bbfebcf4-36cc-46f8-b96b-7bee410e327b20210202053744188wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS1991-874710.37394/232011http://wseas.org/wseas/cms.action?id=4006212202021220201510.37394/232011.2020.15http://wseas.org/wseas/cms.action?id=23180On Solutions of the Nonlinear Heat Equation Using Modified Lie Symmetries and Differentiable Topological ManifoldsJacobManaleDepartment of Mathematical Sciences, University of South Africa, Corner Christiaan de Wet Street and Pioneer Avenue, 1709 Florida, Johannesburg, Gauteng Province, REPUBLIC OF SOUTH AFRICAIn this paper, we present three simple analytical techniques for obtaining solutions of the nonlinear heat equations. The heat equations, both linear and nonlinear, are very important to the mathematical sciences. This is because they are reduced forms of many models, hard to solve directly. The techniques are based on Lie’ symmetry group theoretical methods. The first is the pure Lie approach, followed by our modified Lie approach. The third is our differentiable topological manifolds approach. As an application, we determine the separation distance, in the quantum superposition principle, relevant to nanoscience.1019202010192020132139https://www.wseas.org/multimedia/journals/mechanics/2020/a305111-413.pdf10.37394/232011.2020.15.15https://www.wseas.org/multimedia/journals/mechanics/2020/a305111-413.pdfLie S., On integration of a class of linear partial differential equations by means of definite integrals, Arch. Math., Vol.3, 1881, pp. 328-368Ovsiannikov, L.V., Group properties of nonlinear heat equation, Dokl. AN SSSR, VOL.125, NO.3, 1959, pp. 492–495.10.14311/ap.2020.60.0098Amlan K.H., Andronikos Paliathanasis, P. and Leach, P.G.L., Similarity solutions and conservation laws for the Beam Equations: a complete study, math-ph, 2020, arXiv.10.3934/dcdss.2018040Aeeman F., Mahomed, F.M., Khalique, C.M., Conditional symmetries of nonlinear third-order ordinary differential equations,Discrete & Continuous Dynamical Systems – S, VOL.11, 2018, pp. 655 10.1080/10586458.2018.1516582Wafo Soh, C., Automatic classification of automorphisms of lower-dimensional Lie algebras, Experimental Mathematics, 2017, DOI : 10.1080/10586458.2018.1516582.10.1017/s0956792517000055Anco, S. and Kara, A.Symmetry-invariant conservation laws of partial differential equations,European Journal of Applied Mathematics. VOL. 1, NO. 29, 2020, pp. 78-11710.1016/j.jksus.2016.07.001M.A. Latif, M.A., Dragomir, S.S, and Momoniat, E. Some q-analogues of Hermite–Hadamard inequality of functions of two variables on finite rectangles in the plane, Journal of King Saud University – Science, VOL. 29, NO. 29, 2017, pp. 263-273. 10.33048/semi.2020.17.039Gainetdinova, A. and Gazizov, R., Integrating of systems of two second-order ordinary differential equations with a small parameter that admit four essential operators, Sibirskie Elektronnye Matematicheskie Izvestiya, NO. 17, 2020, pp. 604-614, DOI: 10.33048/semi.2020.17.039. 10.14311/ap.2020.60.0098Amlan K.H., Andronikos Paliathanasis, P. andLeach, P.G.L., Similarity solutions and conservation laws for the Beam Equations: a complete study, math-ph, 2020, arXiv. 10.1002/mma.6333Edelstein, RM, Govinder, KS. On the method of preliminary group classification applied to the nonlinear heat equation u t = f ( x , u x ) u x x + g ( x , u x ) . Math Meth Appl Sci., 2020, 43: 5927? 5940. https://doi.org/10.1002/mma.6333 Manale, J.M., On Errors in Euler’s Formula for Solving ODEs, International Journal of Mathematical and Computational Methods, VOL.5, 2020, pp.1-3. 10.1088/1742-6596/1564/1/012021Manale, J.M., (2020). On errors in Euler’s complex exponent and formula for solving ODEs, Journal of Physics: Conference Series, 2020, DOI: 1564. 012021. 10.1088/1742-6596/1564/1/012021.