We investigate 𝑓 (𝑅) gravity using the Noether symmetry approach. For this purpose, we consider Friedmann Robertson-Walker (FRW) universe and spherically symmetric spacetimes. The Noether symmetry generators are evaluated for some specific choice of 𝑓 (𝑅) models in the presence of the gauge term. Further, we calculate the corresponding conserved quantities in each case. Moreover, the importance and stability criteria of these models are discussed.

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The space-time fractional non-linear Phi-4 equation is a significant equation to describes the fission and fusion process that ensued in chemical kinematics, solid-state physics, astrophysical fusion plasma, plasma physics, and electromagnetic interactions, etc. The Phi-4 non-linear partial differential equation is reshaped utilizing the three different fractional-order derivatives and constructed transformations corresponding to every fractional-order derivative to convert the partial differential equation into an ordinary differential equation. A new extended direct algebraic equation method was successfully applied to extract the solitons solutions. The solitons solutions are developed with the exponential, trigonometric, rational, and hyperbolic functions including different unknown constant parameters. The graphical interpretations of obtained solutions are also depicted by allocating the feasible values to unknown constant parameters. The proposed scheme is an effective and functional scientific technique to investigate different fractional systems and models in engineering and physics referenced to real physical problems. RECEIVED