SYMBOLIC MANIPULATION FOR SOLVING PHYSICAL PROBLEMS

An introduction is made to symbolic manipulation and its application in physical education. Some examples for the effective use of the general purpose software tool Mathematica are presented.

Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.

EXACT SOLITARY WAVE AND SOLITONSOLUTIONS OF THE FIFTH ORDER MODEL EQUATION

With the aid of nonlinear transformations, and using the symbolic manipulation system, the exact solitary wave and soliton solutions to a fifth order nonlinear evolution equation with general coefficients are obtained, and the corresponding sufficient conditions that the equation admits of these type of solutions are given. From the results one can see how the apparently changes in the coefficients would effect the solutions.

EXACT SOLITRAY WAVE SOLUTIONS ANDSINGULAR SOLUTIONS TO THETWO-DIMENSIONAL NONLINEARDISSIPATIVE-DISPERSIVE SYSTEM

An algorithm for constructing exact solitary wave solutions and singular solutions for a class of nonlinear dissipative-dispersive system is presented. With the aid of symbolic manipulation system Maple, some explicit solutions are obtained for the system in physically interesting but non-integrable cases.

A new model for verification

Formal verification is playing a significant role in IC design.However,the common models for verification either have their complexity problems or have applicable limitations.In order to overcome the deficiencies,a novel model-WGL(Weighted Generalized List)is proposed,which is based on the general-list decomposition of polynomials,with three different weights and manipulation rules introduced to effect node sharing and the canonicity.Timing parameters and operations on them are also considered.Examples show the word-level WGL is the only model to linearly represent the common word-level functions and the bit-level WGL is especially suitable for arithmetic intensive circuits.The model is proved to be a uniform and efficient model for both bit-level and word-level functions.Then based on the WGL model,a backward-construction verification approach is proposed,which reduces time and space complexity for multipliers to polynomial complexity(time complexity is less than O(n3.6)and space complexity is less than O(n1.5))without hierarchical partitioning.Both the model and the verification method show their theoretical and applicable significance in IC design.

COMPUTATIONALINTELLECTUAL ANALYTICAL THEORYOF COMPUTATIONAL ANALYTICAL APPROACH TOROTATING FLOW OF NON_NEWTONIAN FLUID

A combination of the computational symbolic calculation, mathematical approach and physico-mechanical model lends to a computational intellectual analytical approach developed by the author. There is a principal difference between the computer proof and the computer derivation completed by the computer, also difference between the numerical and symbolic calculations. In this investigation the computational analytical approach is extended, and an unsteady flow of non-Newtonian fluid in the gap between two rotating coaxial cylinders is studied. The Oldroyd fluid B model is used by which the Weissenberg effects are explained in a good comparison with the experiments. The governing equations are reduced to a partial differential equation of 3 rd order for the dimensionless velocity. Using the computer software Macsyma and an improved variational approach the problem with the initial and boundary conditions is then reduced to a problem of an ordinary differential equation for different approximations. The analytical solutions are given for the 1 st, 2 nd and 3 rd approximations. The present investigation shows the ability of the computational symbolic manipulation in solving the problems of non-Newtonian fluid flows. There is a possibility of that to solve the problems in mathematics and mechanics. An important conclusion can be drawn from the results that the transition from a steady state to another steady state is non-unique.

An Intelligent Approach in Design Optimzation Based on Determining Constraints Compatibility

Based on monotonicity analysis and computer symbolic manipulating technique,a procedure for determining constraints compatibility in design optimization hasbeen proposed in this paper. By using the proposed method relationshipsbetween constrains can be determined and the optimization is greatly simplifid.The method is code with intelligent production systems.