The complexity of checking the correctness of the problem solution on the Turing machine There is a powerful simulation of Turing machine. At the input it takes problems of class P. At the output it gives solution for the problem. So, is it easier (in complexity theory) to test the correctness solution of the given problem than to solve this problem? If checking the solution is really easier than solving the problem, how much?

The practical side of my question is based on the fact that there is a large network of distributed computing. People for the award offer the power of their computers (their GPUs) to solve various problems from other people (from customers). We need an effective algorithm of protection against fraud when a person sends the wrong solution. If checking the correctness of the solution is really much easier than solving the problem, the customers themselves could easily check on their computers the correctness of the sent answers for which they pay. The most common problems are related to the solution of linear homogeneous integral equations and solved in a fraction of seconds, but in general the problem can be any of the class P. 
Thank you! I really hope for an answer.
What you might be looking for is an application of probabilistically checkable proofs (PCPs) for verifying computation. People have been working lately on making this practical, see for example Computational integrity with a public random string from quasi-linear PCPs, and other works by Eli Ben-Sasson and Alessandro Chiesa. At the moment this is unfortunately still a mostly theoretical concept.