Bright optical solution for fractional Lakshmanan–Porsezian–Daniel with spatio temporal dispersion by improved Adomian decomposition method

The objective of this paper is investigating the numerical solution and optical phenomena of highly nonlinear fractional-order partial differential equation namely fractional order Lakshmanan–Porsezian–Daniel model, incorporating spatio-temporal dispersion effects. The fractional derivative is considered in Caputo sense because this is the most crucial tool for working with integer order models in a fractional sense under the right subsidiary conditions, it has many benefits. To achieve the objective of this study, the Improved Adomian decomposition method has been utilized. This method is known for effectively handling nonlinearities and fractional derivatives iteratively. The obtained soliton solutions are presented graphically. We conduct a comparative analysis at \(\rho = 1\) $\rho =1$ , focusing on the bright optical soliton solutions. To the best of our knowledge, there have not been previously reported identical or related findings. By comparing the obtained results with the exact solution, the accuracy of the considered method is examined. The applied method is explicit, efficient, and user-friendly for managing a wider range of fractional order nonlinear models, as demonstrated by the numerical outcomes.