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Solution approximations for a mathematical model of relativistic electrons with beta derivative
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作者 Ibrahim Yalcinkaya Orkun Tasbozan +1 位作者 Ali Kurt Hijaz Ahmad 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期469-485,共17页
The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with... The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative.For this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are used.While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution.The advantage of this procedure is that discretization or normalization is not required.By applying the mETF,the exact solutions are obtained.Also,by applying the HAM semi-analytical results for the considered equation are acquired.In HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution series.Also,comparative graphical representations are given to show the effectiveness,reliability of the methods.The results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative.This study can bring a new perspective for studies on fractional differential equations.On the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and etc.which involves fractional differentiation.Briefly the results may cause a new insight who studies on relativistic electron modelling. 展开更多
关键词 Klein-Fock-Gordon(KFG)equation beta derivative semi-analytical solution exact solution
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Thermodynamic theory of flotation for a complex multiphase solid -liquid system and high-entropy flotation
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作者 Shuming Wen 《International Journal of Minerals,Metallurgy and Materials》 SCIE EI CAS CSCD 2024年第6期1177-1197,共21页
The flotation of complex solid–liquid multiphase systems involve interactions among multiple components,the core problem facing flotation theory.Meanwhile,the combined use of multicomponent flotation reagents to impr... The flotation of complex solid–liquid multiphase systems involve interactions among multiple components,the core problem facing flotation theory.Meanwhile,the combined use of multicomponent flotation reagents to improve mineral flotation has become an important issue in studies on the efficient use of refractory mineral resources.However,studying the flotation of complex solid–liquid systems is extremely difficult,and no systematic theory has been developed to date.In addition,the physical mechanism associated with combining reagents to improve the flotation effect has not been unified,which limits the development of flotation theory and the progress of flotation technology.In this study,we applied theoretical thermodynamics to a solid–liquid flotation system and used changes in the entropy and Gibbs free energy of the reagents adsorbed on the mineral surface to establish thermodynamic equilibrium equations that de-scribe interactions among various material components while also introducing adsorption equilibrium constants for the flotation reagents adsorbed on the mineral surface.The homogenization effect on the mineral surface in pulp solution was determined using the chemical potentials of the material components of the various mineral surfaces required to maintain balance.The flotation effect can be improved through synergy among multicomponent flotation reagents;its physical essence is the thermodynamic law that as the number of compon-ents of flotation reagents on the mineral surface increases,the surface adsorption entropy change increases,and the Gibbs free energy change of adsorption decreases.According to the results obtained using flotation thermodynamics theory,we established high-entropy flotation theory and a technical method in which increasing the types of flotation reagents adsorbed on the mineral surface,increasing the adsorption entropy change of the flotation reagents,decreasing the Gibbs free energy change,and improving the adsorption efficiency and stability of the flotation reagents improves refractory mineral flotation. 展开更多
关键词 multicomponent flotation reagent adsorption entropy change equations of thermodynamic equilibrium adsorption equilibri-um constant high-entropy flotation
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NEW SCALAR-TYPE EQUILIBRIUM EQUATIONS OF THE GENERAL SPACE FORCE SYSTEM
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作者 汤昕燕 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2000年第10期1169-1176,共8页
Using the principle of analytical geometry, several properties of the space straight lille are proved. Based on these properties, the equilibrium of general space force system is considered and its four new scalar-typ... Using the principle of analytical geometry, several properties of the space straight lille are proved. Based on these properties, the equilibrium of general space force system is considered and its four new scalar-type equilibrium equations are derived which are equivalent to the vector-type necessary and sufficient conditions far equilibrium. 展开更多
关键词 general space force system method of analytical geometry new scalar-type equilibrium equations
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Existence of Multiple Equilibrium Points in Global Attractor for the Strongly Damped Wave Equations
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作者 孟凤娟 汪永海 刘存才 《Journal of Donghua University(English Edition)》 EI CAS 2017年第3期448-452,共5页
The aim of this paper is to study the long-term behavior of strongly damped wave equations with a Lyapunov function. Using the theory established by estimating the Z2 index of some sets and the idea of invariant sets ... The aim of this paper is to study the long-term behavior of strongly damped wave equations with a Lyapunov function. Using the theory established by estimating the Z2 index of some sets and the idea of invariant sets of semi-flow,the properties of the global attractor for strongly damped wave equation are discussed. The existence of multiple equilibrium points in global attractor for strongly damped wave equations with critical growth of nonlinearity is obtained. And under some additional condition, the infinite dimension of the attractor is proven. 展开更多
关键词 infinite invariant nonlinearity estimating assumption proven proof regularity stationary neighborhood
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On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method
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作者 Rania Saadeh Ahmad Qazza +1 位作者 Aliaa Burqan Shrideh Al-Omari 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第9期3121-3139,共19页
This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,w... This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach. 展开更多
关键词 Caputo derivative fractional differential equations formable transform time-fractional klein-gordon equation decomposition method
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THE OPTIMAL DEDUCTIBLE AND COVERAGE IN INSURANCE CONTRACTS AND EQUILIBRIUM RISK SHARING POLICIES
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作者 蹇玲玲 《Acta Mathematica Scientia》 SCIE CSCD 2023年第3期1347-1364,共18页
In this paper, we consider the optimal risk sharing problem between two parties in the insurance business: the insurer and the insured. The risk is allocated between the insurer and the insured by setting a deductible... In this paper, we consider the optimal risk sharing problem between two parties in the insurance business: the insurer and the insured. The risk is allocated between the insurer and the insured by setting a deductible and coverage in the insurance contract. We obtain the optimal deductible and coverage by considering the expected product of the two parties' utilities of terminal wealth according to stochastic optimal control theory. An equilibrium policy is also derived for when there are both a deductible and coverage;this is done by modelling the problem as a stochastic game in a continuous-time framework. A numerical example is provided to illustrate the results of the paper. 展开更多
关键词 deductible and coverage equilibrium policy stochastic optimal control Hamilton-Jacobi-Bellman equation
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Analysis and Numerical Computations of the Multi-Dimensional,Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation
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作者 Yuming Chu Saima Rashid +3 位作者 Khadija Tul Kubra Mustafa Inc Zakia Hammouch M.S.Osman 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第9期3025-3060,共36页
The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is an... The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology. 展开更多
关键词 Caputo derivative Elzaki transform time-fractional Navier-Stokes equation decomposition method
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On the partial stability of nonlinear impulsive Caputo fractional systems
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作者 Boulbaba Ghanmi Saifeddine Ghnimi 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2023年第2期166-179,共14页
In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptot... In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results. 展开更多
关键词 impulsive fractional differential equations Mittag-Leffler function partial stability Caputo derivative
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The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique
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作者 Hassan Khan Poom Kumam +2 位作者 Asif Nawaz Qasim Khan Shahbaz Khan 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第4期259-273,共15页
In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(... In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems. 展开更多
关键词 Caputo derivative fractional partial differential equations fornberg-whitham residual power series method
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SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES
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作者 刘曼莉 高凌云 房少梅 《Acta Mathematica Scientia》 SCIE CSCD 2023年第2期821-838,共18页
In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing sm... In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing small functions.As an application of the value distribution result,we study the defect relation of a nonconstant solution to the partial differential equation.In particular,we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation. 展开更多
关键词 meromorphic function in several variables Nevanlinna theory partial differ-ential equation total derivative
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New Configurations of the Fuzzy Fractional Differential Boussinesq Model with Application in Ocean Engineering and Their Analysis in Statistical Theory
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作者 Yu-Ming Chu SaimaRashid +1 位作者 Shazia Karim Anam Sultan 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第11期1573-1611,共39页
The fractional-order Boussinesq equations(FBSQe)are investigated in this work to see if they can effectively improve the situation where the shallow water equation cannot directly handle the dispersion wave.The fuzzy ... The fractional-order Boussinesq equations(FBSQe)are investigated in this work to see if they can effectively improve the situation where the shallow water equation cannot directly handle the dispersion wave.The fuzzy forms of analytical FBSQe solutions are first derived using the Adomian decomposition method.It also occurs on the sea floor as opposed to at the functionality.A set of dynamical partial differential equations(PDEs)in this article exemplify an unconfined aquifer flow implication.Thismethodology can accurately simulate climatological intrinsic waves,so the ripples are spread across a large demographic zone.The Aboodh transform merged with the mechanism of Adomian decomposition is implemented to obtain the fuzzified FBSQe in R,R^(n) and(2nth)-order involving generalized Hukuhara differentiability.According to the system parameter,we classify the qualitative features of the Aboodh transform in the fuzzified Caputo and Atangana-Baleanu-Caputo fractional derivative formulations,which are addressed in detail.The illustrations depict a comparison analysis between the both fractional operators under gH-differentiability,as well as the appropriate attributes for the fractional-order and unpredictability factorsσ∈[0,1].A statistical experiment is conducted between the findings of both fractional derivatives to prevent changing the hypothesis after the results are known.Based on the suggested analyses,hydrodynamic technicians,as irrigation or aquifer quality experts,may be capable of obtaining an appropriate storage intensity amount,including an unpredictability threshold. 展开更多
关键词 Fuzzy set theory aboodh transform adomian decomposition method boussinesq equation fractional derivative operators analysis of variance test
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CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 被引量:9
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作者 杨银 陈艳萍 黄云清 《Acta Mathematica Scientia》 SCIE CSCD 2014年第3期673-690,共18页
We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorou... We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results. 展开更多
关键词 Spectral Jacobi-collocation method fractional order integro-differential equations Caputo derivative
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Calculation of the Phase Equilibrium of CO2–Hydrocarbon Binary Mixtures by PR-BM EOS and PR EOS 被引量:2
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作者 Chuanyan Li Yahui Gao +2 位作者 Shuqian Xia Qiaoyan Shang Peisheng Ma 《Transactions of Tianjin University》 EI CAS 2019年第5期540-548,共9页
The phase equilibrium data of CO2 hydrocarbon binary mixtures are important for the design and operation of CO 2 ood- ing, coal liquefaction, and supercritical extraction processes. Numerous pieces of binary phase equ... The phase equilibrium data of CO2 hydrocarbon binary mixtures are important for the design and operation of CO 2 ood- ing, coal liquefaction, and supercritical extraction processes. Numerous pieces of binary phase equilibrium data have been obtained. Thus, models for the accurate calculation of binary and multicomponent mixtures must be developed on the basis of existing data. In this work, 3578 vapor liquid phase equilibrium data points for 10 CO 2 hydrocarbon binary mixtures, including CO2 butane, CO 2 pentane, CO 2 isopentane, C O 2 hexane, CO 2 benzene, CO 2 heptane, CO 2 octane, C O 2 non- ane, CO 2 decane, and C O 2 undecane, were collected. The PR and PR-BM equations of state (EOS) in combination with relevant mixing rules were used to calculate the phase equilibrium data of the CO 2 hydrocarbon binary mixtures. The binary interaction parameter k ij in the PR EOS was temperature independent, whereas parameters in the PR-BM EOS were functions of temperature. Thus, the phase equilibrium data and other thermodynamic properties of the binary and multicomponent mixtures at di erent temperatures and pressures can be calculated by using the parameters obtained in this work. The PR-BM EOS performed better than the PR EOS, and the average absolute deviations over the temperature range of 255.98 408.15 K calculated by the PR EOS and PR-BM EOS were less than 5.74% and 3.36%, respectively. The results calculated by the two EOS were compared with those calculated by other models, such as PPR78, PR + LCVM + UNIFAC, KIE + PR EOS + HV, and PSRK. The phase equilibrium data of CO 2 butane decane, CO 2 hexane decane, and C O 2 octane decane ternary mixtures were calculated by the two EOS. The average overall deviations for the CO 2 mole fractions calculated by the two EOS were less than 7.66%. 展开更多
关键词 Phase equilibrium EQUATION of state MIXING RULE HYDROCARBON CO 2
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On global behavior of the system of rational difference equations x_(n+1)=p+y_(n-k)/x_n,y_(n+1)=q+x_(n-k)/y_n 被引量:1
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作者 Xia An1,Tai-xiang Sun2,Yan-fang Zhang11. Department of Basic Courses,Institute of Disaster Prevention Science and Technology,Beijing 065201 2. Department of Mathematics,Guangxi University,Nanning 530004,China. 《Journal of Pharmaceutical Analysis》 SCIE CAS 2009年第3期189-191,共3页
In this paper,we investigate the boundedness character,the global attractivity and the periodic nature of the system of rational difference equations:x=p+y/x,y=q+x/y,n=0,1,2…,where p>0,q>0,k∈{1,2,…} and the i... In this paper,we investigate the boundedness character,the global attractivity and the periodic nature of the system of rational difference equations:x=p+y/x,y=q+x/y,n=0,1,2…,where p>0,q>0,k∈{1,2,…} and the initial values xi,yi∈(0,∞),i=-k,-k+1,…,0. Some new results are obtained. 展开更多
关键词 difference equation global behavior equilibrium solution boundedness
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Derivation of baroclinic Ertel–Rossby invariant-based thermally-coupled vorticity equation in moist flow 被引量:1
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作者 杨帅 高守亭 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第11期700-705,共6页
For the potential vorticity (PV) invariant, there is a PV-based complete-form vorticity equation, which we use heuris- tically in the present paper to answer the following question: for the Ertel-Rossby invariant ... For the potential vorticity (PV) invariant, there is a PV-based complete-form vorticity equation, which we use heuris- tically in the present paper to answer the following question: for the Ertel-Rossby invariant (ERI), is there a corresponding vorticity tendency equation? Such an ERI-based thermally-coupled vorticity equation is derived and discussed in detail in this study. From the obtained new vorticity equation, the vertical vorticity change is constrained by the vertical velocity term, the term associated with the slope of the generalized momentum surface, the term related to the horizontal vorticity change, and the baroclinic or solenoid term. It explicitly includes both the dynamical and thermodynamic factors' influence on the vorticity change. For the ERI itself, besides the traditional PV term, the ERI also includes the moisture factor, which is excluded in dry ERI, and the term related to the gradients of pressure, kinetic energy, and potential energy that reflects the fast-manifold property. Therefore, it is more complete to describe the fast motions off the slow manifold for severe weather than the PV term. These advantages are naturally handed on and inherited by the ERI-based thermally-coupled vorticity equation. Then the ERI-based thermally-coupled vorticity equation is further transformed and compared with the traditional vorticity equation. The main difference between the two equations is the term which describes the contribution of the solenoid term to the vertical vorticity development. In a barotropic flow, the solenoid term disappears, then the ERI-based thermally-coupled vorticity equation can regress to the traditional vorticity equation. 展开更多
关键词 vorticity equation derivation Ertel-Rossby invariant
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Computation of the stability derivatives via CFD and the sensitivity equations 被引量:1
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作者 Guo-Dong Lei Yu-Xin Ren 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2011年第2期179-188,共10页
The method to calculate the aerodynamic stability derivates of aircrafts by using the sensitivity equations is ex- tended to flows with shock waves in this paper. Using the newly developed second-order cell-centered f... The method to calculate the aerodynamic stability derivates of aircrafts by using the sensitivity equations is ex- tended to flows with shock waves in this paper. Using the newly developed second-order cell-centered finite volume scheme on the unstructured-grid, the unsteady Euler equations and sensitivity equations are solved simultaneously in a non-inertial frame of reference, so that the aerodynamic stability derivatives can be calculated for aircrafts with complex geometries. Based on the numerical results, behavior of the aerodynamic sensitivity parameters near the shock wave is discussed. Furthermore, the stability derivatives are analyzed for supersonic and hypersonic flows. The numerical results of the stability derivatives are found in good agree- ment with theoretical results for supersonic flows, and variations of the aerodynamic force and moment predicted by the stability derivatives are very close to those obtained by CFD simulation for both supersonic and hypersonic flows. 展开更多
关键词 Aerodynamic stability derivatives Non-inertial frame of reference unstructured-grid Finite volume scheme Sensitivity equations
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Application of the Improved Kudryashov Method to Solve the Fractional Nonlinear Partial Differential Equations 被引量:2
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作者 Md. Abdus Salam Umme Habiba 《Journal of Applied Mathematics and Physics》 2019年第4期912-920,共9页
Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Bur... Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13. 展开更多
关键词 IMPROVED Kudryashov METHOD Time-Space FRACTIONAL KdV-Burger Equation TRAVELLING Wave Solutions Jumarie’s Modified Riemann-Liouville Derivative
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The Isobaric Vapor-Liquid Equilibrium of Butanone-DMF and Toluene-DMF at Reduced Pressures
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作者 刘畅 王丽华 +1 位作者 何潮洪 叶向群 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2006年第6期825-828,共4页
The isobaric vapor-liquid equilibrium data of butanone(1)-N, N-dimethylformamide (DMF)(2) at 100.92kPa, 93.32kPa, and 79.99kPa and of toluene(1)-DMF(2) at 100.92kPa were measured using a modified Rose-Williams still. ... The isobaric vapor-liquid equilibrium data of butanone(1)-N, N-dimethylformamide (DMF)(2) at 100.92kPa, 93.32kPa, and 79.99kPa and of toluene(1)-DMF(2) at 100.92kPa were measured using a modified Rose-Williams still. The above data met the thermodynamic consistency test and were correlated with the Wilson,NRTL, and UNIQUAC equations. These data can be used in the analysis and design of the process that involves separating DMF from butanone and toluene in the leather synthesis industry. 展开更多
关键词 ISOBARIC vapor-liquid equilibrium N N-dimethylformamide DMF butanone toluene Wilson equation NRTL equation UNIQUAC EQUATION
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BASIC EQUATIONS OF THE PROBLEM OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE WITH VARIABLE THICKNESS
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作者 黄家寅 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2004年第7期812-816,共5页
Under the case of ignoring the body forces and considering components caused by diversion of membrane in vertical direction (z-direction),the constitutive equations of the problem of the nonlinear unsymmetrical bendin... Under the case of ignoring the body forces and considering components caused by diversion of membrane in vertical direction (z-direction),the constitutive equations of the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with variable thickness are given;then introducing the dimensionless variables and three small parameters,the dimensionaless governing equations of the deflection function and stress function are given. 展开更多
关键词 orthotropic rectangular thin plate with variable thickness nonlinear unsymmetrical bending equilibrium equation compatibility equation basic equation dimensionless equation
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Investigations on if-E Curve in Cyclic Derivative Chronopotentiometry(Ⅵ)——Theoretical Equations of if-E Curves in the Case of Quasi-reversible and Irreversible Electrode Reactions
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作者 SHEN Xue-ming, CHEN Hong-yuan and GAO Hong (Department of Chemistry, Nanjing University, Nanjing, 210008) 《Chemical Research in Chinese Universities》 SCIE CAS CSCD 1992年第2期71-74,共4页
The theory of if-E curve in cyclic derivative chronopotentiometry is presented. Theoretical equations of if-E curves in the case of quasi-reversible and irreversible electrode reactions are deduced respectively.
关键词 Cyclic derivative chronopotentiometry if-E curve theoretical equation Quasi-reversible IRREVERSIBLE
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