Analyzing solution of the fractional Lorenz system and fractional Riccati equation via the modified Laplace power series residual method

‎The series approach is commonly used to obtain approximate analytic solutions for differential equations‎, ‎but it often converges for a short time‎. ‎To address this limitation‎, ‎a new algorithm has been developed that enables the solution to be carried out over a longer period‎. ‎The Laplace residual power series method (LRPSM) is a technique that generates a solution for fractional differential equations in terms of FOPS via simulation generalized Taylors' series in the Laplace space‎. ‎To apply the LRPSM for a long time space‎, ‎a new Modified LRPSM (MLRPSM) algorithm is introduced which divides the time into shorter intervals and applies the LRPSM to each interval‎. ‎The algorithm investigates the continuity of the solution to ensure that the obtained solution for each interval is smoothly connected to the solution for the previous interval‎. ‎The effectiveness of the proposed algorithm is demonstrated through its application to the Riccati equation and the Lorenz chaotic system‎. ‎To comprehend the physical features of studied models‎, ‎the 2D‎, ‎and 3D graphical representations of the acquired results for some parameters had been drawn‎. ‎Especially‎, ‎at the critical value of the fractional derivative‎, ‎which marks the transition of the solution behavior for the Lorenz system from a chaotic to a non-chaotic attractor‎. ‎The efficacy‎, ‎accuracy‎, ‎and feasibility of this technique are verified numerically‎. ‎From this viewpoint‎, ‎the simulations of gained results indicate that the future iterative technique is indeed robust‎, ‎effective‎, ‎and convenient in gaining the approximation solutions over a longer period of a wide range of linear and nonlinear fractional physical problems‎.