Turing-Computability Of Solutions Of Differential Equations And The Matrix

Many physicists believe that:given an initial value for the physical equation,a change it reflects in the system over time can be described with any precision by computers.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 the combined KdV equation and the fourth-order Schr(o|¨)dinger equation. Firstly,we study the certain position of their solutions by the conservation equations or the energy function.Secondly,we change the equations into the equivalent integral equations on Sobolev space by Fourier transform. Thirdly,we prove that the integral operators are computable by use of the certain position of the solution,the Schwartz functions,contraction principle and TTE.Finaly,by the computable functions constructed,we extend the solution from the internal to the entire space.These results lay the theoretical foundation for computing the solution of the combined KdV equation and the fourth-order Schr(o|¨)dinger equation exactly,and extend the application of digital computers to solve differential equations.In addition,The Turing computability of the real matrix is studied and the computable definition of the real matrix and two representations are given,and then prove they are equivalent.It is proved that some calculation results of the computable real matrix remain computable real matrix by virtue of Type-2 theory of effectivity.