Computer Algebra To Solve The Differential Equation Method And Its Machine Realization

Computer Algebra is also called symbolic or algebra computation(or mathematics mechanization).The subject tries to deal with mathematics in a constructive and algorithmic manner so that the reasoning become mechanical, automated and so much as possible as to be intelligence lacking, with the result of lessening the painstaking heavy brain-labor. It is a new development orientation in the field of mathematics and computer to conduct science calculation by computer. For a long time, the mathematicians and computer scientists have dreamed of replacing human brain with computer to conduct symbolic operation and any kind of mathematical processes, leading the mathematics to a mechanizing way .Nowadays, differential equations is playing a more and more important role in modern science and engineering field, and at the same time, in practice, most solution of differential equations are extremely sophisticated but crucial. Seeking for the simple and practicable computation method is of great concern in the practical application and research of the solution of the model of the differential equations.In this paper, it researches the theories, methods and computation mechanization of solving the differential equations with computer algebra knowledge and computer algebra systems. We research the tanh-method on the base of Wu-characteristic method, and give the interchangeable process of solving the more complicated partial differential equations in the third chapter. Furthermore, we give the algorithm of the initial problems of differential equations which include singularity and its procedure in Maple by the use of the method of Adomian decomposition and accomplish the solution which is unable to achieve by hand-work in the third chapter. We research the Power series and the Pade series approximated solution of differential-algebraic equations in the fourth chapter. It gives the solving process by computer and its figures of exact solution and approximated numerical solution of Pade series under the same coordinate axis and we give the analysis and comparison about asymptotic approximation of them. In the fifth chapter, we give the method of solving the partial differential equations with the combine use of symbolic and numerical method, which is a new way of solving the extremely complicated partial differential equations. Its significance is to conduct us to seeking for the new idea and new sight of solving the complicated partial differential equations. In the process of solving, we use the computer algebra knowledge and Maple or Maple software of computer systems. By the comparison of solving the differential equations with computer and hand-work, the tremendous advantage is opened out and the great practical value is reflected by the using of computer algebra.