Abstract
In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. The method has been successively provided for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ħ. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.
Highlights
In this paper, we consider the fractional nonlinear Klein-Gordon equation∂αu ∂ u ∂tα – ∂x + au + bg(u) = f (x, t),∂ u(x, ) = f (x), u(x, ) = g(x), ∂t≤ x, t
The Klein-Gordon equation plays an important role in mathematical physics
The homotopy perturbation method (HPM) has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation which can be used as a numerical algorithm [ ]
Summary
≤ x, t < , ≤ α < , where u is a function of x and t, a and b are real, g is a nonlinear function, and f is a known analytic function. Analytical approach that can be applied to solve nonlinear differential equations is to employ the homotopy analysis method (HAM) [ – ]. The homotopy analysis method is applied to solve linear and nonlinear fractional partial differential equations (fPDEs) [ ]. Xu et al [ ] applied HAM to linear, homogeneous oneand two-dimensional fractional heat-like PDEs subject to the Neumann boundary conditions They implemented relatively new, exact series method of solution known as the differential transform method for solving linear and nonlinear Klein-Gordon equations [ ]. The Riemann-Liouville fractional derivative is mostly used by mathematicians, but this approach is not suitable for physical problems of the real world since it requires the definition of fractional order initial conditions which have no physically meaningful explanation yet. These equations can be solved using software such as Maple, Mathlab and so on
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