Reducing the PDEs to ODEs through lie vectors using the integrated factors
We reduce the PDEs to ODEs through Lie vectors as previously done through two reduction stages. Some of these ODEs have no solution. Some researchers in this step, use the SMM, power series method or Riccati equation method to solve non-solvable equations. We use the integrating factors as a tool to reduce the order and the nonlinearity in an ODE. This explores new solutions as it appears for the (2+1)-dimensional (CBS) and (3+1)-dimensional generalized BKP solutions compared results.
Ahmad, F., Tohidi, E., Ullah, M. Z., & Carrasco, J. A. (2015). Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs. Computers & Mathematics with Applications, 70(4), 624-636. https://doi.org/10.1016/j.camwa.2015.05.012
Arcak, M. (2011). Certifying spatially uniform behavior in reaction–diffusion PDE and compartmental ODE systems. Automatica, 47(6), 1219-1229. https://doi.org/10.1016/j.automatica.2011.01.010
Bridges, T. J., & Reich, S. (2001). Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Physics Letters A, 284(4-5), 184-193. https://doi.org/10.1016/S0375-9601(01)00294-8
Budd, C., Koch, O., & Weinmüller, E. (2006). From nonlinear PDEs to singular ODEs. Applied Numerical Mathematics, 56(3-4), 413-422. https://doi.org/10.1016/j.apnum.2005.04.012
Hasan, A., Aamo, O. M., & Krstic, M. (2016). Boundary observer design for hyperbolic PDE–ODE cascade systems. Automatica, 68, 75-86. https://doi.org/10.1016/j.automatica.2016.01.058
Krstic, M. (2009). Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5), 372-377. https://doi.org/10.1016/j.sysconle.2009.01.006
Li, X., & Wang, M. (2007). A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms. Physics Letters A, 361(1-2), 115-118.
Moghadam, A. A., Aksikas, I., Dubljevic, S., & Forbes, J. F. (2013). Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica, 49(2), 526-533. https://doi.org/10.1016/j.automatica.2012.11.016
Ren, B., Wang, J. M., & Krstic, M. (2013). Stabilization of an ODE–Schrödinger cascade. Systems & Control Letters, 62(6), 503-510. https://doi.org/10.1016/j.sysconle.2013.03.003
Simsen, J., & Simsen, M. S. (2011). PDE and ODE limit problems for p (x)-Laplacian parabolic equations. Journal of Mathematical Analysis and Applications, 383(1), 71-81. https://doi.org/10.1016/j.jmaa.2011.05.003
Susto, G. A., & Krstic, M. (2010). Control of PDE–ODE cascades with Neumann interconnections. Journal of the Franklin Institute, 347(1), 284-314. https://doi.org/10.1016/j.jfranklin.2009.09.005
Tang, S., & Xie, C. (2011). Stabilization for a coupled PDE–ODE control system. Journal of the Franklin Institute, 348(8), 2142-2155. https://doi.org/10.1016/j.jfranklin.2011.06.008
Tang, S., & Xie, C. (2011). State and output feedback boundary control for a coupled PDE–ODE system. Systems & Control Letters, 60(8), 540-545. https://doi.org/10.1016/j.sysconle.2011.04.011
Wang, J. M., Liu, J. J., Ren, B., & Chen, J. (2015). Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance. Automatica, 52, 23-34. https://doi.org/10.1016/j.automatica.2014.10.117
Zgliczynski, P. (2003). On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach. Journal of Differential Equations, 195(2), 271-283. https://doi.org/10.1016/j.jde.2003.07.009
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