This study explores the solutions of fourth-order
Lane–Emden–Fowler (LEF) equations by employing a refined Modified
Adomian Decomposition Method (MADM). We introduce a novel framework
that features seven specialized differential operators, specifically developed
and utilized to analyze the equations under specific initial and boundary
conditions. Our findings demonstrate that the solutions derived from this
approach not only effectively converge to the exact solutions but also offer
unparalleled accuracy and reliability. A key strength of this methodology lies
in its exceptional flexibility; solutions can be accurately obtained by applying
at least one of these newly developed operators. This work significantly
enhances our comprehension of these intricate equations and highlights the
remarkable efficacy of the MADM in yielding precise solutions across diverse
scenarios, thereby establishing a robust and versatile analytical tool

This study explores the solutions of fourth-order
Lane–Emden–Fowler (LEF) equations by employing a refined Modified
Adomian Decomposition Method (MADM). We introduce a novel framework
that features seven specialized differential operators, specifically developed
and utilized to analyze the equations under specific initial and boundary
conditions. Our findings demonstrate that the solutions derived from this
approach not only effectively converge to the exact solutions but also offer
unparalleled accuracy and reliability. A key strength of this methodology lies
in its exceptional flexibility; solutions can be accurately obtained by applying
at least one of these newly developed operators. This work significantly
enhances our comprehension of these intricate equations and highlights the
remarkable efficacy of the MADM in yielding precise solutions across diverse
scenarios, thereby establishing a robust and versatile analytical tool