Search results for: semilinear parabolic equation

Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: Yuan Yan Tang, Lina Yang, Hong Li

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Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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Authors: Jurriaan Gillissen

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This work is concerned with stabilizing the numerical integration of the Navier-Stokes equation (NSE), backwards in time. Applications involve the detection of sources of, e.g., sound, heat, and pollutants. Stable reverse numerical integration of parabolic differential equations is also relevant for image de-blurring. While the literature addresses the reverse integration problem of the advection-diffusion equation, the problem of numerical reverse integration of the NSE has, to our knowledge, not yet been addressed. Owing to the presence of viscosity, the NSE is irreversible, i.e., when going backwards in time, the fluid behaves, as if it had a negative viscosity. As an effect, perturbations from the perfect solution, due to round off errors or discretization errors, grow exponentially in time, and reverse integration of the NSE is inherently unstable, regardless of using an implicit time integration scheme. Consequently, some sort of filtering is required, in order to achieve a stable, numerical, reversed integration. The challenge is to find a filter with a minimal adverse affect on the accuracy of the reversed integration. In the present work, we explore an adjoint gradient method (AGM) to achieve this goal, and we apply this technique to two-dimensional (2D), decaying turbulence. The AGM solves for the initial velocity field u0 at t = 0, that, when integrated forward in time, produces a final velocity field u1 at t = 1, that is as close as is feasibly possible to some specified target field v1. The initial field u0 defines a minimum of a cost-functional J, that measures the distance between u1 and v1. In the minimization procedure, the u0 is updated iteratively along the gradient of J w.r.t. u0, where the gradient is obtained by transporting J backwards in time from t = 1 to t = 0, using the adjoint NSE. The AGM thus effectively replaces the backward integration by multiple forward and backward adjoint integrations. Since the viscosity is negative in the adjoint NSE, each step of the AGM is numerically stable. Nevertheless, when applied to turbulence, the AGM develops instabilities, which limit the backward integration to small times. This is due to the exponential divergence of phase space trajectories in turbulent flow, which produces a multitude of local minima in J, when the integration time is large. As an effect, the AGM may select unphysical, noisy initial conditions. In order to improve this situation, we propose two remedies. First, we replace the integration by a sequence of smaller integrations, i.e., we divide the integration time into segments, where in each segment the target field v1 is taken as the initial field u0 from the previous segment. Second, we add an additional term (regularizer) to J, which is proportional to a high-order Laplacian of u0, and which dampens the gradients of u0. We show that suitable values for the segment size and for the regularizer, allow a stable reverse integration of 2D decaying turbulence, with accurate results for more then O(10) turbulent, integral time scales.

Keywords: time reversed integration, parabolic differential equations, adjoint gradient method, two dimensional turbulence

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Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa

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An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

Keywords: Cahn-Hilliard equation, miscibility gap, phase separation, dimensional domains

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Authors: Nelson Ponce Jr., Jonas R. Gazoli, Alessandro Sete, Roberto M. G. Velásquez, Valério L. Borges, Moacir A. S. de Andrade

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The search for increased efficiency in generation systems has been of great importance in recent years to reduce the impact of greenhouse gas emissions and global warming. For clean energy sources, such as the generation systems that use concentrated solar power technology, this efficiency improvement impacts a lower investment per kW, improving the project’s viability. For the specific case of parabolic trough solar concentrators, their performance is strongly linked to their geometric precision of assembly and the individual efficiencies of their main components, such as parabolic mirrors and receiver tubes. Thus, for accurate efficiency analysis, it should be conducted empirically, looking for mounting and operating conditions like those observed in the field. The Brazilian power generation and distribution company Eletrobras Furnas, through the R&D program of the National Agency of Electrical Energy, has developed a plant for testing parabolic trough concentrators located in Aparecida de Goiânia, in the state of Goiás, Brazil. The main objective of this test plant is the characterization of the prototype concentrator that is being developed by the company itself in partnership with Eudora Energia, seeking to optimize it to obtain the same or better efficiency than the concentrators of this type already known commercially. This test plant is a closed pipe system where a pump circulates a heat transfer fluid, also calledHTF, in the concentrator that is being characterized. A flow meter and two temperature transmitters, installed at the inlet and outlet of the concentrator, record the parameters necessary to know the power absorbed by the system and then calculate its efficiency based on the direct solar irradiation available during the test period. After the HTF gains heat in the concentrator, it flows through heat exchangers that allow the acquired energy to be dissipated into the ambient. The goal is to keep the concentrator inlet temperature constant throughout the desired test period. The developed plant performs the tests in an autonomous way, where the operator must enter the HTF flow rate in the control system, the desired concentrator inlet temperature, and the test time. This paper presents the methodology employed for design and operation, as well as the instrumentation needed for the development of a parabolic trough test plant, being a guideline for standardization facilities.

Keywords: parabolic trough, concentrated solar power, CSP, solar power, test plant, energy efficiency, performance characterization, renewable energy

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Authors: Hesamoddin Abdollahpour, Roghayeh Abdollahpour, Elham Rahgozar

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This paper we deal with an analysis of the free vibrations of the governing partial differential equation that it is Danel equation in the shells. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of the hinged shell. A new implementation of the new method is presented to obtain natural frequency and corresponding displacement on the shell. Our purpose is to enhance the ability to solve the mentioned complicated partial differential equation (PDE) with a simple and innovative approach. The results reveal that this new method to solve Danel equation is very effective and simple, and can be applied to other nonlinear partial differential equations. It is necessary to mention that there are some valuable advantages in this way of solving nonlinear differential equations and also most of the sets of partial differential equations can be answered in this manner which in the other methods they have not had acceptable solutions up to now. We can solve equation(s), and consequently, there is no need to utilize similarity solutions which make the solution procedure a time-consuming task.

Keywords: large amplitude, free vibrations, analytical solution, Danell Equation, diagram of phase plane

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Authors: S. Mousavian, F. Mousavian, V. Nikkhah Rashidabad

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Cubic equations of state like Redlich–Kwong (RK) EOS have been proved to be very reliable tools in the prediction of phase behavior. Despite their good performance in compositional calculations, they usually suffer from weaknesses in the predictions of saturated liquid density. In this research, RK equation was modified. The result of this study shows that modified equation has good agreement with experimental data.

Keywords: equation of state, modification, ammonia, genetic algorithm

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Authors: Liqiang Duan, Ma Jingkai, Lv Zhipeng, Haifan Cai

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The integrated solar combined cycle (ISCC) system has a series of advantages such as increasing the system power generation, reducing the cost of solar power generation, less pollutant and CO₂ emission. In this paper, the parabolic trough collectors with direct steam generation (DSG) technology are considered to replace the heat load of heating surfaces in heat regenerator steam generation (HRSG) of a conventional natural gas combined cycle (NGCC) system containing a PG9351FA gas turbine and a triple pressure HRSG with reheat. The detailed model of the NGCC system is built in ASPEN PLUS software and the parabolic trough collectors with DSG technology is modeled in EBSILON software. ISCC-DSG systems with the replacement of single, two, three and four heating surfaces are studied in this paper. Results show that: (1) the ISCC-DSG systems with the replacement heat load of HPB, HPB+LPE, HPE2+HPB+HPS, HPE1+HPE2+ HPB+HPS are the best integration schemes when single, two, three and four stages of heating surfaces are partly replaced by the parabolic trough solar energy collectors with DSG technology. (2) Both the changes of feed water flow and the heat load of the heating surfaces in ISCC-DSG systems with the replacement of multi-stage heating surfaces are smaller than those in ISCC-DSG systems with the replacement of single heating surface. (3) ISCC-DSG systems with the replacement of HPB+LPE heating surfaces can increase the solar power output significantly. (4) The ISCC-DSG systems with the replacement of HPB heating surfaces has the highest solar-thermal-to-electricity efficiency (47.45%) and the solar radiation energy-to-electricity efficiency (30.37%), as well as the highest exergy efficiency of solar field (33.61%).

Keywords: HRSG, integration scheme, parabolic trough collectors with DSG technology, solar power generation

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Authors: Muna Alghabshi, Edmana Krishnan

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A nonlinear Schrodinger equation has been considered for solving by mapping methods in terms of Jacobi elliptic functions (JEFs). The equation under consideration has a linear evolution term, linear and nonlinear dispersion terms, the Kerr law nonlinearity term and three terms representing the contribution of meta materials. This equation which has applications in optical fibers is found to have soliton solutions, shock wave solutions, and singular wave solutions when the modulus of the JEFs approach 1 which is the infinite period limit. The equation with special values of the parameters has also been solved using the tanh method.

Keywords: Jacobi elliptic function, mapping methods, nonlinear Schrodinger Equation, tanh method

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Authors: Benedict Barnes, Anthony Y. Aidoo

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A Divergence Regularization Method (DRM) is used to regularize the ill-posed Helmholtz equation where the boundary deflection is inhomogeneous in a Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes the inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation. This ensures the existence, as well as, uniqueness of solution for the equation. The DRM restores all the three conditions of well-posedness in the sense of Hadamard.

Keywords: divergence regularization method, Helmholtz equation, ill-posed inhomogeneous Cauchy boundary conditions

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Authors: Adel A. Ghoneim

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In most existing buildings in hot climate, cooling loads lead to high primary energy consumption and consequently high CO2 emissions. These can be substantially decreased with integrated renewable energy systems. Kuwait is characterized by its dry hot long summer and short warm winter. Kuwait receives annual total radiation more than 5280 MJ/m2 with approximately 3347 h of sunshine. Solar energy systems consist of PV modules and parabolic trough collectors are considered to satisfy electricity consumption, domestic water heating, and cooling loads of an existing building. This paper presents the results of an extensive program of energy conservation and energy generation using integrated photovoltaic (PV) modules and parabolic trough collectors (PTC). The program conducted on an existing institutional building intending to convert it into a Net-Zero Energy Building (NZEB) or near net Zero Energy Building (nNZEB). The program consists of two phases; the first phase is concerned with energy auditing and energy conservation measures at minimum cost and the second phase considers the installation of photovoltaic modules and parabolic trough collectors. The 2-storey building under consideration is the Applied Sciences Department at the College of Technological Studies, Kuwait. Single effect lithium bromide water absorption chillers are implemented to provide air conditioning load to the building. A numerical model is developed to evaluate the performance of parabolic trough collectors in Kuwait climate. Transient simulation program (TRNSYS) is adapted to simulate the performance of different solar system components. In addition, a numerical model is developed to assess the environmental impacts of building integrated renewable energy systems. Results indicate that efficient energy conservation can play an important role in converting the existing buildings into NZEBs as it saves a significant portion of annual energy consumption of the building. The first phase results in an energy conservation of about 28% of the building consumption. In the second phase, the integrated PV completely covers the lighting and equipment loads of the building. On the other hand, parabolic trough collectors of optimum area of 765 m2 can satisfy a significant portion of the cooling load, i.e about73% of the total building cooling load. The annual avoided CO2 emission is evaluated at the optimum conditions to assess the environmental impacts of renewable energy systems. The total annual avoided CO2 emission is about 680 metric ton/year which confirms the environmental impacts of these systems in Kuwait.

Keywords: building integrated renewable systems, Net-Zero energy building, solar fraction, avoided CO2 emission

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: Ayaz Ahmad

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In this work, we present the effect of noise on the solution of a partial differential equation (PDE) in three different setting. We shall first consider random initial condition for two nonlinear dispersive PDE the non linear Schrodinger equation and the Kortteweg –de vries equation and analyse their effect on some special solution , the soliton solutions.The second case considered a linear partial differential equation , the wave equation with random initial conditions allow to substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, when we shall show that the addition of a multiplicative noise term forbids the blow up of solutions under a very weak hypothesis for which we have finite time blow up of a solution in the deterministic case. Here we consider the problem of wave propagation, which is modelled by a nonlinear dispersive equation with noisy initial condition .As observed noise can also be introduced directly in the equations.

Keywords: drift term, finite time blow up, inverse problem, soliton solution

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Authors: Sufyan Muhammad

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Recently, in achieving highly efficient but at the same time highly accurate solutions has become the major target of numerical analyst community. The concept is termed as compact schemes and has gained great popularity and consequently, we construct compact schemes for fourth order parabolic differential equations used to study vibrations in structures. For the superiority of newly constructed schemes, we consider range of examples. We have achieved followings i.e. (a) numerical scheme utilizes minimum number of stencil points (which means new scheme is compact); (b) numerical scheme is highly accurate (which means new scheme is reliable) and (c) numerical scheme is highly efficient (which means new scheme is fast).

Keywords: central finite differences, compact schemes, Bernoulli's equations, finite differences

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Authors: Praveen Kumar, R. Uma, R. P. Sharma

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This study investigates the generation of turbulence in a deep-fluid environment using a damped 1D-modified nonlinear Schrödinger equation model. The well-known damped modified nonlinear Schrödinger equation (d-MNLS) is solved using numerical methods. Artificial damping is added to the MNLS equation, and turbulence generation is investigated through a numerical simulation. The numerical simulation employs a finite difference method for temporal evolution and a pseudo-spectral approach to characterize spatial patterns. The results reveal a recurring periodic pattern in both space and time when the nonlinear Schrödinger equation is considered. Additionally, the study shows that the modified nonlinear Schrödinger equation disrupts the localization of structure and the recurrence of the Fermi-Pasta-Ulam (FPU) phenomenon. The energy spectrum exhibits a power-law behavior, closely following Kolmogorov's spectra steeper than k⁻⁵/³ in the inertial sub-range.

Keywords: water waves, modulation instability, hydrodynamics, nonlinear Schrödinger's equation

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Authors: Meruyert Zhassybayeva, Kuralay Yesmukhanova, Ratbay Myrzakulov

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Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2+1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form from the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 +1)-dimensional Fokas-Lenells equation were found.

Keywords: Fokas-Lenells equation, integrability, soliton, the Hirota bilinear method

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Authors: Ognjen Vukovic

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Chern-Simons equation represents the cornerstone of quantum physics. The question that is often asked is if the aforementioned equation can be successfully applied to the interaction in international financial markets. By analysing the time series in financial theory, it is proved that Chern-Simons equation can be successfully applied to financial time-series. The aforementioned statement is based on one important premise and that is that the financial time series follow the fractional Brownian motion. All variants of Chern-Simons equation and theory are applied and analysed. Financial theory time series movement is, firstly, topologically analysed. The main idea is that exchange rate represents two-dimensional projections of three-dimensional Brownian motion movement. Main principles of knot theory and topology are applied to financial time series and setting is created so the Chern-Simons equation can be applied. As Chern-Simons equation is based on small particles, it is multiplied by the magnifying factor to mimic the real world movement. Afterwards, the following equation is optimised using Solver. The equation is applied to n financial time series in order to see if it can capture the interaction between financial time series and consequently explain it. The aforementioned equation represents a novel approach to financial time series analysis and hopefully it will direct further research.

Keywords: Brownian motion, Chern-Simons theory, financial time series, econophysics

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Authors: Paschal A. Ochang, Emmanuel C. Oji

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The Duffing’s Equation is a second order system that is very important because they are fundamental to the behaviour of higher order systems and they have applications in almost all fields of science and engineering. In the biological area, it is useful in plant stem dependence and natural frequency and model of the Brain Crash Analysis (BCA). In Engineering, it is useful in the study of Damping indoor construction and Traffic lights and to the meteorologist it is used in the prediction of weather conditions. However, most Problems in real life that occur are non-linear in nature and may not have analytical solutions except approximations or simulations, so trying to find an exact explicit solution may in general be complicated and sometimes impossible. Therefore we aim to find out if it is possible to obtain one analytical fixed point to the non-linear ordinary equation using fixed point analytical method. We started by exposing the scope of the Duffing’s equation and other related works on it. With a major focus on the fixed point and fixed point iterative scheme, we tried different iterative schemes on the Duffing’s Equation. We were able to identify that one can only see the fixed points to a Damped Duffing’s Equation and not to the Undamped Duffing’s Equation. This is because the cubic nonlinearity term is the determining factor to the Duffing’s Equation. We finally came to the results where we identified the stability of an equation that is damped, forced and second order in nature. Generally, in this research, we approximate the solution of Duffing’s Equation by converting it to a system of First and Second Order Ordinary Differential Equation and using Fixed Point Iterative approach. This approach shows that for different versions of Duffing’s Equations (damped), we find fixed points, therefore the order of computations and running time of applied software in all fields using the Duffing’s equation will be reduced.

Keywords: damping, Duffing's equation, fixed point analysis, second order differential, stability analysis

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Authors: Arunkumar Thirugnanasambantham, Velraj Ramalingam

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An experimental attempt has been made to study the effect of system integration by two different concentrator assisted desalting systems. The compound parabolic concentrator (CPC) and compound conical concentrator (CCC) are used in this research work. Two solar desalination systems, the single slope solar still (SSSS) and pyramid solar still (PSS), have been integrated with a CCC and compound parabolic concentrator-concentric circular tubular solar still (CPC-CCTSS). To study the effect of system integration, a thick cloth prevents the entry of sunlight into the solar still top. Additionally, the concentrator assisted desalting systems are equipped with phase change material (PCM) for enhancement. In CCC-SSSS, PCM has been filled inside copper balls and placed on the SSSS basin. The PCM is loaded in the specially designed circular trough of the tubular solar still. Here, the used concentrators and distillers are not the same. Two methodologies are followed here to produce the fresh water even while the distillers are blocked from the sunlight. They are (1) thermosyphon effect in CCC-SSSS and (2) waste heat recovery from CPC-CCTSS. The results showed that the productivity of CCC-SSSS, CCC-SSSS with PCM and CCC-SSSS (PCM) top cover shaded were found as 2680 ml / m² / day, 3240 ml / m² / day, and 1646 ml / m² / day. Similarly, the productivity of the CPC-CCTSS-PSS, CPC-CCTSS (PCM)-PSS and CPC-CCTSS (PCM)-PSS top cover shaded were found as 7160 ml / m² / day, 7346 ml / m² / day, and ml / m² / day. The productivity of the CCC-SSSS and CPC-CCTSS-PSS is examined, and conclusions are drawn such as the solar radiation blocked distillers productivity did not drop to zero.

Keywords: compound conical concentrator, compound parabolic concentrator, desalination, system integration

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Authors: Marziehossadat Moezzi

Abstract:

In this paper, electromagnetically induced transparency (EIT) is investigated in a cylindrical quantum dot (QD) with a parabolic confinement potential. We study the effect of control lasers polarization on absorption coefficient, refractive index and also on the generation of the double transparency windows in this system. Considering an effective mass method, the time-independent Schrödinger equation is solved to obtain the energy structure of the QD. Also, we study the effect of structural characteristics of the QD on refraction and absorption of the QD in the presence of EIT.

Keywords: electromagnetically induced transparency, cylindrical quantum dot, absorption coefficient, refractive index

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Authors: D. Pradeep Mitra, Prafulla

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Looking to the word propellant-less system it makes us to believe that it is an impossible one, but this paper demonstrates the use of microwaves to create a system which makes impossible to be possible, it means a propellant-less propulsion system using microwaves. In these thrusters, microwaves are radiated into a sealed parabolic cavity through a waveguide, which act on the surface of the cavity and follow the axis of the thrusters to produce thrust. The advantages of these thrusters are: (1) Producing thrust without propellant; without erosion, wear, and thermal stress from the hot exhaust gas; and at the same time increasing quality. (2) If the microwave output power is stable, the performance of thrusters is not affected by its working environment. This paper is demonstrated from general maxwell equations. These equations are used to create the mathematical model of the thrusters. These mathematical model helps us to calculate the Q factor and calculate the approximate thrust which would be generated in the system.

Keywords: propellant less, microwaves, parabolic wave guide, propulsion system

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Authors: Ayda Nikkar, Roghayye Ahmadiasl

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In this letter, a new analytical method called homotopy perturbation method, which does not need small parameter in the equation is implemented for solving the nonlinear Whitham–Broer–Kaup (WBK) partial differential equation. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of exact solution has led us to significant consequences. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the HPM can be found widely applicable in engineering.

Keywords: homotopy perturbation method, Whitham–Broer–Kaup (WBK) equation, Modified Boussinesq, Approximate Long Wave

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Authors: Mohamed Gar Alalm, Ahmed Tawfik

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In this study, photo-catalytic degradation of phenol by titanium dioxide (TiO2) in aqueous solution was evaluated. The UV energy of solar light was utilized by compound parabolic collectors (CPCs) technology. The effect of irradiation time, initial pH, and dosage of TiO2 were investigated. Aromatic intermediates (catechol, benzoquinone, and hydroquinone) were quantified during the reaction to study the pathways of the oxidation process. 94.5% degradation efficiency of phenol was achieved after 150 minutes of irradiation when the initial concentration was 100 mg/L. The dosage of TiO2 significantly affected the degradation efficiency of phenol. The observed optimum pH for the reaction was 5.2. Phenol photo-catalytic degradation fitted to the pseudo-first order kinetic according to Langmuir–Hinshelwood model.

Keywords: compound parabolic collectors, phenol, photo-catalytic, titanium dioxide

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Authors: Sachin Kumar

Abstract:

Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy.

Keywords: fractional PDE, fuzzy valued function, diffusion equation, Legendre polynomial, spectral method

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Authors: Chor Nejmeddine, Ines Ben Omrane, Mohamed Abdelwahed

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In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler's implicit scheme in time and spectral method in space. We propose two families of error indicators, both of which are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Keywords: heat equation, spectral elements discretization, error indicators, Euler

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Authors: Emad K. Jaradat, Ala’a Al-Faqih

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Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

Keywords: non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two-dimensional Schrodinger equation

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Authors: A. Suparmi, C. Cari, M. Yunianto, B. N. Pratiwi

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D-dimensional Dirac equations of q-deformed shape invariant potentials were solved using supersymmetric quantum mechanics (SUSY QM) in the case of exact spin symmetry. The D dimensional radial Dirac equation for shape invariant potential reduces to one-dimensional Schrodinger type equation by an appropriate variable and parameter change. The relativistic energy spectra were analyzed by using SUSY QM and shape invariant properties from radial D dimensional Dirac equation that have reduced to one dimensional Schrodinger type equation. The SUSY operator was used to generate the D dimensional relativistic radial wave functions, the relativistic energy equation reduced to the non-relativistic energy in the non-relativistic limit.

Keywords: D-dimensional dirac equation, non-central potential, SUSY QM, radial wave function

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Authors: Meng-Yu Lin, Wan-Ju Lee

Abstract:

Pore-water pressure, which mediates effective stress and shear strength at grain contacts, has a great influence on the motion of debris flow. The factors that control the diffusion of excess pore-water pressure play very important roles in the debris-flow motion. This research investigates these effects by solving the distribution of pore-water pressure numerically in an unsteady, surging motion of debris flow. The governing equations are the depth-averaged equations for the motion of debris-flow surges coupled with the one-dimensional diffusion equation for excess pore-water pressures. The pore-pressure diffusion equation is solved using a Fourier series, which may improve the accuracy of the solution. The motion of debris-flow surge is modelled using a Lagrangian particle method. From the computational results, the effects of pore-pressure diffusivities and the initial excess pore pressure on the formations of debris-flow surges are investigated. Computational results show that the presence of pore water can increase surge velocities and then changes the profiles of depth distribution. Due to the linear distribution of the vertical component of pore-water velocity, pore pressure dissipates rapidly near the bottom and forms a parabolic distribution in the vertical direction. Increases in the diffusivity of pore-water pressure cause the pore pressures decay more rapidly and then decrease the mobility of the surge.

Keywords: debris flow, diffusion, Lagrangian particle method, pore-pressure diffusivity, pore-water pressure

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Authors: Weiping Liu

Abstract:

This paper presents an equation to calculate stock prices of different growth model. This equation is mathematically derived by using discounted cash flow method. It has the advantages of being very easy to use and very accurate. It can still be used even when the first stage is lengthy. This equation is more generalized because it can be used for all the three popular stock price models. It can be programmed into financial calculator or electronic spreadsheets. In addition, it can be extended to a multistage model. It is more versatile and efficient than the traditional methods.

Keywords: stock price, multistage model, different growth model, discounted cash flow method

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