Turing-Computability Of Solutions Of Some Partial Differential Equations

Recently people more and more are concerned of solving equation on computer .As a consequence, some software of mathematics are thriving. Nevertheless, the question whether it is always possible to solve equation on computer remains largely open. In this paper we study linear Klein—Gordon equation, heat equation and nonlinear Schrodinger equation and prove solutions of these equations are Turing computable. First, we transform Klein—Gordon equation to the integral equation by Fourier transform. Then, we prove the solution operation of the integral equation is Turing-computable. Furthermore, the question is solved accordingly. Second, we get the fundamental solution of heat equation from the theory of generalized functions. By use of computability of convolutions, we prove the generalized solution of heat equation is computable. Finally, we study computability of the solution operator of nonlinear Schrodinger equation. We mainly apply the contraction principle in analysis and properties of some spaces. By the computable functions constructed, we extend the solution from the internal to the entire space. These results enlarge the application in computing differential equations on digital computers.