Exact Solutions > Interesting Papers > N.A. Kudryashov. Seven common errors in finding exact solutions of nonlinear differential equations > 2. First error... Seven common errors in finding exact solutions of nonlinear differential equations

 (2.1) 
Substituting in the form [9]
 (2.2) 
into the KdV  Burgers equation Eq.(2.1) and finding the order of the pole for the solution and functions and we obtain the transformation [12]
 (2.3) 
Assuming [12]
 (2.4) 
(where C_{1} and C_{2} are arbitrary constants), we have the system of algebraic equations with respect to k and . Solving this algebraic system we have the values of k and as follows
 (2.5) 
For these values of k and the solitary wave solutions for the KdV  Burgers equation takes the form [12]
 (2.6) 
We can see that solution (2.6) satisfies the nonlinear ordinary differential equation in the form
 (2.7) 
We have found exact solution of the KdV — Burgers equation in essence using the travelling wave solutions although we tried to look for more general solutions. Many researches look for the exact solutions taking directly into account the travelling wave solutions.
Example 1b. Application of the Riccati equation as the simplest equation (the first variant) [22, 23, 24, 25, 26].
Solution (2.6) can be written as
 (2.8) 
Consider the expression
 (2.9) 
Taking the derivative of the function H with respect to we get
 (2.10) 
We obtain that solution (2.8) can be written in the form
 (2.11) 
Eq.(2.11) means that one can look for the solution of the KdV  Burgers equation using the expression
 (2.12) 
where H is the solution of Eq.(2.10). It is seen, that the application of the simplest equation method with the Riccati equation gives the same results as the truncated expansion method.
Example 1c. Application of the tanh  function method [27, 28, 29, 30].
Expression (2.9) can be transformed as follows
 (2.13) 
From (2.13) we have that solution (2.8) of the KdV  Burgers equation can be written in the form
 (2.14) 
Substituting and into solution (2.14) we have the solution of the KdV  Burgers equation in the form [12]
 (2.15) 
We obtain that the solution of the KdV  Burgers equation can be found as the sum of the hyperbolic tangents
 (2.16) 
where m is unknown parameter.
We have obtained that the tanh  function method is equivalent in essence to the truncated expansion method. As this fact takes place we can see that the maximum power of the hyperbolic tangent in (2.16) coincides with the order of the pole for the solution of the KdV  Burgers equation.
Example 1d. Application of the Riccati equation as the simplest equation (the second variant). [22, 23, 24].
Note that the function
 (2.17) 
is the general solution of the Riccati equation
 (2.18) 
From (2.16) we obtain, that we can look for the solution of the KdV  Burgers equation taking into account the formula
 (2.19) 
where Y satisfies Eq.(2.18). The maximum power of the function Y in (2.19) coincides with the pole of the solution for the KdV  Burgers equation.
Example 1e. Application of the method [31, 32, 33].
Taking into account the transformation
 (2.20) 
we reduce Eq.(2.18) to the linear equation
 (2.21) 
As this fact takes place expression (2.19) can be written in the form
 (2.22) 
This means, that we can search for the exact solutions of the KdV  Burgers equation in the form (2.22), where satisfies Eq.(2.21). We obtain, that application of the method is equivalent to application of the simplest equation method with the Riccati equation, to the tanh  function method and to the truncated expansion method.
Note that the method follows directly from the truncated expansion method because solution (2.3) can be written as
 (2.23) 
Assuming that the function satisfies linear equation of the second order
 (2.24) 
where m_{0}, m_{1} and m_{2} are unknown parameters we have the expression (2.22) for . So, the  method coincides with the truncated expansion method if we use the travelling wave solutions.
Example 1f. Application of the tanh — coth method [34, 35, 36].
Assuming in (2.16) , we obtain the formula
 (2.25) 
This means that we can search for the solution of the KdV  Burgers equation in the form (2.25).
Using the identity for the hyperbolic functions in the form
 (2.26) 
and substituting (2.26) into (2.25) we have the  method to look for the solitary wave solutions of the KdV  Burgers equation
 (2.27) 
We have obtained, that the tanh  coth method is equivalent to the truncated expansion method too and we cannot find new exact solutions of the KdV  Burgers equation by the tanh  coth method.
Example 1g. Application of the Exp  function method [37, 38, 39, 40, 41].
Solution (2.8) of the KdV  Burgers equation can be written in the form
 (2.28) 
This solution can be found if we look for a solution of the KdV — Burgers equation using the Exp  function method in the form
 (2.29) 
Using the Exp  function method to search for the exact solutions of the KdV  Burgers equation one can have again the solitary wave solutions (2.3) with the functions (2.4) and the parameters (2.5). Studying the application of the Exp  function method we obtain that this method provides no new exact solutions of nonlinear differential equations in comparison with other methods. Moreover the Exp  function method do not allow us to find the order of the pole for the solution of nonlinear differential equation [42]. Usually the authors use a few variants of fractions with the sum of exponential functions to look for the solitary wave solutions.
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