A reliable spectral method based on Andre Jeannin polynomials for solving a nonlinear fractional-order Rosenau–Hyman equation with Caputo–Hadamard derivative

The Rosenau–Hyman equation is a Korteweg–De Vries (KdV)-type equation that admits compacton solutions. It is named after Philip Rosenau and James M. Hyman, who introduced it in their 1993 study of compactons. In this paper, we consider the fractional-order Rosenau–Hyman equation with the Caputo–Hadamard derivative, which models nonlinear wave dynamics in dispersive, memory-dependent media, and we develop an efficient spectral method based on André Jeannin polynomials. To this end, the highest-order partial derivative of the unknown solution is expressed as a truncated series of multivariable André Jeannin polynomials, and the terms of the nonlinear fractional-order Rosenau–Hyman equation are approximated accordingly. After performing the necessary integration, we derive an expression for the approximate solution of the original equation. By applying the Caputo–Hadamard derivative to this expression, an approximation for the fractional derivative is obtained based on the André Jeannin basis polynomials. We then prove that the numerical scheme converges to the exact solution. To validate the proposed method, numerical results are compared with those obtained using other approaches on representative test problems.