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Approximate solutions and nonlinear scalarizations for set optimization with a co-radiant set 认领 引用
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作者 ZHANG Qi XU Yi-hong 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2026年第2期367-385,共19页
In normed linear spaces,by applying the Minkowski difference,the concepts of efficient solutions,weakly efficient solutions,proper efficient solutions and Henig efficient solutions are introduced for set optimization ... In normed linear spaces,by applying the Minkowski difference,the concepts of efficient solutions,weakly efficient solutions,proper efficient solutions and Henig efficient solutions are introduced for set optimization with respect to co-radiant sets,and the relationships among them is discussed.Optimality conditions for minimal solutions and strictly minimal solutions of scalar set optimization are established by using the generalized oriented distance functions,respectively.The relationship between the solutions of set optimization problem and the solutions of scalar problem is studied.Several examples are given to explain our result.Properties of Gerstewitz’s functions with respect to co-radiant sets are discussed,which are used to establish sufficient conditions for weakly efficient solutions of set optimization. 展开更多
关键词 set optimization nonlinear function approximate solution co-radiant set
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Approximate solutions of the Alekseevskii–Tate model of long-rod penetration 认领 引用 被引量：8
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作者 W.J.Jiao X.W.Chen 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2018年第2期334-348,共15页
The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, pe... The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetratorarget combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration. 展开更多
关键词 Long-rod penetration Alekseevskii–Tate model Theoretical solution Approximate solution Perturbation solution
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Computational method for analytical solution with finite elements(CMAS-FE):Deriving approximate analytical solution for an isotropic homogeneous elastic medium with linear finite element method 认领 引用
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作者 Jiajia Yue Zifeng Yuan 《Theoretical & Applied Mechanics Letters》 EI CAS CSCD 2025年第6期540-550,共11页
This study presents a novel methodology to obtain an approximate analytical solution for an isotropic homo-geneous elastic medium with displacement and traction boundary conditions.The solution is derived through solv... This study presents a novel methodology to obtain an approximate analytical solution for an isotropic homo-geneous elastic medium with displacement and traction boundary conditions.The solution is derived through solving a specific numerical problem under the scope of the linear finite element method(LFEM),so the method is termed computational method for analytical solutions with finite elements(CMAS-FE).The primary objective of the CMAS-FE is to construct analytical expressions for displacements and reaction forces at nodes,as well as for strains and stresses at elemental quadrature points,all of which are formulated as infinite series solutions of various orders of Poisson’s ratios.Like the conventional LFEM,the CMAS-FE forms global sparse linear equations,but the Young’s modulus and Poisson’s ratio remain variables(or symbols).By employing a direct inverse method to solve these symbolic linear systems,an analytical expression of the displacement field can be constructed.The CMAS-FE is validated via patch and bending tests,which demonstrate convergence with mesh and term refine-ment.Furthermore,the CMAS-FE is applied to obtain the bending stiffness of a beam structure and to estimate an approximate stress intensity factor for a straight crack within a square-shaped plate. 展开更多
关键词 CMAS-FE Finite element method Linear elastic problem Analytical solution
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Classification and Approximate Solutions to a Class of Perturbed Nonlinear Wave Equations 认领 引用 被引量：1
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作者 ZHANG Zhi-Yong CHEN Yu-Fu YONG Xue-Lin 《Communications in Theoretical Physics》 SCIE CAS 2009年第11期769-772,共4页
A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutio... A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutions of the equations based on the Lie group method are constructed. 展开更多
关键词 approximate symmetry Lie reduction approximate solution
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Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method 认领 引用
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作者 Xiao-Yu Jiao 《Chinese Physics B》 SCIE EI CAS CSCD 2018年第10期123-129,共7页
In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step proce... In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished. 展开更多
关键词 approximate symmetry method （2＋1）-dimensional perturbed Boussinesq equation series solutions convergence of series solutions
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Explicit approximate solutions to two transcendental equations in two-phase stratified 认领 引用
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作者 Baisheng WU Yixin ZHOU +1 位作者 Zeyao CHEN Siukai LAI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2025年第10期2007-2016,共10页
Stratified flow is a common phenomenon in horizontal tubes of two-phase flow systems. However, the existing methods for calculating the wetted angle of the flat interface model and the central angle of the two-circle ... Stratified flow is a common phenomenon in horizontal tubes of two-phase flow systems. However, the existing methods for calculating the wetted angle of the flat interface model and the central angle of the two-circle model rely on solving implicit transcendental equations, which require iterative numerical root-finding methods,thereby introducing computational complexity and inefficiency. This paper proposes the high-precision explicit approximate solutions for the two models, directly correlating the geometric parameters with the flow parameters, thus significantly enhancing the efficiency and accuracy of two-phase flow analysis. 展开更多
关键词 two-phase stratified flow horizontal circular tube wetted angle central angle Padé(rational)approximation Schröder iteration
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Classification and Approximate Solutions to Perturbed Nonlinear Diffusion-Convection Equations 认领 引用 被引量：4
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作者 WANG Yong ZHANG Shun-Li 《Communications in Theoretical Physics》 SCIE CAS 2008年第1期17-21,共5页
This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admi... This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained. 展开更多
关键词 perturbed nonlinear diffusion-convection equation approximate generalized conditional symme-try approximate invariant solution
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Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution 认领 引用 被引量：2
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作者 樊尚春 李艳 +2 位作者 郭占社 李晶 庄海涵 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第5期58-65,共8页
Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the ap... Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope. 展开更多
关键词 resonant gyroscopes dynamic characteristics Mathieu equation approximate solution
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Approximate Analytic Solution of Solitary Wave for a Class of Nonlinear Disturbed Long-Wave System 认领 引用 被引量：6
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作者 莫嘉琪 姚静荪 唐荣荣 《Communications in Theoretical Physics》 SCIE CAS 2010年第7期27-30,共4页
In this paper, the approximate expressions of the solitary wave solutions for a class of nonlinear disturbed long-wave system are constructed using the homotopie mapping method.
关键词 nonlinear long-wave equation solitary wave approximate analytic solution
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Approximate Solution for Mechanism of Thermally and Wind-driven Ocean Circulation 认领 引用 被引量：4
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作者 MO Jiaqi LIN Wantao LIN Yihua 《Chinese Geographical Science》 SCIE 2010年第5期383-388,共6页
The thermally and wind-driven ocean circulation is a complicated natural phenomenon in the atmospheric physics. Hence we need to reduce it using basic models and solve the models using approximate methods. A non-linea... The thermally and wind-driven ocean circulation is a complicated natural phenomenon in the atmospheric physics. Hence we need to reduce it using basic models and solve the models using approximate methods. A non-linear model of the thermally and wind-driven ocean circulation is used in this paper. The results show that the zero solution of the linear equation is a stable focus point, which is the path curve trend origin point as time (t) trend to infinity. By using the homotopic mapping perturbation method, the exact solution of the model is obtained. The homotopic mapping perturbation method is an analytic solving method, so the obtained solution can be used for analytic operating sequentially. And then we can also obtain the diversified qualitative and quantitative behaviors for corresponding physical quantities. 展开更多
关键词 global climate atmosphere-ocean oscillation homotopic mapping approximate solution
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Approximate solution for the Klein Gordon-Schrdinger equation by the homotopy analysis method 认领 引用 被引量：4
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作者 王佳 李彪 叶望川 《Chinese Physics B》 SCIE EI CAS 2010年第3期83-89,共7页
The Homotopy analysis method is applied to obtain the approximate solution of the Klein-Gordon Schrodinger equation. The Homotopy analysis solutions of the Klein-Gordon Schrodinger equation contain an auxiliary parame... The Homotopy analysis method is applied to obtain the approximate solution of the Klein-Gordon Schrodinger equation. The Homotopy analysis solutions of the Klein-Gordon Schrodinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution. 展开更多
关键词 Klein-Gordon-Schrodinger equation homotopy analysis method approximate solution
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STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION 认领 引用 被引量：1
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作者 Helge Holden Kenneth H. Karlsen +1 位作者 Darko Mitrovic Evgueni Yu. Panov 《Acta Mathematica Scientia》 SCIE 2009年第6期1573-1612,共40页
Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux ... Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes. 展开更多
关键词 degenerate hyperbolic-elliptic equation degenerate convection-diffusion equation conservation law discontinuous flux approximate solutions compactness
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Approximate Analytical Solutions for Scattering States of D-dimensional Hulthen Potentials 认领 引用 被引量：1
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作者 CHEN Chang-Yuan SUN Dong-Sheng +1 位作者 LIU Cheng-Lin LU Fa-Lin 《Communications in Theoretical Physics》 SCIE CAS CSCD 2011年第3期399-404,共6页
Using the exponential function transformation approach along with an approximation for the centrifugal potential,the radial Schr6dinger equation of D-dimensional Hulthen potential is transformed to a hypergeometric di... Using the exponential function transformation approach along with an approximation for the centrifugal potential,the radial Schr6dinger equation of D-dimensional Hulthen potential is transformed to a hypergeometric differential equation.The approximate analytical solutions of scattering states are attained.The normalized wave functions expressed in terms of hypergeometrie functions of scattering states on the"k/2πscale"and the calculation formula of phase shifts are given.The physical meaning of the approximate analytical solutions is discussed. 展开更多
关键词 D-dimensional Hulthen potential Schrodinger equation scattering states approximate analytical solutions
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Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation 认领 引用 被引量：3
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作者 YAO Ruo-Xia JIAO Xiao-Yu LOU Sen-Yue 《Communications in Theoretical Physics》 SCIE CAS 2009年第5期785-788,共4页
Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symm... Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here. 展开更多
关键词 perturbed Kuramoto-Sivashinsky equation approximate symmetry reduction series reduction solution
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New approximate solution for time-fractional coupled KdV equations by generalised differential transform method 认领 引用 被引量：2
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作者 刘金存 侯国林 《Chinese Physics B》 SCIE EI CAS 2010年第11期41-47,共7页
In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The pr... In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations. 展开更多
关键词 fractional coupled KdV equations Caputo fractional derivative differential transform method approximate analytic solution
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Approximate analytical solutions for threedimensional ascent trajectory of a solid-fuel launch vehicle with time-varying mass flow rate 认领 引用 被引量：1
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作者 Qi YU Wanchun CHEN Wenbin YU 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2024年第10期275-293,共19页
In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management,Approximate Analytical Solutions(AAS)for the threedimensional(3D)ascent flight stat... In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management,Approximate Analytical Solutions(AAS)for the threedimensional(3D)ascent flight states are derived,which are the only solutions capable of considering time-varying Mass Flow Rate(MFR)at present.The uneven MFR makes the thrust vary nonlinearly and thus increases the difficulty of the problem greatly.The AAS are derived based on a 3D Generalized Ascent Dynamics Model(GADM)with a normalized mass as the independent variable.To simplify some highly nonlinear terms in the GADM,several approximate functions are introduced carefully,while the errors of the approximations relative to the original terms are regarded as minor perturbations.Notably,a finite series with positive and negative exponents,called Exponent-Symmetry Series(ESS),is proposed for function approximation to decrease the highest exponent in the AAS so as to reduce computer round-off errors.To calculate the ESS coefficients,a method of seeking the Optimal Interpolation Points(OIP)is proposed using the leastsquares-approximation theory.Due to the artful design of the approximations,the GADM can be decomposed into two analytically solvable subsystems by a perturbation method,and thus the AAS are obtained successfully.Finally,to help implement the AAS,two indirect methods for measuring the remaining mass and predicting the burnout time in flight are put forward using information from accelerometers.Simulation results verify the superiority of the AAS under the condition of time-varying MFR. 展开更多
关键词 Approximate analytical solutions 3D ascent trajectory Solid-fuel launch vehicle Uneven thrust Energy management
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Approximate Solutions of Primary Resonance for Forced Duffing Equation by Means of the Homotopy Analysis Method 认领 引用 被引量：1
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作者 YUAN Peixin LI Yongqiang 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2011年第3期501-506,共6页
Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approxima... Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method （HAM）.Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy. 展开更多
关键词 homotopy analysis method（HAM） series solution forced Duffing equation numerical solution
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WAVELET APPROXIMATE INERTIAL MANIFOLD AND NUMERICAL SOLUTION OF BURGERS' EQUATION 认领 引用
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作者 田立新 许伯强 刘曾荣 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2002年第10期1140-1152,共13页
The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation ... The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation has a good localization property of the numerical solution distinguishably. 展开更多
关键词 wavelet wavelet approximate inertial manifold (WAIM) wavelet Galerkin solution infinite dimensional dynamic system
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Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation 认领 引用
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作者 赵国忠 蔚喜军 +2 位作者 徐云 朱江 吴迪 《Chinese Physics B》 SCIE EI CAS 2010年第8期46-54,共9页
This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV）... This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions. 展开更多
关键词 approximate analytic solutions generalized Hirota-Satsuma coupled KdV equation coupled mKdV equation variational iteration method
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A New Kind of Iteration Method for Finding Approximate Periodic Solutions to Ordinary Diferential Equations 认领 引用
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作者 Wu Dong-xu Wang Cai-ling +1 位作者 Xu Xu Li Yong 《Communications in Mathematical Research》 CSCD 2013年第4期297-304,共8页
In this paper, a new kind of iteration technique for solving nonlinear ordinary differential equations is described and used to give approximate periodic solutions for some well-known nonlinear problems. The most inte... In this paper, a new kind of iteration technique for solving nonlinear ordinary differential equations is described and used to give approximate periodic solutions for some well-known nonlinear problems. The most interesting features of the proposed methods are its extreme simplicity and concise forms of iteration formula for a wide range of nonlinear problems. 展开更多
关键词 iteration method approximate periodic solution ordinary differentialequation
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