M.S. Osman

Abstract The present article uses a modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo accent="false">′</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:mfenced> </mml:math> -expansion method and the generalized Kudryashov method on Klein-Fock-Gordon (KFG) equation and receives some stable and functional solutions. The obtained results are checked by putting backwards into the physical model and are very beneficial over various existing processes. The diverse variety of stable and functional outcomes such as kink-type shape, bright and dark lump shape, bright and dark singular kinky shape, periodic bright and dark lump shape, multiple bright and dark lump shape, the lump with rough wave shape, the rough wave shape and the kinky shape are taken. The above procedure could also be employed to get stable and functional solutions for other integral and fractional nonlinear models in physics, mathematics, and engineering.