Search results for: closed form

Authors: Lina Wu, Ye Li

Abstract:

An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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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: H. Nouri, I. E. Achouri, A. Grimes, H. Ait Said, M. Aissou, Y. Zebboudj

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This paper aims to analysis the behaviour of DC corona discharge in wire-to-plate electrostatic precipitators (ESP). Current-voltage curves are particularly analysed. Experimental results show that discharge current is strongly affected by the applied voltage. The proposed method of current identification is to use the method of least squares. Least squares problems that of into two categories: linear or ordinary least squares and non-linear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. A closed-form solution (or closed form expression) is any formula that can be evaluated in a finite number of standard operations. The non-linear problem has no closed-form solution and is usually solved by iterative.

Keywords: electrostatic precipitator, current-voltage characteristics, least squares method, electric field, magnetic field

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Authors: Damir Latypov

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A closed-form solution for 4-D double surface integrals arising in boundary integrals equations of a potential theory is obtained for arbitrary coplanar polygonal surfaces. The solution method is based on the construction of exact differential forms followed by the application of Stokes' theorem for each surface integral. As a result, the 4-D double surface integral is reduced to a 2-D double line integral. By an appropriate change of variables, the integrand is transformed into a separable function of integration variables. The closed-form solutions to the corresponding 1-D integrals are readily available in the integration tables. Previously closed-form solutions were known only for the case of coincident triangle surfaces and coplanar rectangles. Solutions for these cases were obtained by surface-specific ad-hoc methods, while the present method is general. The method also works for non-polygonal surfaces. As an example, we compute in closed form the 4-D integral for the case of coincident surfaces in the shape of a circular disk. For an arbitrarily shaped surface, the proposed method provides an efficient quadrature rule. Extensions of the method for non-coplanar surfaces and other than 1/R integral kernels are also discussed.

Keywords: boundary integral equations, differential forms, integration, stokes' theorem

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Authors: Asha Gupta, Ramandeep Kaur

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The purpose of this paper is to study a class of closed sets, called generalized pre-closed sets with respect to an ideal (briefly Igp-closed sets), which is an extension of generalized pre-closed sets in general topology. Then, by using these sets, the concepts of Igp- compact spaces along with some classes of maps like continuous and closed maps via ideals have been introduced and analogues of some known results for compact spaces, continuous maps and closed maps in general topology have been obtained.

Keywords: ideal, gp-closed sets, gp-closed maps, gp-continuous maps

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Authors: Farida Buniatova

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Testamentary succession rules in the Russian Federation have been developing intensively since the collapse of the Soviet Union. The article analyses specific aspects of the closed will in Russian civil law. It discusses advantages and drawbacks of the closed will. In addition to that, the paper focuses on the will drafting and attestation procedures. The research provides ways to improve and enhance Russian legislation governing the closed will.

Keywords: closed will, testamentary succession, testator, will

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Authors: Francesco Marotti de Sciarra, Raffaele Barretta

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Starting from nonlocal continuum mechanics, a thermodynamically new nonlocal model of Euler-Bernoulli nanobeams is provided. The nonlocal variational formulation is consistently provided and the governing differential equation for transverse displacement are presented. Higher-order boundary conditions are then consistently derived. An example is contributed in order to show the effectiveness of the proposed model.

Keywords: Bernoulli-Euler beams, nanobeams, nonlocal elasticity, closed-form solutions

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Authors: Mohammed Ali Hjaji, Magdi Mohareb

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This paper develops the exact solutions for coupled flexural-lateral-torsional static response of thin-walled asymmetric open members subjected to general loading. Using the principle of stationary total potential energy, the governing differential equations of equilibrium are formulated as well as the associated boundary conditions. The formulation is based on a generalized Timoshenko-Vlasov beam theory and accounts for the effects of shear deformation due to bending and warping, and captures the effects of flexural–torsional coupling due to cross-section asymmetry. Closed-form solutions are developed for cantilever and simply supported beams under various forces. In order to demonstrate the validity and the accuracy of this solution, numerical examples are presented and compared with well-established ABAQUS finite element solutions and other numerical results available in the literature. In addition, the results are compared against non-shear deformable beam theories in order to demonstrate the shear deformation effects.

Keywords: asymmetric cross-section, flexural-lateral-torsional response, Vlasov-Timoshenko beam theory, closed form solution

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Authors: Arunee Wongkhao, Suparat Niwitpong, Sa-aat Niwitpong

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In this paper, we propose two new confidence intervals for the difference between the inverse of normal means with known coefficients of variation. One of these two confidence intervals for this problem is constructed based on the generalized confidence interval and the other confidence interval is constructed based on the closed form method of variance estimation. We examine the performance of these confidence intervals in terms of coverage probabilities and expected lengths via Monte Carlo simulation.

Keywords: coverage probability, expected length, inverse of normal mean, coefficient of variation, generalized confidence interval, closed form method of variance estimation

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Authors: Bahman Gorjian

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This study investigated the effects of closed-captioning on vocabulary learning and listening comprehension of English-language movies. Captioning is thus an effective language-learning tool for persons learning English as a second language. Because students may learn a foreign language "passively," utilizing subtitles on television could make learning English enjoyable for them. Closed captioning is an electrical technique that converts spoken words from a television program's audio into written text that mimics subtitles in another language. The findings of this study showed the importance of using closed-captioning software when learning a foreign language. As a result, these must be considered when teaching EFL/ESL. The influence of watching movies with closed captions on vocabulary and hearing is compared in this study. This goal can be reached by employing a closed-captioned movie as a teaching tool in the classroom. This research was critical because it demonstrates the advantages of closed-captioning programs in EFL classrooms for both teachers and students. The study's findings assisted teachers in better understanding how to employ closed captioning as a teaching tool in the classroom. The effects will be seen as even more significant for language learners who use the method.

Keywords: closed-captions, listening, comprehension, vcabulary

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Authors: Arpit Bhardwaj, Koushik Roy

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The work includes derivation of the exact-closed form solution for simply supported quasicrystal and crystal plates by using propagator matrix method under surface loading and free vibration. As a numerical example a quasicrystal and a crystal plate are considered, and after investigation, the variation of displacement and stress fields along the thickness of these two plates are presented. Further, it includes analyzing the displacement and stress fields for two plates having two different stacking arrangement, i.e., QuasiCrystal/Crystal/QuasiCrystal and Crystal/QuasiCrystal/Crystal and comparing their results. This will not only tell us the change in the behavior of displacement and stress fields in two different materials but also how these get changed after trying their different combinations. For the free vibration case, Crystal and Quasicrystal plates along with their different stacking arrangements are considered, and displacements are plotted in all directions for different Mode Shapes.

Keywords: free vibration, multilayered plates, surface loading, quasicrystals

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Authors: M. A. Tayfun, M. A. Alkhalidi

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The exact theoretical expression describing the probability distribution of nonlinear sea-surface elevations derived from the second-order narrowband model has a cumbersome form that requires numerical computations, not well-disposed to theoretical or practical applications. Here, the same narrowband model is re-examined to develop a simpler closed-form approximation suitable for theoretical and practical applications. The salient features of the approximate form are explored, and its relative validity is verified with comparisons to other readily available approximations, and oceanic data.

Keywords: ocean waves, probability distributions, second-order nonlinearities, skewness coefficient, wave steepness

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Authors: Septimia Sarbu

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The entropy rate of a stochastic process is a fundamental concept in information theory. It provides a limit to the amount of information that can be transmitted reliably over a communication channel, as stated by Shannon's coding theorems. Recently, researchers have focused on developing new measures of information that generalize Shannon's classical theory. The aim is to design more efficient information encoding and transmission schemes. This paper continues the study of generalized entropy rates, by deriving a closed-form solution to the Sharma-Mittal entropy rate for Gaussian processes. Using the squeeze theorem, we solve the limit in the definition of the entropy rate, for different values of alpha and beta, which are the parameters of the Sharma-Mittal entropy. In the end, we compare it with Shannon and Rényi's entropy rates for Gaussian processes.

Keywords: generalized entropies, Sharma-Mittal entropy rate, Gaussian processes, eigenvalues of the covariance matrix, squeeze theorem

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Authors: Alessandro Marano

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Aim: The author describes the aesthetic modification using orthognathic surgery and closed rhinoplasty. Methods: Series of case study. After orthognathic surgery we can observe a dramatical change of aesthetic especially in the mid-face and nose projection. The advancement of maxillary bone through Le Fort I osteotomy will change the nasal tip projection and lips roundness; combining orthognathic surgery with closed approach rhinoplasty will manage both function and aesthetic of all mid face district. Results: Combining Le Fort I osteotomy with closed approach rhinoplasty resulted in good objective results with high patient satisfaction. Le Fort I osteotomy will increase projection of mid face and the closed approach rhinoplasty will modify the nasal shape to be more harmonic with the new maxillary district. The scars are not visible because hidden inside the mouth and nose. Conclusions: The orthognathic surgery combined with closed approach rhinoplasty are very effective for changing the aesthetic of the mid face. The results illustrate the difference between the use of orthognathic surgery only and to use it in association of closed approach rhinoplasty. Using both will allow to obtain a long lasting and pleasing results.

Keywords: orthognathic, rhinoplasty, aesthetic, face

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Authors: Slobodan B. Tričković

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We present the results concerned with trigonometric series that include sine and cosine functions with a parameter appearing in the denominator. We derive two types of closed-form formulas for trigonometric series. At first, for some integer values, as we know that Riemann’s ζ function and Dirichlet η, λ equal zero at negative even integers, whereas Dirichlet’s β function equals zero at negative odd integers, after a certain number of members, the rest of the series vanishes. Thus, a trigonometric series becomes a polynomial with coefficients involving Riemann’s ζ function and Dirichlet η, λ, β functions. On the other hand, in some cases, one cannot immediately replace the parameter with any positive integer because we shall encounter singularities. So it is necessary to take a limit, so in the process, we apply L’Hospital’s rule and, after a series of rearrangements, we bring a trigonometric series to a form suitable for the application of Choi-Srivastava’s theorem dealing with Hurwitz’s zeta function and Harmonic numbers. In this way, we express a trigonometric series as a polynomial over Hurwitz’s zeta function derivative.

Keywords: Dirichlet eta lambda beta functions, Riemann's zeta function, Hurwitz zeta function, Harmonic numbers

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Authors: Mohammed Ali Hjaji, A.K. El-Senussi, Said H. Eshtewi

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The flexural dynamic response of symmetric laminated composite beams subjected to general transverse harmonic forces is investigated. The dynamic equations of motion and associated boundary conditions based on the first order shear deformation are derived through the use of Hamilton’s principle. The influences of shear deformation, rotary inertia, Poisson’s ratio and fibre orientation are incorporated in the present formulation. The resulting governing flexural equations for symmetric composite Timoshenko beams are exactly solved and the closed form solutions for steady state flexural response are then obtained for cantilever and simply supported boundary conditions. The applicability of the analytical closed-form solution is demonstrated via several examples with various transverse harmonic loads and symmetric cross-ply and angle-ply laminates. Results based on the present solution are assessed and validated against other well established finite element solutions and exact solutions available in the literature.

Keywords: analytical solution, flexural response, harmonic forces, symmetric laminated beams, steady state response

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Authors: Alessandro Marano

Abstract:

Aim: The author describes a strategy for choosing between two different rhinoplasty approaches for outcomes treatment. Methods: Series of the case study. There are advantages and disadvantages on both approaches for rhinoplasty. On the side of the open approach, we are be able to better manage the techniques for shaping and restoring nasal structures in rhinoplasty outcomes; on the other side, the closed approach requires more practice and experience to achieve good results. Results: Author’s choice is the closed approach on rhinoplasty outcomes. Anyway, the open approach is most commonly preferred due to superior management and better vision on nasal structures. Conclusions: Both approaches are valid for the treatment of rhinoplasty outcomes, author's preferred approach is closed, with minimally invasive modification focused on restoring outcomes in nasal function and aesthetics.

Keywords: rhinoplasty, aesthetic, face, outcomes

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Authors: Yuyang Cheng, Marcos Escobar-Anel

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This paper introduces a multivariate 4/2 stochastic covariance process generalizing the one-dimensional counterparts presented in Grasselli (2017). Our construction permits stochastic correlation not only among stocks but also among volatilities, also known as co-volatility movements, both driven by more convenient 4/2 stochastic structures. The parametrization is flexible enough to separate these types of correlation, permitting their individual study. Conditions for proper changes of measure and closed-form characteristic functions under risk-neutral and historical measures are provided, allowing for applications of the model to risk management and derivative pricing. We apply the model to an expected utility theory problem in incomplete markets. Our analysis leads to closed-form solutions for the optimal allocation and value function. Conditions are provided for well-defined solutions together with a verification theorem. Our numerical analysis highlights and separates the impact of key statistics on equity portfolio decisions, in particular, volatility, correlation, and co-volatility movements, with the latter being the least important in an incomplete market.

Keywords: stochastic covariance process, 4/2 stochastic volatility model, stochastic co-volatility movements, characteristic function, expected utility theory, verication theorem

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Authors: Rasha Kumari Panda

Abstract:

The crime against women in closed door is an important burning issue in day to day life. Domestic violence has become daily part of women’s life. It affects the millions of the women throughout the India as it violates their human rights. Crime against women behind closed door is a manifestation of historically unequal power relations between men and women, discrimination against women moreover, when the world is approaching towards modernization, worse the condition of women and girls in our society. This paper examines how the rights of women are being violated and suggests the remedial measures to empower women. Powerlessness of women is the root cause of violence has been specifically addressed.

Keywords: domestic violence, cruelty, dowry, statutes

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Authors: Arnaud Nougues

Abstract:

This paper describes a two-stage methodology derived from internal model control (IMC) for tuning a proportional-integral-derivative (PID) controller for levels or other integrating processes in an industrial environment. Focus is the ease of use and implementation speed which are critical for an industrial application. Tuning can be done with minimum effort and without the need for time-consuming open-loop step tests on the plant. The first stage of the method applies to levels only: the vessel residence time is calculated from equipment dimensions and used to derive a set of preliminary proportional-integral (PI) settings with IMC. The second stage, re-tuning in closed-loop, applies to levels as well as other integrating processes: a tuning correction mechanism has been developed based on a series of closed-loop simulations with model errors. The tuning correction is done from a simple closed-loop step test and the application of a generic correlation between observed overshoot and integral time correction. A spin-off of the method is that an estimate of the vessel residence time (levels) or open-loop process gain (other integrating process) is obtained from the closed-loop data.

Keywords: closed-loop model identification, IMC-PID tuning method, integrating process control, on-line PID tuning adaptation

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Authors: Anat Thanpinta

Abstract:

This research aimed to study on the efficiency of wastewater treatment by comparing the different aeration times of surface aerators in Suan Sunandha Rajabhat University. In doing so, the operation of surface aerators was divided into 2 groups which included the groups of 8 hours (8-0/opened-closed) and 4 hours (2-2/opened-closed) of aeration time per day. As a result of the study, it was found that the efficiency of wastewater treatment in the forms of DO, BOD, turbidity and NO2- by 8 hours (8-0/opened-closed) and 4 hours (2-2/opened-closed) of aeration time per day of surface aerators was not statistically different [Sig. = .644, .488, .716 and .054 > α (.05)] while the efficiency in the forms of NO3- and P was significantly different at the statistical level of .01 [Sig. = .001 and .000 < α (.01)].

Keywords: aeration time, surface aerator, wastewater treatment, efficiency

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Authors: Fares Senouci, Bachir Imine

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This document provides numerical and experimental optimization of the aerodynamic performance of a drone equipped with three types of horizontal stabilizer. To build this optimal configuration, an experimental and numerical study was conducted on three parameters: the geometry of the stabilizer (horizontal form or reverse V form), the position of the horizontal stabilizer (up or down), and the landing gear position (closed or open). The results show that up-stabilizer position with respect to the horizontal plane of the fuselage provides better aerodynamic performance, and that the landing gear increases the lift in the zone of stability, that is to say where the flow is not separated.

Keywords: aerodynamics, drag, lift, turbulence model, wind tunnel

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Authors: U. Schmidt, D. Dannehl, I. Schuch, J. Suhl, T. Rocksch, R. Salazar-Moreno, E. Fitz-Rodrigues, A. Rojano Aquilar, I. Lopez Cruz, G. Navas Gomez, R. A. Abraham, L. C. Irineo, N. G. Gilberto

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The integration of agricultural production systems into urban areas is a challenge for the coming decades. Because of increasing greenhouse gas emission and rising resource consumption as well as costs in animal husbandry, the dietary habits of people in the 21st century have to focus on herbal foods. Intensive plant cultivation systems in large cities and megacities require a smart coupling of information, material and energy flow with the urban infrastructure in terms of Horticulture 4.0. In recent years, many puzzle pieces have been developed for these closed processes at the Humboldt University. To compile these for an urban plant production, it has to be optimized and networked with urban infrastructure systems. In the field of heat energy production, it was shown that with closed greenhouse technology and patented heat exchange and storage technology energy can be provided for heating and domestic hot water supply in the city. Closed water circuits can be drastically reducing the water requirements of plant production in urban areas. Ion sensitive sensors and new disinfection methods can help keep circulating nutrient solutions in the system for a longer time in urban plant production greenhouses.

Keywords: semi closed, greenhouses, urban farming, solar heat collector, closed water cycles, aquaponics

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Authors: Imad Eddine Charif, Sidi Mohamed Mekelleche, Didier Villemin

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The solvent effects on the keto-enol tautomeric equilibriums of acetylacetone and dimedone are theoretically investigated at the correlated Becke-3-parameter-Lee-Yang-Parr (B3LYP) and second-order Møller-Plesset (MP2) computational levels. The present study shows that the most stable keto tautomer of acetylacetone corresponds to the trans-diketo, E,Z form; while the most stable enol tautomer corresponds to the closed cis-enol,Z,Z form. The keto tautomer of dimedone prefers the trans diketo, E, E form; while the most stable enol tautomer corresponds to trans-enol form. The calculated free Gibbs enthalpies indicate that, in polar solvents, the keto-enol equilibrium of acetylacetone is shifted toward the keto tautomer; whereas the keto-enol equilibrium of dimedone is shifted towards the enol tautomer. The experimental trends of the change of equilibrium constants with respect to the change of solvent polarity are well reproduced by both B3LYP and MP2 calculations.

Keywords: acetylacetone, dimedone, solvent effects, keto-enol equilibrium, theoretical calculations

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Authors: Niti Kammuang-lue, Kritsada On-ai, Phrut Sakulchangsatjatai, Pradit Terdtoon

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The objectives of this paper are to investigate effects of dimensionless numbers on thermal performance of the vertical closed-loop pulsating heat pipe (VCLPHP) and to establish a correlation to predict the thermal performance of the VCLPHP. The CLPHPs were made of long copper capillary tubes with inner diameters of 1.50, 1.78, and 2.16mm and bent into 26 turns. Then, both ends were connected together to form a loop. The evaporator, adiabatic, and condenser sections length were equal to 50 and 150 mm. R123, R141b, acetone, ethanol, and water were chosen as variable working fluids with constant filling ratio of 50% by total volume. Inlet temperature of heating medium and adiabatic section temperature was constantly controlled at 80 and 50oC, respectively. Thermal performance was represented in a term of Kutateladze number (Ku). It can be concluded that when Prandtl number of liquid working fluid (Prl), and Karman number (Ka) increases, thermal performance increases. On contrary, when Bond number (Bo), Jacob number (Ja), and Aspect ratio (Le/Di) increases, thermal performance decreases. Moreover, the correlation to predict more precise thermal performance has been successfully established by analyzing on all dimensionless numbers that have effect on the thermal performance of the VCLPHP.

Keywords: vertical closed-loop pulsating heat pipe, working fluid, thermal performance, dimensionless parameter

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Authors: Anusuya Ghosh, Vishnu Narayanan

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The approximation of convex set by semidefinite representable set plays an important role in semidefinite programming, especially in modern convex optimization. To optimize a linear function over a convex set is a hard problem. But optimizing the linear function over the semidefinite representable set which approximates the convex set is easy to solve as there exists numerous efficient algorithms to solve semidefinite programming problems. So, our approximation technique is significant in optimization. We develop a technique to approximate any closed convex set, say K by compactly semidefinite representable set. Further we prove that there exists a sequence of compactly semidefinite representable sets which give tighter approximation of the closed convex set, K gradually. We discuss about the convergence of the sequence of compactly semidefinite representable sets to closed convex set K. The recession cone of K and the recession cone of the compactly semidefinite representable set are equal. So, we say that the sequence of compactly semidefinite representable sets converge strongly to the closed convex set. Thus, this approximation technique is very useful development in semidefinite programming.

Keywords: semidefinite programming, semidefinite representable set, compactly semidefinite representable set, approximation

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Authors: Hyunsang Kim, Younghun Kim, Younghee Kim, Sangku Lee

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We investigated ecotoxicity and performed an experiment for removing ZnO nanoparticles in water. Short-term exposure of hatching test using fertilized eggs (O. latipes) showed deformity in 5 ppm of ZnO nanoparticles solution, and in 10ppm ZnO nanoparticles solution delayed hatching was observed. Herein, chemical precipitation method was suggested for removing ZnO nanoparticles in water. The precipitated ZnO nanoparticles showed the form of ZnS after addition of Na2S, and the form of Zn3(PO4)2 for Na2HPO4. The removal efficiency of ZnO nanoparticles in water was closed to 100% for two case. In ecotoxicity evaluation of as-precipitated ZnS and Zn3(PO4)2, they did not cause any acute toxicity for D. magna. It is noted that this precipitation treatment of ZnO is effective to reduce the potential cytotoxicity.

Keywords: ZnO nanopraticles, ZnS, Zn3(PO4)2, ecotoxicity evaluation, chemical precipitation

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Authors: Jochen Edmund Kerschenbauer, Roman Salinger, Daniel Strametz

Abstract:

Insufficient informal communication systems can lead to the first crisis (‘Crisis of Leadership’) for start-ups. Management Control Systems (MCS) are one way for high-technology start-ups to successfully overcome these problems. So far the literature has investigated the incubation of a start-up, but focused less on the post-incubation stage. This paper focuses on the use of MCS in post-incubation and, if failed start-ups agree, on how MCS are used. We conducted 14 semi-structured interviews for this purpose, to obtain our results. The overall conclusion is that the majority of the companies were closed down due to a combination of strategic, operative and financial reasons.

Keywords: closed down, high-technology, incubation, levers of control, management control systems, post-incubation, start-ups

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Authors: Ying Yi Wu

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In this paper, we present a proof of the fact that a finite morphism is proper. We show that a finite morphism is universally closed and of finite type, which are the conditions for properness. Our proof is based on the theory of schemes and involves the use of the projection formula and the base change theorem. We first show that a finite morphism is of finite type and then proceed to show that it is universally closed. We use the fact that a finite morphism is also an affine morphism, which allows us to use the theory of coherent sheaves and their modules. We then show that the map induced by a finite morphism is flat and that the module it induces is of finite type. We use these facts to show that a finite morphism is universally closed. Our proof is constructive, and we provide details for each step of the argument.

Keywords: finite, morphism, schemes, projection.

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Authors: Mei-Hsiu Chi, Jyh-Yang Wu, Sheng-Gwo Chen

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The diffusion-reaction equations are important Partial Differential Equations in mathematical biology, material science, physics, and so on. However, finding efficient numerical methods for diffusion-reaction systems on curved surfaces is still an important and difficult problem. The purpose of this paper is to present a convergent geometric method for solving the reaction-diffusion equations on closed surfaces by an O(r)-LTL configuration method. The O(r)-LTL configuration method combining the local tangential lifting technique and configuration equations is an effective method to estimate differential quantities on curved surfaces. Since estimating the Laplace-Beltrami operator is an important task for solving the reaction-diffusion equations on surfaces, we use the local tangential lifting method and a generalized finite difference method to approximate the Laplace-Beltrami operators and we solve this reaction-diffusion system on closed surfaces. Our method is not only conceptually simple, but also easy to implement.

Keywords: closed surfaces, high-order approachs, numerical solutions, reaction-diffusion systems

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