An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator...An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.展开更多
The magnetic interface forward and inversion method is realized using the Taylor series expansion to linearize the Fourier transform of the exponential function. With a large expansion step and unbounded neighborhood,...The magnetic interface forward and inversion method is realized using the Taylor series expansion to linearize the Fourier transform of the exponential function. With a large expansion step and unbounded neighborhood, the Taylor series is not convergent, and therefore, this paper presents the magnetic interface forward and inversion method based on Pade approximation instead of the Taylor series expansion. Compared with the Taylor series, Pade＇s expansion＇s convergence is more stable and its approximation more accurate. Model tests show the validity of the magnetic forward modeling and inversion of Pade approximation proposed in the paper, and when this inversion method is applied to the measured data of the Matagami area in Canada, a stable and reasonable distribution of underground interface is obtained.展开更多
Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be a...Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.展开更多
In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and ...In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.展开更多
To evaluate the water storage and project the future evolution of glaciers, the ice-thickness of glaciers is an essential input. However, direct measurements of ice thickness are labo- rious, not feasible everywhere, ...To evaluate the water storage and project the future evolution of glaciers, the ice-thickness of glaciers is an essential input. However, direct measurements of ice thickness are labo- rious, not feasible everywhere, and necessarily restricted to a small number of glaciers. In this article, we develop a simple method to estimate the ice-thickness along flow-line of mountain glaciers. Different from the traditional method based on shallow ice approximation （SIA）, which gives a relationship be- tween ice thickness, surface slope, and yield stress of glaciers, the improved method considers and pre- sents a simple way to calibrate the influence of valley wall on ice discharge. The required inputs are the glacier surface topography and outlines. This shows the potential of the method for estimating the ice-thickness distribution and volume of glaciers without using of direct thickness measurements.展开更多
The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the bas...The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.展开更多
Multidisciplinary feasible method (MDF) is conventional method to multidisciplinary optimization (MDO) and well-understood by users. It reduces the dimensions of the multidisciplinary optimization problem by using the...Multidisciplinary feasible method (MDF) is conventional method to multidisciplinary optimization (MDO) and well-understood by users. It reduces the dimensions of the multidisciplinary optimization problem by using the design variables as independent optimization variables. However, at each iteration of the conventional optimization procedure, multidisciplinary analysis (MDA) is numerously performed that results in extreme expense and low optimization efficiency. The intrinsic weakness of MDF is due to the times that it loop fixed-point iterations in MDA, which drive us to improve MDF by building inexpensive approximations as surrogates for expensive MDA. An simple example is presented to demonstrate the usefulness of the improved MDF. Results show that a significant reduction in the number of multidisciplinary analysis required for optimization is obtained as compared with original MDF and the efficiency of optimization is increased.展开更多
The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and math...The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.展开更多
The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory;however,it still faces the challenges of large deviation from the optimal solution and inability to satisfy...The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory;however,it still faces the challenges of large deviation from the optimal solution and inability to satisfy the specific flight time and fuel mass constraints.In this paper,a modified shape approximation low-thrust model is presented,and a novel constrained optimization algorithm is developed to solve this problem.The proposed method aims at settling the bi-objective optimization orbit involving the twin objectives of minimum flight time and low fuel consumption and enhancing the accuracy of optimized orbit.In particular,a transformed high-order polynomial model based on finite Fourier series is proposed,which can be characterized as a multi-constraint optimization problem.Then,a novel optimization algorithm is specifically developed to optimize the large-scale multi-constraint dynamical equations of shape trajectory.The key performance indicators of the index include minimum flight time,low fuel consumption and bi-objective optimization of the two.Simulation results prove that this approach possesses both the high precision achievable by numerical methods and low computational complexity offered by shape approximation techniques.Besides,the Pareto front of the fuel-time bi-objective optimization orbit is firstly introduced to analyze an intact optimal solution set.Furthermore,we have demonstrated that our proposed approach is appropriate to generate the preliminary orbit for pseudo-spectral method.展开更多
To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the s...To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.展开更多
In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpin...In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/(l + l/2n-3)s≤Hs(S)≤ Pn(S). An algorithm is presented to get Pn(S) for n ≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.展开更多
The aim of this paper is to employ fractional order proportional integral derivative(FO-PID)controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system(M...The aim of this paper is to employ fractional order proportional integral derivative(FO-PID)controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system(MLS),which is inherently nonlinear and unstable system.The proposal is to deploy discrete optimal pole-zero approximation method for realization of digital fractional order controller.An approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within given bandwidth is explored.The controller parameters are tuned using dynamic particle swarm optimization(d PSO)technique.Effectiveness of the proposed control scheme is verified by simulation and experimental results.The performance of realized digital FO-PID controller has been compared with that of the integer order PID controllers.It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.展开更多
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With t...Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With the help of symbolic computation Maple, the results obtained by ADM-Pade technique are compared with those obtained by using ADM alone. The numerical results demonstrate that ADM-Pade technique give the approximate solution with faster convergence rate and higher accuracy and relative in larger domain of convergence than using ADM.展开更多
Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K ...Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.展开更多
A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discusse...A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.展开更多
In many industrial applications,heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention,owing to their immense importance in technology,engineering,and science.These processe...In many industrial applications,heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention,owing to their immense importance in technology,engineering,and science.These processes are relevant for polymer solutions,porous industrial materials,ceramic processing,oil recovery,and fluid beds.The present tangent hyperbolic fluid flow and heat transfer model accurately predicts the shear-thinning phenomenon and describes the blood flow characteristics.Therefore,the entropy production analysis of a non-Newtonian tangent hyperbolic material flow through a vertical microchannel with a quadratic density temperature fluctuation(quadratic/nonlinear Boussinesq approximation)is performed in the present study.The impacts of the hydrodynamic flow and Newton’s thermal conditions on the flow,heat transfer,and entropy generation are analyzed.The governing nonlinear equations are solved with the spectral quasi-linearization method(SQLM).The obtained results are compared with those calculated with a finite element method and the bvp4c routine.In addition,the effects of key parameters on the velocity of the hyperbolic tangent material,the entropy generation,the temperature,and the Nusselt number are discussed.The entropy generation increases with the buoyancy force,the pressure gradient factor,the non-linear convection,and the Eckert number.The non-Newtonian fluid factor improves the magnitude of the velocity field.The power-law index of the hyperbolic fluid and the Weissenberg number are found to be favorable for increasing the temperature field.The buoyancy force caused by the nonlinear change in the fluid density versus temperature improves the thermal energy of the system.展开更多
By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive...By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.展开更多
A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in th...A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.展开更多
The purpose of this paper is to propose and study local spline approximation methods for singular product integration,for which;i)the precision degree is the highest possible using splint approximation; ii) the nodes ...The purpose of this paper is to propose and study local spline approximation methods for singular product integration,for which;i)the precision degree is the highest possible using splint approximation; ii) the nodes fan be assumed equal to arbitrary points,where the integrand function f is known; iii) the number of the requested evaluations of f at the nodes is low,iv) a satisfactory convergence theory can be proved.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11172093 and 11372102)the Hunan Provincial Innovation Foundation for Postgraduate,China(Grant No.CX2012B159)
文摘An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.
基金supported by Sino Probe-09-01-Integrated geophysical data processing and integrated system for moving platform(No.201311192)Graduate innovation fund of Jilin University(No.2015025)
文摘The magnetic interface forward and inversion method is realized using the Taylor series expansion to linearize the Fourier transform of the exponential function. With a large expansion step and unbounded neighborhood, the Taylor series is not convergent, and therefore, this paper presents the magnetic interface forward and inversion method based on Pade approximation instead of the Taylor series expansion. Compared with the Taylor series, Pade＇s expansion＇s convergence is more stable and its approximation more accurate. Model tests show the validity of the magnetic forward modeling and inversion of Pade approximation proposed in the paper, and when this inversion method is applied to the measured data of the Matagami area in Canada, a stable and reasonable distribution of underground interface is obtained.
基金Project supported by the National Natural Science Foundation of China(Grant No.10271074)
文摘Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.
文摘In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.
基金supported by the National Basic Research Program of China (No. 2007CB411501)the Knowledge Innovation Project of the Chinese Academy of Sciences (No. KZCX2-EW-311)+1 种基金the National Natural Science Foundation of China (Nos. 91025012, J0930003/J0109)the Project for Outstanding Young Scientists of the National Natural Science Foundation of China (No. 40121101)
文摘To evaluate the water storage and project the future evolution of glaciers, the ice-thickness of glaciers is an essential input. However, direct measurements of ice thickness are labo- rious, not feasible everywhere, and necessarily restricted to a small number of glaciers. In this article, we develop a simple method to estimate the ice-thickness along flow-line of mountain glaciers. Different from the traditional method based on shallow ice approximation （SIA）, which gives a relationship be- tween ice thickness, surface slope, and yield stress of glaciers, the improved method considers and pre- sents a simple way to calibrate the influence of valley wall on ice discharge. The required inputs are the glacier surface topography and outlines. This shows the potential of the method for estimating the ice-thickness distribution and volume of glaciers without using of direct thickness measurements.
文摘The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.
文摘Multidisciplinary feasible method (MDF) is conventional method to multidisciplinary optimization (MDO) and well-understood by users. It reduces the dimensions of the multidisciplinary optimization problem by using the design variables as independent optimization variables. However, at each iteration of the conventional optimization procedure, multidisciplinary analysis (MDA) is numerously performed that results in extreme expense and low optimization efficiency. The intrinsic weakness of MDF is due to the times that it loop fixed-point iterations in MDA, which drive us to improve MDF by building inexpensive approximations as surrogates for expensive MDA. An simple example is presented to demonstrate the usefulness of the improved MDF. Results show that a significant reduction in the number of multidisciplinary analysis required for optimization is obtained as compared with original MDF and the efficiency of optimization is increased.
基金Project supported by the National Key Basic Research Project of China (Grant No 2004CB318000)the National Natural Science Foundation of China (Grant Nos 10771072 and 10735030)Shanghai Leading Academic Discipline Project of China (Grant No B412)
文摘The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.
基金supported by the National Natural Science Foundation of China(Nos.61627810,61790562,61403096).
文摘The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory;however,it still faces the challenges of large deviation from the optimal solution and inability to satisfy the specific flight time and fuel mass constraints.In this paper,a modified shape approximation low-thrust model is presented,and a novel constrained optimization algorithm is developed to solve this problem.The proposed method aims at settling the bi-objective optimization orbit involving the twin objectives of minimum flight time and low fuel consumption and enhancing the accuracy of optimized orbit.In particular,a transformed high-order polynomial model based on finite Fourier series is proposed,which can be characterized as a multi-constraint optimization problem.Then,a novel optimization algorithm is specifically developed to optimize the large-scale multi-constraint dynamical equations of shape trajectory.The key performance indicators of the index include minimum flight time,low fuel consumption and bi-objective optimization of the two.Simulation results prove that this approach possesses both the high precision achievable by numerical methods and low computational complexity offered by shape approximation techniques.Besides,the Pareto front of the fuel-time bi-objective optimization orbit is firstly introduced to analyze an intact optimal solution set.Furthermore,we have demonstrated that our proposed approach is appropriate to generate the preliminary orbit for pseudo-spectral method.
基金Project supported by the National Natural Science Foundation of China (No. 10271074)
文摘To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.
文摘In this paper, we firstly define a decreasing sequence {Pn(S)} by the generation of the Sierpinski gasket where each Pn(S) can be obtained in finite steps. Then we prove that the Hausdorff measure Hs(S) of the Sierpinski gasket S can be approximated by {Pn(S)} with Pn(S)/(l + l/2n-3)s≤Hs(S)≤ Pn(S). An algorithm is presented to get Pn(S) for n ≤5. As an application, we obtain the best lower bound of Hs(S) till now: Hs(S)≥0.5631.
基金supported by the Board of Research in Nuclear Sciences of the Department of Atomic Energy,India(2012/36/69-BRNS/2012)
文摘The aim of this paper is to employ fractional order proportional integral derivative(FO-PID)controller and integer order PID controller to control the position of the levitated object in a magnetic levitation system(MLS),which is inherently nonlinear and unstable system.The proposal is to deploy discrete optimal pole-zero approximation method for realization of digital fractional order controller.An approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within given bandwidth is explored.The controller parameters are tuned using dynamic particle swarm optimization(d PSO)technique.Effectiveness of the proposed control scheme is verified by simulation and experimental results.The performance of realized digital FO-PID controller has been compared with that of the integer order PID controllers.It is observed that effort required in fractional order control is smaller as compared with its integer counterpart for obtaining the same system performance.
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
基金supported by the National Natural Science Foundation of China under Grant No. 10735030Shanghai Leading Academic Discipline Project under Grant No. B412Program for Changjiang Scholars and Innovative Research Team in University under Grant No. IRT0734
文摘Combining Adomian decomposition method （ADM） with Pade approximants, we solve two differentiaidifference equations （DDEs）： the relativistic Toda lattice equation and the modified Volterra lattice equation. With the help of symbolic computation Maple, the results obtained by ADM-Pade technique are compared with those obtained by using ADM alone. The numerical results demonstrate that ADM-Pade technique give the approximate solution with faster convergence rate and higher accuracy and relative in larger domain of convergence than using ADM.
文摘Let K be a closed convex subset of a real reflexive Banach space E, T：K→K be a nonexpansive mapping, and f：K→K be a fixed weakly contractive （may not be contractive） mapping. Then for any t∈（0, 1）, let x1∈K be the unique fixed point of the weak contraction x1→tf（x）＋（1-t）Tx. If T has a fixed point and E admits a weakly sequentially continuous duality mapping from E to E^＊, then it is shown that {xt} converges to a fixed point of T as t→0. The results presented here improve and generalize the corresponding results in （Xu, 2004）.
文摘A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.
文摘In many industrial applications,heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention,owing to their immense importance in technology,engineering,and science.These processes are relevant for polymer solutions,porous industrial materials,ceramic processing,oil recovery,and fluid beds.The present tangent hyperbolic fluid flow and heat transfer model accurately predicts the shear-thinning phenomenon and describes the blood flow characteristics.Therefore,the entropy production analysis of a non-Newtonian tangent hyperbolic material flow through a vertical microchannel with a quadratic density temperature fluctuation(quadratic/nonlinear Boussinesq approximation)is performed in the present study.The impacts of the hydrodynamic flow and Newton’s thermal conditions on the flow,heat transfer,and entropy generation are analyzed.The governing nonlinear equations are solved with the spectral quasi-linearization method(SQLM).The obtained results are compared with those calculated with a finite element method and the bvp4c routine.In addition,the effects of key parameters on the velocity of the hyperbolic tangent material,the entropy generation,the temperature,and the Nusselt number are discussed.The entropy generation increases with the buoyancy force,the pressure gradient factor,the non-linear convection,and the Eckert number.The non-Newtonian fluid factor improves the magnitude of the velocity field.The power-law index of the hyperbolic fluid and the Weissenberg number are found to be favorable for increasing the temperature field.The buoyancy force caused by the nonlinear change in the fluid density versus temperature improves the thermal energy of the system.
文摘By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.
基金This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada(Grant OGPIN-336)and by the"Ministere de l'Education du Quebec"(FCAR Grant-ER-0725)
文摘A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.
基金Work sponsored by"Ministero dell' University"CNR of Italy
文摘The purpose of this paper is to propose and study local spline approximation methods for singular product integration,for which;i)the precision degree is the highest possible using splint approximation; ii) the nodes fan be assumed equal to arbitrary points,where the integrand function f is known; iii) the number of the requested evaluations of f at the nodes is low,iv) a satisfactory convergence theory can be proved.
