Algebraic Approximation And Its Application In Solutions Of Differential Equations

The theory of algebaric functions is an old branch of mathemathics. The therory attracted many excellent mathematicians in the late 18th century. But from that time it seemed to have been forgotten for a long time untill it arised again and taking a new form , and being connected with many other important mathematical problems. The classical theory of algebaric function studies the rational function ψ(x, y) of x and y which are connected by the equationf(x,y)=0, (1)where f(x, y) is a polynomial of x and y . In the history, mathematicians were motivated to develop the theory by their tring to give the result ofψ(x,y) (2)in finite terms. The key to the problem is to choose proper transforms likex=ψ1(u,v),y=ψ2(u,v) (3)where ψ1,ψ2 are both rational functions, so that the calculus (2) becomes a calculus of a single variable. The property of calculus (3) that it can be calculated in finite terms is indepedent of the tansforms (3). From this problem, an idea initiated, that is to sdudy the invariant properties of tansforms like (3). An important case is when tansforms (3) and their inverse tansforms are both rational functions. This kind of transforms are called birational transforms. So the classical theory of algebaic functions can be seen as an geometric system in Klein sense. In numerical mathematics, it is interesting and of importance to study the theory of algebraic functions and its applications. But so far this kind of study seems to be neglected. It is known that polynomials or spline functions are often used as approximation functions. Some times, rational functions and rational splines are also used as appoximation tools. However algebaric functions is a kind of generalization of rational functions because in some cases, algebaric functions reduce to rational functions. In the thesis, we study the applications of algebaric approximation in numerical mathematics, especially in the numerical solution of differential equations. the main results are as follows:1.The definition of algebraic approximant of order [n, m] to a given analyticfunction is given. The conditions for its existence as well as its connection with Pade approximant are also studied.2.The exponential function exp(z) satisfies the conditionexp(a+b) = exp(a) exp(b) (4)so it has good properties in practical application. In the thesis, we give the algebraic approximants to exp(z) of order [1,2] , [2,2] , [1,3] that satisfies (4), and give the error estimates of these approximants.3.The approximation to exp(z) is of great importance in numerical solution of differential equations. A approximation to exp(z) often leads to a difference scheme for differential equations. In the thesis, we give many difference schemes for the initial problemby the use of algebraic approximants to exp(z). Some of these schems are nonlinear multistep schemes. The covergence and stability of these schems are studied. we also give many difference schemes for some partial differential equations.4.Another study for problem (5) is to observe what its solution would be like when t So problem (5) can be seen as a dynamic system. In the thesis, we give some numerical methods for these dynamic systems by using the results of algebraic approximations. and we prove that these methods do not give rise to spurious orbits. Some numerical experiments are also given.5.In mathematical physics, the Hamiltonian system plays a important role.In the thesis, we put forward some symplectic difference schemes for numerically solving linear Hamiltonian systems by using the algebraic approximants to exp(z).7. The Numerical approximation to the solution of stiff ODE equations is an very important problem in numerical mathematics that far from being solved. In the thesis, we develop some explicit and A-stable methods for the problem. Numerical experiments are also given.8.We prove the existence of the b...