Computing The Solution Of Two Kinds Of Nonlinear Development Equation With Arbitrary Precision On Turing Machines

In engineering and other scientific computation there are a lot of practical applications which are related to finding solutions of some kind of differential equations.It is always a great challenge for mathematicians to determine whether the equations of certain type have solution and,if it is the case,how these solutions can be computed.Unfortunately,this task cannot always be satisfactorily fulfilled for all equations.However,there are a lot of special equations whose solutions do exist and can be exactly and precisely specified.Therefore,finding the computability of solution operators of some partial differential equations is very important and practical significant.In this paper,we study the computability of the solution operators of a kind of Cauchy problem and KdV-Burgers equation. Firstly,we change the equations into the equivalent integral equations on H~s(R)by Fourier transform.Secondly,we prove that the integral operators are computable by using the Schwartz functions contraction principle and TTE.Thirdly,by the computable functions constructed,we extend the solution from the internal to the entire space.These results extend the application of digital computers to solve differential equations and lay the theoretical foudation to the application of the two equations.The method can be extended to other type of the differential equations.