Baranwal Vipul K, Pandey Ram K, Singh Om P
Department of Applied Mathematics, Maharaja Agrasen Institute of Technology, Rohini, Delhi 110086, India.
Department of Mathematics and Statistics, Dr. Hari Singh Gaur University, Sagar 470003, India.
Int Sch Res Notices. 2014 Oct 15;2014:847419. doi: 10.1155/2014/847419. eCollection 2014.
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
我们提出了最优变分渐近方法来求解时间分数阶非线性偏微分方程。在所提出的方法中，在标准变分迭代法的校正泛函中引入任意数量的辅助参数γ₀、γ₁、γ₂…以及辅助函数H₀(x)、H₁(x)、H₂(x)…。通过最小化平方残差误差来获得这些参数的最优值。为了检验该方法，我们将其应用于求解两类重要的非线性偏微分方程：(1) 具有非线性源项的分数阶平流扩散方程和(2) 分数阶Swift-Hohenberg方程。对于第一个和第二个问题，仅需很少的迭代次数就能获得相当精确的解。
