Time-Fractional Shallow-Water Model for Atmospheric Fluid Layers: Analysis and Semi-Analytical Solution

Oscillatory motions in stratified atmospheric fluid layers significantly influence weather and climate dynamics. Shallow-water equations effectively describe these motions. This study extends the shallow-water model to the time-fractional domain using the conformable fractional derivative. This derivative preserves the local differential structure while introducing tunable time scaling in the dynamics. Approximate analytical solutions were obtained using the conformable Laplace Adomian Decomposition Method (CLADM). This method combines the conformable Laplace transform with Adomian decomposition. Numerical results for fractional orders ϑ∈(0, 1] demonstrate that the fractional parameter systematically modulates the system dynamics. The solutions at ϑ=1 align well with the established Elzaki Adomian Decomposition Method (EADM), Homotopy Analysis Method (HAM), Fractional Reduced Differential Transform Method (FRDTM), and reference numerical solutions (NUM). This fractional framework offers a flexible approach to modeling atmospheric fluid-layer dynamics.