New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach

• Main Authors: Wei Gao, Pundikala Veeresha, Doddabhadrappla Gowda Prakasha, Haci Mehmet Baskonus, Gulnur Yel
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Symmetry
• Subjects: q-homotopy analysis method; laplace transform; phi-four equation; caputo fractional derivative.
• Online Access: https://www.mdpi.com/2073-8994/12/3/478

This manuscript investigates the fractional Phi-four equation by using <inline-formula> <math display="inline"> <semantics> <mi>q</mi> </semantics> </math> </inline-formula>-homotopy analysis transform method (<inline-formula> <math display="inline"> <semantics> <mi>q</mi> </semantics> </math> </inline-formula>-HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and <inline-formula> <math display="inline"> <semantics> <mi>q</mi> </semantics> </math> </inline-formula>-homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach&#8217;s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations.