Parallel Solving For Nonlinear Differential Equations Based On The Riccati Equations And Development Of The Online Knowledge Database For Equations

Nonlinear evolution equations play a very important role in the field of nonlinear science.The exact analytical solutions of these equations,especially the traveling wave solutions,can accurately describe the inherent laws of many physical phenomena,such as vibration,propagating waves and solitons and so on.With the rapid development of computer science and symbolic calculation methods,many methods for constructing analytical solutions of nonlinear evolution equations,such as hyperbolic tangent function method and elliptic function method,have emerged.These algorithms are direct algebraic methods.However,due to the accuracy of symbolic computation and the rapid expansion of expressions,the direct algebraic method has the problem of low computational efficiency.In order to improve the efficiency of calculation,this paper applies the idea of parallel computing to the process of solving nonlinear evolution equations by direct algebra methods.Based on the Riccati equations method and the idea of parallel computation,a new parallel algorithm for constructing exact traveling wave solutions of nonlinear evolution equations is proposed.In particular,in the process of solving the parallel equation,we use both polynomial factorization and load balancing technology to effectively improve the utilization of multiple CPUs.Moreover,compared to other methods,we can get more solutions by polynomial factorization and running time constraints.The parallel algorithm proposed in this paper has been implemented on Maple 18 and encapsulated into a package with a flexible interface and input and output.By applying the software to some specific examples and comparing the results,it is verified that the parallel algorithm proposed in this paper not only improves the efficiency of the algorithm,but also has better ability than the original algorithm.From natural sciences to social sciences,almost all subject areas are increasingly inclined to use differential equation to solve problems.In order to facilitate the learning of differential equation as well as academic exchange and cooperation of scholars and engineering technicians in different fields,we have developed an online open system of knowledge database.The knowledge database uses a lot of new technologies to improve the storage,editing and display of formulas,besides,all formulas are displayed using two-dimensional formula.In addition,for each equation,we not only use a record to save its important information,such as equation type,equation name,equation expression,etc.,but also establish an equation homepage where the basic information,background knowledge and the relevant research results are shown.We also built a comment area for academic discussion in the equation homepage.The system is not only fully functional,but also quick in response to a variety of operations with a good user experience.