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 every differential equation has a well behaved solution. For those equations whose well behaved solutions exist, we are interested in how they can be computed. Thus, the computability of the solution operators for different types of nonlinear differential equations becomes one of the most exciting topics in effective analysis. This answers questions of the type: is it possible to calculate the solutions of some real word 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 operators are computable. This means that, there are Turing machines which can transfer the initial data to the solutions 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 line R: ut +uux +uxxx = 0, t,x ∈R, u(x,0) = φ(x) has a computable solution operator. In this paper, we investigate a variation of Korteweg-de Vries equation: ut + uux + uux + uxxx = 0. This is often called Combined Korteweg-de Vries (CKdV) equation. The Combined KdV equation is also an important equation which is frequently used as a mathematical model in physics, hydrodynamics, biological and chemical fields. We will show that the solution operator of the CKdV equation is also computable. This extends the results of [2, 7]. The proof of the main theorem is given in Section 2.