An efficient q-procedures to solve q-generalized quintic complex Ginzburg-Landau equations

Abstract In this work, q-Residual power series method (q-RPSM) is extended for the first time to solve q-generalized quintic complex Ginzburg-Landau (q-GCGL) equations. In 2019, Arqub (2019, Fundamenta Informaticae 166 87–110), has investigated the solutions of time fractional Schr <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>o</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̈</mml:mo> </mml:mrow> </mml:mover> </mml:math> dinger equations without q-derivative by using RPSM. However, in this article, the RPSM procedure is extended to non-linear q-fractional partial differential equations and thus some useful results are obtained under the definition of q-derivative, q-gamma function, q-Caputo derivative and q-factorial function. The solutions to few numerical examples under the q-Caputo operator are presented. In the current procedure, the targeted q-problems are first converted into system of two non-linear q-fractional partial differential equations and then solved by using q-RPSM and q-LADM. In particular, we considered two illustrative examples of fractional q-GCGL for analytical solutions using q-RPSM and q-LADM. The numerical simulations for the suggested numerical problems are done successfully. It is also investigated that higher terms series solution have the higher degree of accuracy. In the current q-RPSM and q-LADM simulations, the higher accuracy is achieved by taking sixth terms series solution. The fractional-order solutions are convergent toward integer-order solution as the fractional-order approaches to integer-order. The solutions curve at different q-values is also calculated and obtain the useful dynamics of the targeted problems. The q-analogy can be used effectively in the field of quantum physics and thus keep the higher norm for researchers working in the field.