A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

By Shijun Liao, Y. Tan. 2007

474 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b4b15dda8c2a3c35c819830056fd01a8a398185d

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Citations in the corpus

A general approach to get series solution of non-similarity boundary-layer flows (2009, methodology): “Besides, the order of L might be even not equal to 3, as pointed out by Liao and Tan [41].”
Homotopy Analysis Method in Nonlinear Differential Equations (2012, methodology): “For example, the two-dimensional 2nd-order Gelfand equation was solved by means of the HAM in a rather easy way by transferring it into an infinite number of linear PDEs governed by a very simple 4th-order linear operator, as shown by Liao and Tan (2007).”
A one-step optimal homotopy analysis method for nonlinear differential equations (2010, methodology): “The introduction of the convergence-control parameter h greatly improves the homotopy analysis method in theory, as shown by Liao and Tan [5], Abbasbandy [7], Sajid and Hayat [8], and Liang and Jeffrey [9].”
On Convergence of q-Homotopy Analysis Method (2013, unknown): “In a series of papers[3-14], Liao developed and applied the homotopy analysis method(HAM) to deal with a lot of nonlinear problems.”
An optimal homotopy-analysis approach for strongly nonlinear differential equations (2010, background): “In fact, the guarantee of the convergence of homotopy-series solutions provides us much larger freedom to choose the auxiliary linear operator L and initial guess: with such kind of guarantee in the frame of the HAM, a nonlinear ODE with variable coefficients can be transferred into a sequence of li”
Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation (2009, unknown): “[12], Bataineh et al [14], Van Gorder et al [15], Hayat and Sajid [16][17], and Song et al [18].”
On the relationship between the homotopy analysis method and Euler transform (2010, methodology): “Thereafter, the HAM has been improved step by step [14–20] and has been widely applied in science, engineering and finance [21–29].”
Notes on the homotopy analysis method: Some definitions and theorems (2009, methodology): “For details, please refer to Liao and Tan [26].”