An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

NUMERICAL RESOLVENT METHODS FOR CONSTRAINED PROBLEMS IN MECHANICS

Resolvent methods are presented for generating systematically iterative numerical algorithms for constrained problems in mechanics.The abstract framework corresponds to a general mixed finite element subdif-ferential model,with dual and primal evolution versions,which is shown to apply to problems of fluid dynamics,transport phenomena and solid mechanics,among others.In this manner,Uzawa's type methods and penalization-duality schemes,as well as macro-hybrid formulations,are generalized to non necessarily potential nanlinear mechanical problems.

AN APPLICATION OF THE EXPONENTIAL CUBIC SPLINES TO NUMERICAL SOLUTION OF A SELF-ADJOINT PERTURBATION PROBLEM

We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM FOR THE HYPERBOLIC EQUATION WITH INITIAL JUMP

In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.

A Novel Hybrid Vortex Search and Artificial Bee Colony Algorithm for Numerical Optimization Problems

Though vortex search（VS） algorithm has good performance in solving global numerical optimization problems, it cannot fully search the whole space occasionally. Combining the vortex search algorithm and the artificial bee colony algorithm（ABC） which has good performance in exploration, we present a HVS（hybrid vortex search） algorithm to solve the numerical optimization problems. We first use the employed bees and onlooker bees of ABC algorithm to find a solution, and then adopt the VS algorithm to find the best solution. In the meantime, we cannot treat the best solution so far as the center of the algorithm all the time. The algorithm is tested by 50 benchmark functions. The numerical results show the HVS algorithm has superior performance over the ABC and the VS algorithms.

NUMERICAL ANALYSIS OF NONSTATIONARY THERMISTOR PROBLEM

The thermistor problem is a coupled system of nonlinear PDEs which consists of the heat equation with the Joule heating as a source, and the current conservation equation with temperature dependent electrical conductivity. In this paper we make a numerical analysis of the nonsteady thermistor problem. L(infinity)(OMEGA), W1,infinity(OMEGA) stability and error bounds for a piecewise linear finite element approximation are given.

NUMERICAL SOLUHON TO THE PROBLEM ON DlFFUSION OF CATALYTIC AGENTS RELEASED FROM AN AIRPLANE

A solution to the problem on diffusion of catalytic agents released from an airplane is sought.The variation of falling velocities of agent particles with the altitudes is taken into account in the study of the problem.A comparison is also made between the calculated results obtained by using the finite-difference method and those by using the analytic method,the similarities and the differences between the two methods are revealed.

THE APPLICATION OF INTEGRAL EQUATIONS TO THE NUMERICAL SOLUTION OF NONLINEAR SINGULAR PERTURBATION PROBLEMS

The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method.

NUMERICAL STUDIES FOR GAS SOLID TWO PHASE STEADY MIXED CONVECTION PROBLEMS WITH PHASE CHANGE

A mathematical model for describing gas solid two phase steady mixed convection with phase change has been developed and numerical calculation methods presented.A melting liquid droplet failing a counter gas currenl expe- riences three processes,cooling of liquid droplet,solidification and cooling of the solid particle.The turbulent model used for Rayleigh number greater than 10~6 is a two equation(k—ε)model of turbulence.For phase change,an improved enthalpy method with varied time step is proposed.The gas particle two phase flow is described by using Eulerian-Lagrangian approach.Modified SIMPLE algorithm and Runge-Kutta method are used in interative calcu- lation.As an example of calculation,the flow in a special 2-dimensional axi-symmetrical prilling tower of diameter 20 m and height 50 m has been performed.Buoyancy effect is important for moving droplet with phase change. The model to be developed and analysis of results obtained in this paper are useful for engineering design in indus- try.

BIFURCATION SOLUTION BRANCHESAND THEIR NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM WITH TWO PARAMETERS

In this paper the bifurcation solution branches of a semi-linear elliptic problem are studied.

The optimization of architectural shape based on Genetic Algorithm

Genetic Algorithm （GA） is widely adopted in optimization and the improvement of its optimization performance is attracting many researchers＇ attentions. In solving practical probtems in the process of architectural design, the ways of converting design problems into mathematical models that can be addressed by GA are of great significance in achieving final optimal results. However, no such rute that can be applied to such conversion has been devetoped so far. In general, problems which can be addressed by GA can be divided into combinatorial problems and numerical probtems. In this paper, by means of attempting to disintegrate a complicated architectural probtem into combinatorial and numericat probtems, the author discusses feasibitity and practicality of sotving these two types of problems simultaneousty utitizing GA and discloses both advantages and disadvantages of GA by comparing with other algorithms.

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|^(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.

On the clean numerical simulation(CNS) of chaotic dynamic systems

According to Lorenz, chaotic dynamic systems have sensitive dependence on initial conditions（SDIC）, i.e., the butterfly-effect： a tiny difference on initial conditions might lead to huge difference of computer-generated simulations after a long time. Thus, computer-generated chaotic results given by traditional algorithms in double precision are a kind of mixture of ＂true＂（convergent） solution and numerical noises at the same level. Today, this defect can be overcome by means of the ＂clean numerical simulation＂（CNS） with negligible numerical noises in a long enough interval of time. The CNS is based on the Taylor series method at high enough order and data in the multiple precision with large enough number of digits, plus a convergence check using an additional simulation with even smaller numerical noises. In theory, convergent（reliable） chaotic solutions can be obtained in an arbitrary long（but finite） interval of time by means of the CNS. The CNS can reduce numerical noises to such a level even much smaller than micro-level uncertainty of physical quantities that propagation of these physical micro-level uncertainties can be precisely investigated. In this paper, we briefly introduce the basic ideas of the CNS, and its applications in long-term reliable simulations of Lorenz equation, three-body problem and Rayleigh-Bénard turbulent flows. Using the CNS, it is found that a chaotic three-body system with symmetry might disrupt without any external disturbance, say, its symmetry-breaking and system-disruption are ＂self-excited＂, i.e., out-of-nothing. In addition, by means of the CNS, we can provide a rigorous theoretical evidence that the micro-level thermal fluctuation is the origin of macroscopic randomness of turbulent flows. Naturally, much more precise than traditional algorithms in double precision, the CNS can provide us a new way to more accurately investigate chaotic dynamic systems.

A novel hybrid FEM-BEM method for 3D eddy current field calculation using current density J

This paper introduces a novel hybrid FEM-BEM method for calculating 3D eddy cur-rent field. In the eddy current region, the eddy current density J is solved by the finite element method (FEM) which is discretized by brick finite element mesh, while in the eddy current free re-gion, the magnetic field intensity H is solved by the boundary element method (BEM) which is dis-cretized by rectangular boundary element mesh. Under the boundary conditions, an algebraic equation group is obtained that only includes J by eliminating H. This method has many advan-tages over traditional ones, such as fewer variables, more convenient coupling between the FEM and the BEM and wider application to multiply-connected regions. The calculated values of two models are in good agreement with experimental results. This shows the validity of our method.