Classifications of different dimensional partial differential equations and their invariant solutions via symmetry reductions and optimal systems

Irshad, Muhammad; Hussan, Mhammad; Ali, Zulfiqar; Jahangir, Iram; Sufyan, Abu; Javed, Muhammad Tayyab

doi:10.22034/cmde.2025.64053.2882

Classifications of different dimensional partial differential equations and their invariant solutions via symmetry reductions and optimal systems

Document Type : Research Paper

Authors

¹ Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 38000, Pakistan.

² Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan.

10.22034/cmde.2025.64053.2882

Abstract

The study analyzes the (1+2)-dimensional modified Breaking Soliton equation using classical symmetries and explores the (1+1)-dimensional heat equation and modified Boussinesq equation through Lie symmetry analysis and Lie subalgebras. Classical symmetries are derived from the solutions of nonlinear partial differential equations, and optimal systems are constructed using commutator relationships and adjoint representations. The research presents new invariant solutions and their graphical analyses, which are valuable for applied sciences and numerical simulations. The solutions explain phenomena such as circular membrane vibrations, heat conduction, and electromagnetic waves. Wave, contour, and patch contour solutions are used in sound, light, weather forecasting, medical imaging, and materials science. This paper provides a comprehensive analysis of the generalized single and double reduction methods, highlighting the significance of inherited symmetries at each stage of the reduction process.

Keywords

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References

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