Turing-Computability Of Solutions Of Two Nonlinear Partial Differential Equations

As an important part of modern mathematics, nonlinear partial differential equation has always been an important field of study, but its solution is a problem in the research process, and blocking the application of it. Therefore, the study of the existence of solutions and its computability problems have become important topic. And we urgently need to explore the Turing-computability of the solution.In this paper, we mainly discuss the Turing-computability of the solution operation of nonlinear Pseudo-parabolic equation and generalized shallow water wave equation. First two chapters, we introduce the emergence and development history of computability, some basic concepts, theorems, lemmas and representation spaces in Type-2theory of effectivity. In the next two chapters, we discuss the computability of the solution operators of the Pseudo-parabolic equation and generalized shallow water wave equation. First, we change the differential equations into equivalent integral equations by Fourier transform and Duhamel principle. Next, by contraction principle, TTE theory, conservation of equations and properties of Schwartz functions,we prove that the integral operators are computable in a short interval.Finally by constructing computable function, we extend the solution frompartial internal to the entire space, so we can get the conclusion that the solution of the original equation are also computable.