Graphical Method Of Solution Research Articles

Simulation of Time Independent Schrödinger Equation for Finite Potential Well Using the Graphical Solution Method

This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.

A concise analytical model for the ideal reverse osmosis desalination processes.

Permeate recovery is a key parameter that plays a central role in the performance assessment and optimization of reverse osmosis processes. It remains a great challenge for engineers in the field to determine the recovery conveniently and accurately from the basic parameters of a membrane system. A concise analytical model is presented here that, without the need for the empirical or fitting coefficients and tedious numerical calculation, links the recovery of a reverse osmosis process rigorously to three basic parameters: the feed water salt concentration, the characteristic of membrane process, and the driving pressure. A graphical solution method to the model is also introduced to find out the recovery effortlessly. The concise model and graphical method are demonstrated under various conditions as a powerful tool for the performance simulations and improvement of reverse osmosis processes. PRACTITIONER POINTS: A concise model is presented for permeate recovery in an ideal RO desalination system. The recovery of an RO system is analytically related to a few system parameters. A graphical solution method is introduced to find out the explicit recovery. The ideal RO model can be a powerful tool for simulation of RO processes.

Изучение диэлектрических, поляризационных свойств циклопентанола и его растворов в циклопентане новым вариационным методом

Results of calculating the theoretical principles of the variational method for measuring the dielectric parameters of polar liquids: cyclopentanol and its solutions in cyclopentane have been shown in the paper. Their dielectric constant ξ' and dielectric losses ξ'' are calculated. Solutions to the equations were found and a graphical solution method and an automated method for calculating ξ' and ξ'' were developed on the basis of this method. Comparison with the results of other methods revealed that these indicators are at the same time minimal within 1.5–2.0%.

Mathematical design and graphical solution of the multiple bifurcation equations of a 4-DoF benchmark model

A 4 degree-of-freedom (DoF) benchmark model of multiple bifurcation (MB) in compound stability problems of nonlinear structures is proposed. In the MB model, the governing equations are designed to be highly simplified when two critical modes exist in the singular Jacobian matrix. The resulting MB equations can be solved analytically through manual calculation. To verify the applicability, graphical solution methods are applied to solve the MB equations and visualize the multiple path branching through a graphical monitor. The proposed benchmark model and graphical strategies can help understand the stability phenomenology and MB in the stability design of large-scale finite element models.

Theoretical Investigation of a Long Period Grating Sensor Employing Multilayer Fiber Geometry

A theoretical formulation of the long-period grating sensor is implemented with respect to efficient refractive indices of core and cladding. The obtained result is utilized further for determination of ERIs of core and cladding using graphical analytical method and multilayer fiber geometry. ERI values of primary guided core, bi and tri layer fiber geometry using MATLAB software. In the present paper effective refractive index of core and cladding for the two and three layers have been investigated. The cladding modes have also been calculated using Erdogan’s proposed three-layer geometry as well as two-layer graphical solution method. We have compared both the techniques, discussed the limitations of graphical solution method and the benefits of three-layer geometry.

“The joy that engineers and mathematicians have come together.”1

“The joy that engineers and mathematicians have come together.”¹

Development of mathematical model of extraction from lupine with cheese serum by applying low-frequency mechanical vibrations

It is noted that the model is designed to create the largest possible pressure change in the cheese whey in the extractor, since the rate of transfer of the target components is proportional to the pressure difference at the ends of the capillaries. The mathematical description of impregnation as the main or important auxiliary operation is given in detail. The equations for the impregnated part of the capillary, the ratio of impregnation rates at different times are given. From the above dependencies, the equation Washburne regarding the time of impregnation. The formulas for calculating the volume of extractant passed through the capillary, serum and forced out of the capillary air taking into account the viscous resistance of the latter. After integration of the equation of the speed of capillary impregnation of the obtained expression allows to estimate the final value of the impregnation in the initial stage. For different cases of capillary impregnation expressions are written at atmospheric pressure, vacuuming and overpressure. The introduction of dimensionless values allowed to simplify the solution and to obtain an expression for calculating the time of pore impregnation. The analysis of the equation of dimensionless impregnation time taking into account the application of low-frequency mechanical vibrations is made. It is noted that the processes of impregnation and extraction occur simultaneously, so the impregnation time is often neglected, which impoverishes the understanding of the physics of the process, reduces the accuracy of the calculation. Taking into account the diffusion unsteadiness of the process of substance transfer due to hydrodynamic unsteadiness, the equation containing the effective diffusion coefficient is written. The equation of unsteady diffusion for a spherical lupine particle in a batch extractor is supplemented with initial and boundary conditions. Taking into account the balance equation, the kinetic equation of the process is obtained. We studied the distribution of pores in the particle lupine along the radii and squares, the calculated value of the porosity of the particle. The values of De and Bi are determined by the method of graphical solution of the balance equation, the equation of kinetics and the parameters included in these equations. Conclusions on the work.

Graphical solution for system optimum dynamic traffic assignment with day-based incentive routing strategies

This paper analyzes the dynamic traffic assignment problem on a three-alternative network with day-based incentive routing strategies by using graphical solution method. It is assumed that the cumulative count curve of vehicles is known and that the arrival rate is unimodal. The dynamic system optimum (DSO) allocation lines are first drawn based on calculus of variations. Three possible optimal allocation lines are analyzed. A day-based incentive routing strategy is designed and conditions that when and how to implement the incentive scheme to realize DSO are then derived. Extension to general parallel networks is also given. Examples are presented to demonstrate the effectiveness of the scheme.

Propagation Characteristics of Optical Long Period Fiber Gratings using Graphical Solution Methods

In this paper, we explicitly show a complete analysis and detailed calculation of propagation characteristics of the effective refractive indices of the core and cladding for a long period grating employing both the Two-Layer and Three-Layer Geometries. The results are achieved employing graphical solution methods and presented in terms of the effective refractive indices of the core and cladding. Then, we compare the results with OptiGrating (Optiwave Systems Inc., Ottawa, Canada). Finally, we point the best model to evaluate the effective refractive indices, avoiding incorrect designs of the long period gratings.

Summary on Research Advances on the Reservoir Bank Instability

The reservoir bank instability is the outcome combined the action of the external factors with the action of the internal factors. The limit equilibrium analysis, the strength reduction finite element method, the saturated/unsaturated seepage coupling and so on were applied to analyze the reservoir bank stability. The condition analogy graphical solution methods, the mathematical analysis method, the balanced cross-section method and so on were applied to forecast the reservoir bank stability. The reservoir bank instability mechanism depends on the physical action, the chemical action and the mechanical action of the water in the reservoir bank rock and soil. At present, the drainage measure, the cutting slope and reducing load measure, the retaining and supporting measure , the anti-slide piles measure and so on have been widely used to control the reservoir banks instability. Aiming at the research status of the reservoir bank instability, the problems in study and the suggestions for future work are pointed out in this paper.

Calculation of cladding modes using both two layer and three layer geometry in LPFG

In this paper we have calculated cladding modes of long period fiber grating. We have calculated cladding modes using both two layer (Graphical solution method) geometry and three layer geometry proposed by Erdogan. We have compared both these techniques, discussed the limitations of Graphical solution method and benefits of three layer geometry.

The Effectiveness of Using a Web-Based Applet to Teach Concepts of Linear Programming: An Experiment in Active Learning

The graphical solution method to a two-variable linear program (LP) provides valuable insights about the general nature of multivariable linear programming models. As a result, introductory operations research/management science textbooks typically present a graphical solution method to a two-variable LP as a prelude to the presentation of more complex problems. Construction of a two-dimensional feasible region combined with iso-profit (or iso-cost) lines on a blackboard or overhead projector can be tedious, at best. Even with tools such as PowerPoint, that include drawing tools, it is difficult to show students what happens in a graphical LP as constraint lines and iso-profit lines shift around on a graph. To overcome this hurdle, this paper presents a Web-based Java script applet that was used by both instructors and students to graphically illustrate/learn fundamental concepts of LP models. It then describes the results of a study that compares student performance on exams of those who did use the applet versus those who did not. Results show that the students who used the applet to learn about LP concepts performed significantly better than those who did not. Implications for using such active learning techniques and models in the classroom are discussed.

The Product-Mix Auction: A New Auction Design for Differentiated Goods

The “Product-Mix Auction” is a single-round auction that can be used whenever an auctioneer wants to sell (or buy) multiple differentiated goods. It allows all participants to express their preferences between varieties, as well as for alternative quantities of specific varieties. Bidders simultaneously make sets of bids. Each of a bidder’s bids can be an OR” bid that offers a different price for each different variety, and the auction then accepts at most one of the offers from each bid (whichever is best for the bidder given the prices that the auction sets). The graphical solution method makes the operation and merits of the auction easy to explain. The auction is more efficient, and less sensitive to market power, than either running a separate auction for each different variety or running a combined auction with predetermined price differences between varieties. It is also faster, less vulnerable to collusion, and can be easier to use and understand than a simultaneous multiple round auction (SMRA). Moreover, unlike in an SMRA, the auctioneer can specify how the quantities to be sold will depend on the auction prices, by choosing supply functions across varieties. Related material is at www.paulklemperer.org.I invented the auction for the Bank of England when it consulted me at the beginning of the financial crisis in 2007. It now uses it regularly to auction loans of funds secured against different varieties of collateral. The Bank’s first implementation corresponds closely to the description in section 2; the then-Governor, Mervyn King, told the Economist that “[the Product-Mix Auction] is a marvellous application of theoretical economics to a practical problem of vital importance to financial markets” I also advised the Bank on the updated version it introduced in 2014. This permits more dimensions (i.e., more “varieties” of goods), and also allocates an endogenous total quantity. Simple linear-programming methods are now used to solve the auction.The current Governor, Mark Carney, announced plans for greater use of the auction.Future auctions may use others of the enhancements described in section 3.The Guardian newspaper published a 5 minute video including an interview with then-Deputy Governor, Paul Tucker.

Graphing Calculator Use in Algebra Teaching

This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed. Forty of the 75 schools (53%) returned a total of 109 individual surveys. Results indicated that: (1) While 78% of teachers have some access to the technology, only 28% use it regularly. (2) Statistically significant relationships exist between use and age, years of experience, teaching assignment, and teaching level. (3) Respondents view graphical solution methods as secondary to symbolic methods. (4) Teachers are more receptive to using technology to supplement rather than expand the curriculum.

A New Auction for Substitutes: Central-Bank Liquidity Auctions, 'Toxic Asset' Auctions, and Variable Product-Mix Auctions

This paper was revised as: The Product-Mix Auction: a New Auction Design for Differentiated Goods The “Product-Mix Auction” is a single-round auction that can be used whenever an auctioneer wants to sell (or buy) multiple differentiated goods. It allows all participants to express their preferences between varieties, as well as for alternative quantities of specific varieties. Bidders simultaneously make sets of bids. Each of a bidder’s bids can be an OR” bid that offers a different price for each different variety, and the auction then accepts at most one of the offers from each bid (whichever is best for the bidder given the prices that the auction sets). The graphical solution method makes the operation and merits of the auction easy to explain. The auction is more efficient, and less sensitive to market power, than either running a separate auction for each different variety or running a combined auction with predetermined price differences between varieties. It is also faster, less vulnerable to collusion, and can be easier to use and understand than a simultaneous multiple round auction (SMRA). Moreover, unlike in an SMRA, the auctioneer can specify how the quantities to be sold will depend on the auction prices, by choosing supply functions across varieties. Related material is at www.paulklemperer.org. I invented the auction for the Bank of England when it consulted me at the beginning of the financial crisis in 2007. It now uses it regularly to auction loans of funds secured against different varieties of collateral. The Bank’s first implementation corresponds closely to the description in section 2; the then-Governor, Mervyn King, told the Economist that “[the Product-Mix Auction] is a marvellous application of theoretical economics to a practical problem of vital importance to financial markets” I also advised the Bank on the updated version it introduced in 2014. This permits more dimensions (i.e., more “varieties” of goods), and also allocates an endogenous total quantity. Simple linear-programming methods are now used to solve the auction. The current Governor, Mark Carney, announced plans for greater use of the auction. Future auctions may use others of the enhancements described in section 3. The Guardian newspaper published a 5 minute video including an interview with then-Deputy Governor, Paul Tucker here: www.economics.ox.ac.uk/index.php/General-News/how-geometry-came-to-the-rescue-during-the-banking-crisis-video

Achieving Stability under Inaccessible Conversions Using CSTR Cascades

Multiple steady states in an isothermal CSTR for reaction rates of the form R=kC/(1+KC)2 exhibit concentration multiplicity for a certain parameter range of the Damkohler number, Da. The unstable steady state of the above reaction rate form is very difficult to be observed and maintained. In this work the graphical solution method as proposed by Chen and Crynes (1987) is modified to attain a cascade of CSTRs that are capable of achieving this inaccessible steady state. The modified numerical scheme is based on optimization search. This method was successful in achieving the inaccessible steady state by two CSTRs cascade. The comparison with the graphical method showed that the modified method is faster and more accurate in solving the CSTR problem. The CSTR cascade gives better stability and tolerance to a probable control system failure.

The use of surface strain data and a neural networks solution method to determine lumbar facet joint loads during in vitro spine testing

A new method for determining facet loads during in vitro spine loading using strain gauges and a neural networks solution method was investigated. A test showed that the new solution method was more robust than and as accurate as a previously presented graphical solution method for computing facet loads using surface strain. The technique was subsequently utilized to assess facet loads at L1–L2 during flexibility testing [7.5 N m pure moments in flexion (FL), extension (EX), right and left axial rotation (AR), and right and left lateral bending (LB)], and stiffness testing (FL-EX with 400 N compressive follower load) of six human lumbar spine segments (T12–L2). In contrast to other techniques, such as thin film sensors or pressure-sensitive film, the strain-gauge method leaves the facet joint capsule intact during data collection, presumably allowing more natural load transmission. During flexibility tests, the mean (±standard deviation) calculated facet loads (in N) were 46.1±41.3 (FL), 51.5±39.0 (EX), 70.3±43.2 (AR-contralateral side), 31.3±33.4 (AR-ipsilateral side), 30.6±29.1 (LB-contralateral side), and 32.0±44.4 (LB-ipsilateral side). During stiffness tests, the calculated facet loads were 45.5±40.4 (upright), 46.6±41.9 (full FL), and 75.4±39.0 (full EX), corresponding to an equivalent of 11.4%, 11.6%, and 18.8% of the compressive follower load (upright, full FL and EX, respectively). The error associated with this technique, which was below 11 N for loads up to 125 N, is comparable to that reported with other techniques. The new method shows promise for assessing facet load during in vitro spine testing, an important parameter when evaluating new implant systems and surgical techniques.

Graphical solution and continuum approximation for the single destination dynamic user equilibrium problem

This paper studies the user optimum dynamic traffic assignment in a network consisting of a congested freeway section connected to a surface street grid via capacitated off-ramps. It is shown that although the network is simple, it is in general complicated to obtain the optimum solution analytically because (i) it has two different alternating diversion patterns and (ii) it can only be obtained sequentially in time. A simplified graphical solution method is proposed to cope with (i) but the method remains sequential. Finally, a continuum approximation is proposed that enables finding closed-form solutions without the above limitations.

Theory of two-layer hydraulic exchange flows with rotation

Two-layer rotating exchange flows through channels of rectangular cross-section are modelled using semi-geostrophic, zero-potential-vorticity theory. For a given channel cross-section the full range of possible flow states is considered. The interface always has a uniform slope across the channel, but may separate from one or both of the sidewalls to attach to the upper or lower boundary. The flow may be subcritical, critical or supercritical. These different states are identified in a pseudo-Froude-number plane analogous to that developed by Armi (1986) for non-rotating flows. If the ratio of the channel width to the Rossby radius is constant along the length of the channel, then the solution may be traced along the entire channel using a single diagram. Several examples of maximal and submaximal exchanges are considered. This graphical method of solution is contrasted with the functional approach of Dalziel (1988, 1990).The exchange flux is determined as a function of the channel geometry, the strength of rotation and the difference in Bernoulli potential between the two layers.

System optimum dynamic traffic assignment graphical solution method for a congested freeway and one destination

This paper studies the system optimum dynamic traffic assignment in a network consisting of a hypothetical surface street grid and a congested freeway section. Vehicles can be diverted through off-ramps, and on-ramps can be metered. The family of solutions are identified graphically using Newell’s queueing diagrams. Because enforcing diversion is still a technological puzzle, these results provide a benchmark for future ITS applications, and a building-block for including both departure time choice and several destinations. It is also shown that pricing according to marginal cost would be difficult to implement in this case, that eliminating all queues from the freeway is always suboptimal, and that ramps near the bottleneck should be metered more severely.