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In this paper, by using the theories and methods of mathematical analysis and computer algebra, a new algorithm of symbolic integral was established, a new Maple procedure myint for it was established, too. And it was applied to the automatic proving the mean-value theorems for derivatives. The results indicated that the algorithm of symbolic integral had been established in this paper had advantage of simple idea, excellent property for operation and powerful competence. This would be useful for the problem of solving differential equations, automatic proving some mathematical theorems, and so on. [Copyright &y& Elsevier]

Abstract: In this paper, by using the theories and methods of mathematics analysis, ordinary differential equation and computer algebra, a reliable algorithm for solving ordinary differential equation (ODE) was established, and a new Maple procedure mechodeproc was established, too. The results of the examples indicated that the procedure mechodeproc based on Taylor polynomial method was simple and effective, and could provide an accuracy approximate solution or exact solution of the ordinary differential equation. This would be useful for solving differential equation. [Copyright &y& Elsevier]

Abstract: In this paper, by using theories of stochastic process and computer algebra, a reliable algorithm for computing the high-order transition probability matrix of finite Markov chain is established, a new mechanical procedure markovproc is established, too. The procedure markovproc give not only the exact expression of the Markov chains, but also the limiting probability. Some examples are presented to illustrate the implementation of the algorithm. [Copyright &y& Elsevier]

Abstract: Resultant is a classical algebraic tool for determining whether or not a polynomial systems have a common root without explicitly solving for the roots. But, the complexity of resultants can be very high in practice. In this paper, by using the theories and methods of computer algebra, two reliable algorithms for computing multi-resultant problems-Dixon resultant and Macaulay resultant were established, two new Maple procedures dixonres and macaulayres were established, too. Some examples are presented to illustrate the implementations of the programs. This would be useful for solving a system of polynomial equations. [Copyright &y& Elsevier]

Abstract: In this paper, by using the theories and methods of mathematical analysis and computer algebra, a reliable algorithm of Padé approximation method for solving nonlinear singular perturbation problems was established, a new Maple procedure spp was established, too. The procedure spp give not only the approximate analytic solutions but also the numerical solutions of SPP. Six examples are presented to illustrate the implementation of the program. The method is very simple and effective for most of singular perturbation problems. This would be useful for solving differential equations. [Copyright &y& Elsevier]