Reducing the PDEs to ODEs through lie vectors using the integrated factors

  • Tony Brian Fogel Cornell University, Ithaca, US
  • Steven Segan Lucky Cornell University, Ithaca, US
  • James Arnold Mandy Columbia University, New York, US
  • Berth Bryan Wellbirth Columbia University, New York, US
Keywords: factors, integrating, lie, solutions, transformation


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.


Download data is not yet available.


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.

Arcak, M. (2011). Certifying spatially uniform behavior in reaction–diffusion PDE and compartmental ODE systems. Automatica, 47(6), 1219-1229.

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.

Budd, C., Koch, O., & Weinmüller, E. (2006). From nonlinear PDEs to singular ODEs. Applied Numerical Mathematics, 56(3-4), 413-422.

Hasan, A., Aamo, O. M., & Krstic, M. (2016). Boundary observer design for hyperbolic PDE–ODE cascade systems. Automatica, 68, 75-86.

Krstic, M. (2009). Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5), 372-377.

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.

Ren, B., Wang, J. M., & Krstic, M. (2013). Stabilization of an ODE–Schrödinger cascade. Systems & Control Letters, 62(6), 503-510.

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.

Susto, G. A., & Krstic, M. (2010). Control of PDE–ODE cascades with Neumann interconnections. Journal of the Franklin Institute, 347(1), 284-314.

Tang, S., & Xie, C. (2011). Stabilization for a coupled PDE–ODE control system. Journal of the Franklin Institute, 348(8), 2142-2155.

Tang, S., & Xie, C. (2011). State and output feedback boundary control for a coupled PDE–ODE system. Systems & Control Letters, 60(8), 540-545.

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.

Zgliczynski, P. (2003). On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach. Journal of Differential Equations, 195(2), 271-283.

How to Cite
Fogel, T. B., Lucky, S. S., Mandy, J. A., & Wellbirth, B. B. (2019). Reducing the PDEs to ODEs through lie vectors using the integrated factors. International Research Journal of Management, IT and Social Sciences, 6(5), 193-203.