for ODEs we can use the definition of stability,
for example when the ODE is linear, by calculating the
Eigenvalues or poles of the system the stability characteristic of the system can be achieved. or in nonlinear case the Lyapunov theory of stability can be used.
do such definitions (equilibrium point, stability, Lyaponov theory) exist for PDEs?
Dear Ali,
ODE are differential equations with one independent variable and PDE are differential equations of several independent variables.
Almost all results that study the qualitative behavior of the ODE solutions
can be generalized into PDE.
See for example:
ODE and PDE Stability Analysis - Semantic Scholar
https://pdfs.semanticscholar.org/.../b1f349f6737e3c00823097e7393c94cea488.pdf
(if the link didn't work, copy paste and google the web)
https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.22113
https://ieeexplore.ieee.org/iel7/9/4601496/07268848.pdf
math.bu.edu/people/mabeck/lin_stab_minicourse_2012.pdf
https://pdfs.semanticscholar.org/.../b1f349f6737e3c00823097e7393c94cea488.pdf
etc.
Best regards
Discus stability definition for PDE in following PDF link below:
http://math.bu.edu/people/mabeck/lin_stab_minicourse_2012.pdf
There are several notions of stability of solutions for PDEs that generalise the ones for ODEs. For example there are generalised notions of Lyapunov and asymptotic stability. There is also a notion of linear stability. You can view this as linear stability in infinite dimension. All these notions are discusses in many places. One excellent reference are the lecture notes by Margaret Beck
http://math.bu.edu/people/mabeck/lin_stab_minicourse_2012.pdf
suggested by Ravendra Singh.
Above said methods in given sites, due to Kaddoura, Singh and Santoprete, may be useful to discuss the stability for PDEs.
Besides the the above mentioned internet sites, it could be of certain interest some chapters of the book by Zauderer “Partial Differential Equations of Applied Mathematics”, Third Edition, Wiley, N.Y. (2006).
In certain cases, the analysis of the solutions of a boundary value problem for a nonlinear PDF can be done introducing coupled modes of evolution that reduces the problem to one of stability analysis of the solutions of an ODF, as suggested in the attachment. This approach appears in the book by Wicktor Eckhaus, “Studies in Nonlinear Stability Theory”, Springer, N.Y. (1965).
In Zauderer's book nonlinear modal analysis is also included.
Thank you very much Dr. Kaddoura , Dr. Singh , Dr. Santoprete and Dr. Suárez-Ántola
I have read the mini course:
http://math.bu.edu/people/mabeck/lin_stab_minicourse_2012.pdf
what I am really interested in is the stability of Fokker-Planck equation, which is a linear PDE.
but I have some difficulty understanding the mini course, it seems that like linear ODE, where for stability or convergence of the ODE we analyze the eigenvalues of A matrix (x_dot=A*x), in PDE we consider the eigenvalues of the operator. I could not get it, how we can compute the eigenvalues of the operator?
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