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

https://doi.org/10.21744/irjmis.v6n5.732

Authors

  • 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

Abstract

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.

Downloads

Download data is not yet available.

References

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

Published

2019-09-03

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. https://doi.org/10.21744/irjmis.v6n5.732

Issue

Vol. 6 No. 5 (2019): September

Section

Peer Review Articles
Copyright & Licensing

Articles published in the International Research Journal of Management, IT and Social sciences (IRJMIS) are available under Creative Commons Attribution Non-Commercial No Derivatives Licence (CC BY-NC-ND 4.0). Authors retain copyright in their work and grant IRJMIS right of first publication under CC BY-NC-ND 4.0. Users have the right to read, download, copy, distribute, print, search, or link to the full texts of articles in this journal, and to use them for any other lawful purpose.

Articles published in IRJMIS can be copied, communicated and shared in their published form for non-commercial purposes provided full attribution is given to the author and the journal. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.

This copyright notice applies to articles published in IRJMIS volumes 7 onwards. Please read about the copyright notices for previous volumes under Journal History.