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Zbl 1185.65187
Dehghan, Mehdi; Manafian, Jalil; Saadatmandi, Abbas
Solving nonlinear fractional partial differential equations using the homotopy analysis method. (English)
[J] Numer. Methods Partial Differ. Equations 26, No. 2, 448-479 (2010). ISSN 0749-159X; ISSN 1098-2426/e

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 2000:: *65M70 Spectral, collocation and related methods (IVP of PDE)
35R11
35Q53 KdV-like equations
35C10 Series solutions of PDE

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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