress time and/or improve the number of ow units from source to sink. This thesis sketches a brief survey of models and results on contra ow evacuation planning problems. Continuous time model for maximum dynamic contra ow (MDCF) problem is studied with its e cient solution. Thesis also extends contra ow model for multi-network. Network modi cation strategy is applied to give polynomial time algorithms to solve the problems; namely, MDCF problem and earliest arrival contra ow (EACF) problem based on extended model with discrete as well as continuous time setting. The former problems are considered in general networks whereas the latter problems in two terminal series parallel (TTSP) networks. Arc reversibility is allowed only once at time zero in each of the cases. Evacuation models with intermediate temporary shelters could be extra bene t while implementing them. This thesis formulates, as another contribution, ow model for network with capacitated vertices of given priority order in which ow conservation may be violated. This violation makes possible for ow units to be held at intermediate vertices which turns out to be applicable in modeling an evacuation planning problem with intermediate holding of evacuees at temporary shelters despite sending them into the sink. Based on this model, maximum ow problem is considered and proposed a polynomial solution for static case and pseudo-polynomial solution for dynamic case. Also, polynomial solutions for MDF problem and QF problem modeled on uniform path length (UPL) network and for EAF problem modeled on UPL-TTSP network are proposed. As the nal contribution, contra ow approach is linked to evacuation problems with capacitated prioritized vertices. Keywords: Network ow models, Contra ow, Capacitated vertices, Evacuation planning problem, Disaster management.
Effect of Changing the Dimension of Initial Debris Mass in the Dynamics of Landslide Generated Tsunami
(Department of Mathematics, 2021) Acharya, Grishma
Debris ow is a traveling mass of loose mud, soil, air, water and sand that moves down a slope caused due to gravity. When debris ows, landslides, or any gravitational mass ows hit closed or partially open water sources such as seas, oceans, fjords, hydraulic reservoirs, mountain lakes, bays and landslide dams, it results in tsunami (impulse water waves) by transforming their impact energy to water body, potentially causing damages of infrastructures and human casualties both near eld and the distant coastlines. The degree of hazard depends on the scale, types, location and process of the landslide. Volume or size of the initial debris mass that fails in the slope, is one of the dominant factors in accelerating the splash strength or intensity, the propagation and amplitudes of the subsequent water waves and potential dam breach or water spill over. Here, we numerically integrate the two-phase mass ow model [61] for quasi three dimensional, high-resolution simulation results with variation of size of the two-phase initial landslide or debris both longitudinally and laterally. In our numerical experimental results, we observe fundamentally di erent solid and uid wave structures in the reservoir, and the dynamics of submarine mass ow for di erent volumes of the release mass by extending or contracting the base area along down-slope and/or cross-slope directions. The simulation results show that tsunami amplitudes and run out extents are rapidly increased when the volume of initial release mass in the form of a triangular wedge is enlarged by increasing the base area through the increment of the length and breadth of the release base. This study can be an instructive tool to develop and implement tsunami hazard mitigation measures to enhance public safety and reduce potential loss.
Efficient Dynamic Flow Algorithms for Evacuation Planning
(Department of Mathematics, 2020) Dhungana, Ram Chandra
The large scale calamities caused by different natural or human-created disasters are challenging issues to protect life and their surroundings. A great loss of people and socio-economic damages of our society on such disasters is due to the lack of proper planning and their implementation rather than the disaster itself. These issues draw increasing attention of the researchers towards different aspects of disaster management. It is a complex task to develop a signiﬁcant and universally accepted solution strategy to handle such issues. During such disasters, the primary concern is to protect the life, property, and their surroundings with a minimum loss as far as possible. There are different solution approaches to have a signiﬁcant solution for an evacuation planning problem. Contraﬂow, the lane reversal strategy, is one of the widely accepted solution approaches for evacuation planning as it maximizes the outbound capacities of roads by reversing the required road directions and makes the trafﬁc smooth. This signiﬁcantly increases the ﬂow value and decreases the evacuation time. The abstract ﬂow model deals with the ﬂow paths (routes) that satisﬁes the switching property. This concept can be embedded in the contraﬂow technique to have the mathematical formulation on abstract contraﬂow models with efﬁcient algorithms for solving such abstract contraﬂow problems. In this work, different efﬁcient solution procedures are presented for maximum dynamic, lexicographically maximum, and earliest arrival abstract contraﬂow problems. This approach maximizes the ﬂow value in a given time and seeks to eliminate the crossing conﬂicts. The earliest arrival ﬂow problem is one of the most important aspects of evacuation planning with a given capacity and travel time. The objective of the problem is to send the maximum number of evacuees from the given sources to the sinks as quickly as possible. It maximizes the ﬂow value at each time instances simultaneously. Here, we study the earliest arrival ﬂow problem with the contraﬂow approach having supplies and demands in abstract network. v During the evacuation planning problem, one of the essential components is the facility location as it correlates the pre- and post-disaster management. Appropriate facility locations and transportation facilities play a vital role in the solution of evacuation planning problems. Here, the network facility location and the contraﬂow approach are incorporated into the ﬂow models and some efﬁcient algorithms are presented to locate the facility with an objective of minimum ﬂow loss on the evacuation network. Our facility location contraﬂow solutions obtain optimal plans concerning the given and as well as arbitrary locations. With limited resources, it is not an easy task to develop a universally accepted model to handle different aspects and challenges of the evacuation planning problem. However, the budget-constrained network ﬂow improvement approach plays a signiﬁcant role to evacuate the maximum number of people within the given time horizon for the budget provided. We consider an evacuation planning problem that aims to shift the maximum number of evacuees from a danger area to a safe zone in limited time under the budget constraints for network modiﬁcation. In this work, different ﬂow improvement strategies for ﬁxed switching costs will be investigated namely integral, rational, and either to increase the full capacity of an arc or not at all. A solution technique on a static network is extended to the dynamic one. Moreover, we introduce the static and dynamic maximum ﬂow problems with lane reversal strategy and also propose efﬁcient algorithms for their solutions. Here, the contraﬂow approach reverses the direction of arcs concerning the lane reversal costs to increase the ﬂow value. As an implementation of an evacuation plan may demand a large cost, the solutions proposed in this thesis with budget-constrained problems play an important role in practice. In this thesis, the contraﬂow models and their solutions strategies have been established and investigated in an abstract network topology. To allocate the facility during the evacuation process FlowLoc problems and their solution have been introduced in the evacuation network. The arc switching costs have been considered for the ﬁrst time in the evacuation network. These optimization methods play signiﬁcant roles in maximizing the ﬂow and minimizing the evacuation time, and also have the great support for logistics and emergency vehicle movements in disasters.
Evacuation Optimization with Minimum Clearance Time
(Department of Mathematics, 2020) Adhikari, Iswar Mani
We are under the threats of natural or human-created disasters. Disasters are unavoidable and are mostly uncertain to happen. If occurs, the situation becomes vulnerable, effects badly the society, and its socio-economic status. Its direct impact is on the trafﬁc systems. On the other hand, the increasing number of complex trafﬁc networks brought difﬁculty in managing the rush hour trafﬁc as well as the large events in urban areas. The optimal use of the vehicles and their assignments to the appropriate shelters from the disastrous zones are highly complicated in emergency situations. The maximum efﬁciency and effectiveness of the evacuation planning can be achieved by the appropriate assignment of the transit-vehicles during pre- and post-disaster operations. The evacuation planning problem deals with sending the maximum number of evacuees from the danger zones to the safe zones in minimum time, as efﬁciently as possible. It can be further classiﬁed into microscopic and macroscopic planning. The microscopic planning deals with the individual evacuee’s behavior in which some probabilistic laws for individual evacuees movement are presented and mainly based on the simulation approaches. But in macroscopic planning, it is principally based on optimization approaches where the evacuees are treated as the homogeneous group and only the common characteristics are considered. Optimization approaches on such macroscopic evacuation planning can further be classiﬁed as a heuristic approach, population optimization, modeling as ﬂuid dynamics, mathematical programmings, trafﬁc management, optimal evacuation destination, and network ﬂow formulation. Among them, the dynamic network ﬂow formulation has been found suitable evacuation optimization approach with the variants of ﬂow maximization and/or time minimization problems. In such formulations, time can be considered as discrete or continuous. Evacuation planning problems are handled with different prospectives, namely, the transit-based, car-based, and pedestrian movements depending upon the movement of the evacuees on the evacuation scenarios. The transit-based planning problems are to vi minimize the duration of evacuation by routing and scheduling a ﬂeet of vehicles, say buses, as the bus-based evacuation planning problem. Such a problem is an important variant of the vehicle routing problem. Trafﬁc route guidance, destination optimization, and optimal route choice are some of the approaches to accelerate the evacuation planning process. Their effectiveness depends upon the evacuee arrival patterns at the pickup locations and their appropriate assignment to the transit-vehicles in the available evacuation network. An embedded network is composed of two constituent sub-networks, namely, the primary and the secondary sub-networks. Evacuees are to be collected at the pickup locations of the primary sub-network from the danger zone(s) and are to be assigned to transit-vehicles in the secondary sub-network. For time minimization evacuation planning problems, evacuees are to be collected in the earliest arrival ﬂow pattern at zero transit times and is to be followed by dominant vehicle assignments. Transit-vehicles are provided from the bus depot in the secondary sub-network. Pickup locations are taken as the sources for the subsequent process to minimize the overall network clearance time from the danger zone to safety. In our work, we have proposed an integrated optimization approach in such an embedding to achieve the minimum clearance time. The earliest arrival pattern respects the partial lane reversal strategy, whereas the better assignments are based on the dominance relations concerning the evacuation duration. We use the quickest transshipment partial arc reversal strategy to collect the evacuees in minimum time from the disaster zones to the pickup locations of the primary prioritized sub-network. By treating such pickup locations as sources, the available set of transit-buses is assigned simultaneously in the secondary sub-network to shift the evacuees ﬁnally to the sinks with minimum clearance time. The lane reversal strategy signiﬁcantly reduces the evacuation time and maximizes the ﬂow of evacuees, whereas reversing them only partially has an additional beneﬁt that the unused road capacities can be used for supplying emergency logistics and allocating facilities as well.
The Interplay Between Measure Theory and Topology
(Department of Mathematics, 2016) Rana, Jit Bahadur
A connection between measure theory and topology is established when a eld F is de ned in terms of topological properties. More precisely, we de ned F as the smallest eld containing all the open sets of a topological space , then there are interesting interrelation between measure theory and topology. We study the interrelation between topological space, open sets and continuous functions in one hand and measure space, measurable set and measurable function on the other.
Newton Type Iterative Methods for Solving Nonlinear Equations
(Faculty of Mathematics, 2017) Jnawali, Jivandhar
Solving single variable nonlinear equations efficiently is an important consideration in numerical analysis and has wide range of applications in all elds of science and engineering. Finding the analytic solutions of such equations is not always possible. Newton's method is the most widely used numerical method for solving such equations. In this thesis, we have developed several new Newton type iterative methods for solving nonlinear equations of a single variable. To obtain these methods, we used different techniques such as: (i) amalgamation of existing methods; (ii) amalgamation of existing and our investigated methods with the secant method; (iii) amalgamation of existing methods and modi ed secant method; (iv) idea of integral approximation; and (v) use of inverse function methods. The work done in this thesis is inspired by the work of Potra and Pt ak, Kasturiarachi, Jain, Weerakon and Fernando, Ozban, Dhegain and Hajarian, Ujevi c, Erceg and Laki c, Amit and Basqular, Hasanov, Ivanov and Nedzhibov as well as recent work of McDaugall and Wotherspoon. For each method obtained in this thesis, the order of convergence has been calculated and compared with that of the similar existing methods. Also, most of the methods are supported by numerical examples.
Numerical modeling of influence of source in heat transformation: An application in blacksmithing metal heating
(Department of mathematics, 2022) Kandel, Hari Prapanna
Partial differential equations (PDEs) are used to mimic a variety of real-world physical issues. A standard parabolic PDE of the form u ; ( > 0) is an 1D heat equation. In a regular form of domain, the heat equation has an analytical solution. Computing an analytical solution becomes challenging, if not impossible, any time the domain of such modeled issues has an uneven shape. In this case, numerical methods can be used to ﬁnd the numerical solution of these PDEs. Through the domain’s discretization into a limited number of areas. One of the numerical techniques used to determine the numerical solutions of PDEs is the ﬁnite difference method (FDM). Here, the FTCSSfor the one-dimensional heat equation and the numerical computation of its solution using FTCSS are discussed. Furthermore, numerical solution and analytic solution of heat equation has been compared and analyzed. Additionally, the 1D heat equation with variable starting conditions (ICs) and numerous initial conditions (ICs)has been solved numerically using FDMs. Blacksmiths heated the parts at various temperatures and locations to mold different metals into the necessary shapes. The numerical solution method for the 1D heat problem given here can be used to solve heat equations used in engineering and scientiﬁc disciplines. t = u xx
Numerical Solutions of 2D Incompressible Navier-Stokes Equations in Variable Viscosity Case
(Faculty of Mathematics, 2016) Thapa, Dhak Bahadur
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On the Study of Distribution of Primes and Twin Prime Conjecture
(Faculty of Mathematics, 2016) Adhikari, Khagendra
The distribution of primes, mainly focusing on the Tchebycheff estimates of prime counting function, Mertens Theorem which are most signiﬁcant results for distribution of primes have beeen studied in this thesis. Distribution of Twin Primes, Twin Prime Conjecture and some developments towards the Twin Prime Conjecture is also studied. The alternative approaches for the Twin Prime Conjecture has also been studied in this thesis.
Optimization models with exclusive bus lanes
(Department of Mathematics, 2022) Chand, Gaurab
Exclusively reserved lane for public buses in arterial road of the city is called exclusive bus lane (EBL). In this research study, we survey network optimization EBL models, then we review min-max dynamic optimization EBL model with three modes of vehicles. Major upgraded terms on reviewed model have been taken prior origin count of the bus travel time, bureau of public road (BPR) constraint to the car mode and maximum number of motorcycle rider constraint. Among them, BPR constraint has impacted signi cantly over objective function as well as planning of EBL on the transportation network. Tra c data related to the motorcycle mode had been estimated using statistical tool by increasing the capacity of arcs and without loss of generality with original data of buses and cars. We prefer parallel genetic algorithm (PGA) for the solution of the revised problem and proved that complexity is NP-hard. A numerical example is revealed as a reviewed optimization network model to achieve the feasibility and therefore yield optimal solution.
Orthogonality in Normed Linear Spaces
(Department of Mathematics, 2022) Ojha, Bhuwan Prasad
This thesis deals with the orthogonality in normed linear spaces. The goal is to investigate and study different notions of orthogonality in normed spaces. By utilizing the 2-HH norm and bounded linear operators, some notions of orthogonality are introduced and then, different properties of orthogonality in relation to these orthogonalities are studied. We generalize the Robert, Birkhoff-James, and a new orthogonality in terms of the 2HH norm, and study the main properties of orthogonality. We prove that the Birkoff and Robert orthogonality in terms of the 2-HH norm are equivalent if the underlying space is real inner product space. Further, we prove that the isosceles orthogonality is homogeneous if and only if it is additive. Additionally, we prove that the orthogonality relation of type (I) in terms of 2-HH norm satisﬁes non-degeneracy, simpliﬁcation, continuity, and uniqueness properties. Moreover, we prove that the Carlsson orthogonality in terms of bounded linear operators also satisﬁes non-degeneracy, simpliﬁcation, and continuity properties. In the case of norm attaining bounded linear operator with disjoint support in a Hilbert space H, we prove that two operators are orthogonal in the sense of Pythagoras if and only if they are orthogonal in the sense of isosceles. In terms of buonded linear operators, we prove that the Pythagorean orthogonality and orthogonality relation of type (I), imply the Birkhoff-James orthogonality, but the converse may not be true. Under the restriction of an element belonging to the norm attainment set, we prove that the orthogonality of images also implies the orthogonality of operators in the Carlsson as well as Robert’s sense. Finally, as applications, we prove that the Pythagorean orthogonality implies the best approximation, and the best approximation (resp. best approximation) and Birkhoff orthogonality ( resp. Birkhoff orthogonality) are equivalents. Keywords: Normed linear spaces, Inner-product space, Birkhoff-James orthogonality, Pythagorean orthogonality, p-HH norm, Best approximation
Structure and Properties of Fatou, Julia, Escaping and Fast Escaping Sets of Holomorphic Semigroups
(Faculty of Mathematics, 2020) Subedi, Bishnu Hari
We study the dynamical behavior of a semigroup generated by holomorphic functions in the complex plane. In particular, we concentrate on semigroup dynamics, where semigroups are generated by transcendental entire functions. It is a study of the behavior of the compositions of a ﬁnite set of holomorphic functions in the complex plane. We study Fatou, Julia, escaping and fast escaping sets of such semigroups. The principal aim of this thesis is to investigate the structure and the properties of these sets in the more general settings of holomorphic semigroups. In this thesis, we see to what extent, the structure and the properties of the Fatou, Julia, escaping and fast escaping sets of classical holomorphic dynamics are preserved and generalized to semigroup dynamics, and what new phenomena can occur. A holomorphic semigroup is not abelian in general; however, a cyclic semigroup is abelian, so differences in the dynamics can occur in the structure and the properties of these sets. If a semigroup is abelian, such types of differences will narrow down, and most of the structure and the properties of these sets of classical holomorphic dynamics are preserved and generalized. In this thesis, we generalize the notion of abelian semigroups to nearly abelian semigroups, and we investigate the identical structure and the properties of these sets in such semigroups. On the basis of the algebraic notion of different indices such as ﬁnite index, coﬁnite index and Rees index, we also investigate subsemigroups whose Fatou, Julia and escaping sets coincide with their corresponding parent semigroup. In the holomorphic semigroup setting, there may be empty Fatou sets and empty escaping sets; hence we also investigate certain holomorphic semigroups whose Fatou sets and escaping sets are non-empty on the basis of (partial)fundamental sets and Carlemen sets. Finally, we deﬁne fast escaping sets of transcendental semigroups, and we discuss some fundamental structure and properties of these sets.
A Study on Advances of Hurwitz Theorem
(Faculty of Mathematics, 2016) Gautam, Ramesh
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Visualization, Formulation and Intuitive Explanation of Iterative Methods for Transient Analysis of RLC Circuit
(Department of Mathematics, 2021) Thakur, Bhogendra Kumar
The time-varying currents and voltages resulting from the sudden application of sources usually due to switching are transients. An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The transient response is dependent on the value of the di erent characteristics of the damping factor (i.e., overdamped, critically damped, and underdamped). The numerical solutions of rst and second-order di erential equations with initial value problem (IVP) have been computed by using the Explicit (Forward) Euler method, Implicit (Backward) Euler method, Classical second-order (Heun's or RK2) method, Third-order Runge-Kutta (RK3) method, Fourth order Runge-Kutta method and Butchers fth-order Runge-Kutta (BRK5) method. The observation compares this numerical solution of ODEs obtained by the above-mentioned methods among them with the necessary visualization and analysis of the error. These iterative methods will be extended and implement to analyze the transient analysis of an RLC circuit. The superiority of these methods over one another has been examined. The Butcher's fth-order Runge-Kutta (BRK5) method is found to be the best numerical technique to solve the transient analysis due to its high accuracy of approximations. Moreover, we consider the possibility of discussing and analyze above mentioned iterative methods in the cases of di erent characteristics of damping factors. Di erent methods are used di erent iterative methods to analyze the transient analysis of an RLC circuit and compared among them. The consequences of this work lend some limelight to the modern approaches to solving complicated mathematical problems.

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