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**Solving nonlinear fractional partial differential equations using the homotopy analysis method.**
*(English)*
Zbl 1185.65187

Summary: The homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional Korteweg-de Vries (KdV), \(K(2,2)\), Burgers, Benjamin-Buna-Mahony (BBM)-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like \(B(m,n)\) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35R11 | Fractional partial differential equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

35C10 | Series solutions to PDEs |

### Keywords:

analytical solution; coupled KdV and Boussinesq-like \(B(m; n)\) equations; fractional KdV; \(K(2,2)\); Burgers; BBM-Burgers; cubic Boussinesq; fractional partial differential equations; homotopy analysis method; series solution; numerical examples
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\textit{M. Dehghan} et al., Numer. Methods Partial Differ. Equations 26, No. 2, 448--479 (2010; Zbl 1185.65187)

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