Computing the Solutions of the Combined Korteweg-de Vries Equation by Turing Machines

Differential equations are very popular mathematical models of real world problems. Not everydifferential equation has a well behaved solution. For those equations whose well behaved solutionsexist, we are interested in how they can be computed. Thus, the computability of the solution operatorsfor different types of nonlinear differential equations becomes one of the most exciting topics in effec-tive analysis. This answers questions of the type: is it possible to calculate the solutions of some realword problems algorithmically? The answers to these questions are unfortunately not always positive.However, there are a lot of very interesting equations whose solutions do exist and can be calculated.These equations can be called computably solvable equations, in other words, their solution operatorsare computable. This means that, there are Turing machines which can transfer the initial data to thesolutions of the equation in some particular spaces. For example, Klaus Weihrauch and Ning Zhong [7]have shown that the initial value problem of Korteweg-de Vries (KdV) equation posed on the real lineR: u