The homotopy analysis method (HAM) is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained. The efficiency and accuracy of the method are verified by using two famous examples: coupled Burgers and mKdV equations. The obtained results show that the homotopy perturbation method is a special case of homotopy analysis method.

Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives, and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow. The differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [

The HAM offers certain advantages over routine numerical methods. Numerical methods use discretization which gives rise to rounding off errors causing loss of accuracy and requires large computer memory and time. This computational method yields analytical solutions and has certain advantages over standard numerical methods. The HAM method is better since it does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time.

In this paper, we extend the application of HAM to discuss the explicit numerical solutions of a type of nonlinear-coupled equations with parameters derivative in this form:

The paper has been organized as follows. Notations and basic definitions are given in Section

A real function

The Riemann-Liouville fractional integral operator (

For

For the concept of fractional derivative, there exist many mathematical definitions [

The Mittag-Leffler function

To describe the basic ideas of the HAM, we consider the operator form of (

By means of generalizing the traditional homotopy method, Liao [

The parameters

In order to illustrate the method discussed above, we consider the nonlinear coupled Burgers equations with parameters derivative in an operator form:

The comparison of the results of the HAM (

Realtive error | ||||
---|---|---|---|---|

10 | 0.01 | −5.3861e-001 | −0.9002497662 | |

10 | 0.02 | −5.3325e-001 | −0.8912921314 | |

10 | 0.03 | −5.2794e-001 | −0.8824236265 | |

5 | 0.01 | −0.9493828183 | −0.9493828187 | |

5 | 0.02 | −0.9399363019 | −0.9399363019 | |

5 | 0.03 | −0.9305837792 | −0.9305837793 | |

-2 | 0.01 | −0.5386080102 | −0.5386080104 | |

-2 | 0.02 | −0.5332487712 | −0.5332487712 | |

-2 | 0.03 | −0.5279428571 | −0.5279428572 |

The comparison of the 6th-order HAM and exact solution with

Explicit numerical solutions with

Explicit numerical solutions with

As suggested by Liao [

The

This example has been solved using homotopy perturbation method [

In order to illustrate the method discussed above, we consider the nonlinear coupled mKdV equations with parameters derivative in an operator form:

For application of homotopy analysis method, in view of (

The absolute error of the 6th-order HAM and exact solution with

The comparison of the results of the HAM (

−15 | 0.002 | ||

−12 | 0.002 | ||

−6 | 0.002 | ||

6 | 0.002 | ||

12 | 0.002 | ||

15 | 0.002 |

The comparison of the 6th-order HAM and exact solution with

Explicit numerical solutions with

Explicit numerical solutions with

As suggested by Liao [

The

This example has been solved using homotopy perturbation method [

In this paper, based on the symbolic computation MATLAB, the HAM is directly extended to derive explicit and numerical solutions of the nonlinear coupled equations with parameters derivative. HAM provides us with a convenient way to control the convergence of approximation series by adapting