Abstract
The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein‐Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.
Highlights
The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory 1, 2
The reliable treatment of the classical Homotopy Perturbation Method (HPM) suggested by Odibat and Momani 26 is presented for nonlinear function N u which is assumed to be an analytic function and has the following Taylor series expansion:
The reliable treatment HPM is applied to obtain the solution of the Klien-Gordon partial differential equation of arbitrary fractional orders with spatial and temporal fractional derivatives
Summary
The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory 1, 2. The HPM, proposed by He in 1998, has been the subject of extensive studies and was applied to different linear and nonlinear problems 8–13. This method has the advantage of dealing directly with the problem without transformations, linearization, discretization, or any unrealistic assumption, and usually a few iterations lead to an accurate approximation of the exact solution 13. Some other methods for series solution that are used to solve nonlinear partial differential equations of fractional order include the Journal of Applied Mathematics. Our aim here is to apply the reliable treatment of HPM to obtain the solution of the initial value problem of the nonlinear fractional-order Klein-Gordon equation of the form. T aDxβu x, t bu x, t cuγ x, t f x, t , x ∈ R, t > 0, α, β ∈ 1, 2 , 1.1 subjected to the initial condition u k x, 0 gk x , x ∈ R, k 0, 1, 1.2 where Dtα denotes the Caputo fractional derivative with respect to t of order α, u x, t is unknown function, and a, b, c, and γ are known constants with γ ∈ R, γ / ± 1
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