A Parallel Solution Algorithm To Analytic Approach The Partial Differential Equation

This dissertation is composed of three parts. In the first part we introduce the background of our research, describe the partial differential equation (PDE) problem and give general solutions such as variation principle, finite element method (FEM) and Meshless method. In the second part, we do search on parallel algorithm. A brief analysis is made on evolutionary algorithm and genetic expression programming (GEP). Based on those techniques, we proposed a new parallel algorithm to solve PDE based on analytical approximation. In addition, we made some theory analysis on the algorithm. In the third part, we applied the algorithm into 4 test problems from ordinary differential, parabolic equation, elliptic equation and hyperbolic equation respectively. The experimental results of the new algorithm show the advantageous performance. The main research work and algorithm show innovative points as follows:1,Define the function warehouse and operation warehouse; it is through the Compound Calculations and Four Fundamental Operations to produce shape functions.2,Introduce oo norm as the algorithm's fitness function, make the distance between analytical and accurate solutions as the criterion of solution's level.3,Turn the function solution to a double-object (boundary problem and function solution) optimization problem. The traditional method is to search approximated results of PDE under their satisfying with the boundary constraints; in this case, the result space is very simple. In this paper, we proposed two demands that analytical solutions should approach accurate one as well as boundary constraints, therefore, the results space can be enlarged to a broadened one.4,Make analytical solutions to approach accurate outputs by using the GEP which is strong in paralleled computation. We divided a main task into several sub processes, it is through operations of Partitioning,Communication,Agglomeration and Mapping among processes to parallel computation.5,Analysis the availability of the new algorithm based on intelligence and parallelism. And it is proved to be useful by 4 tests. Due to the independency on function formats, this method is suited to parallel computing, and also deal with most PDE.