Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Abstract

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts.

Keywords

• G ′ G ′ + G + A -expansion method%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> the <named-content content-type="mml:math"><inline-formula> <mml:math id="mm54444"> <mml:semantics> <mml:mrow> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>G</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-expansion method
• the conformable fractional derivative
• the time-fractional Klein–Gordon equation
• the time-fractional Sharma–Tasso–Olever equation
• the time-fractional Clannish Random Walker’s Parabolic equation