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

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	Classifications of different dimensional partial differential equations and their invariant solutions via symmetry reductions and optimal systems
Computational Methods for Differential Equations
مقاله 13، دوره 14، شماره 4، دی 2026، صفحه 1732-1748 اصل مقاله (691.33 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2025.64053.2882
نویسندگان
Muhammad Irshad^* ¹؛ Mhammad Hussan²؛ Zulfiqar Ali¹؛ Iram Jahangir¹؛ Abu Sufyan¹؛ Muhammad Tayyab Javed¹
¹Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 38000, Pakistan.
²Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan.
چکیده
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.
کلیدواژه‌ها
Classifications؛ Symmetry reductions؛ Optimal systems؛ Invariant solutions

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