Solutions of nonlinear Murray equation for blood flow in vessels by Laplace-residual power series

Purpose This study aims to introduce a novel hybrid approach called the Laplace-Residual Power Series Method (L-RPSM) for solving fractional nonlinear problems, specifically the Murray differential equation. This method combines the Residual Power Series Method (RPSM) with the Laplace Transform (LT). Design/methodology/approach The L-RPSM is applied to fractional nonlinear problems, including the Murray equation. The method provides an efficient means of obtaining exact and approximate series solutions for fractional differential equations. Numerical and graphical results are computed for different values of the fractional order parameter μ using Mathematica software. The performance and solutions of L-RPSM are compared with other established methods (Bernoulli wavelet collocation and reduced differential transform method) to demonstrate its effectiveness. Findings The L-RPSM successfully solves two cases of the Murray equation. The results demonstrate that the proposed approach is simple, accurate, and broadly applicable. The numerical and graphical results illustrate the behavior of the L-RPSM solutions and specifically show the influence of the fractional derivative (through parameter μ) on the obtained solutions. Originality/value The primary originality lies in the novel combination of the RPSM with the LT to form the L-RPSM specifically for tackling fractional nonlinear differential equations. The study provides clear evidence of the method's simplicity, accuracy, and broad applicability through solved examples and comparisons. Furthermore, it visually demonstrates the significant impact of the fractional order derivative on the solution behavior using 2D and 3D plots.