Sometimes to solve a PDE, the PDE is converted into an ODE of abstract form and then the solution of the abstract form is said to be the solution of the given PDE. What are the methods available in the literature to convert a PDE into an abstract form ODE?
I reckon, one would apply some finite difference operators (or projection on some finite dimensional spaces in the case of FEM or DG) for few variables (spacial variables for example) and one variable (for example time) is left in differential form.
Whenever one has a problem of the form
u_t = (differential operators, possibly nonlinear, applied to) u, the
'abstract' ODE has already been obtained, you just call A(u) the operator in brackets....This may help for some existence and uniqueness proofs in infinite dimensional functional spaces, but it does not help in general for the practical solution by numerical methods. What was referred to in the previous post is the so called method of lines, in which one replaces u with a finite dimensional approximation u_N, A(u) with a discrete operator A_N and ends up with a (usually very large...) system
d u_N/dt =A_N(u_N), to be solved then by ODE methods. Examples of how to derive these discretized equations by finite difference methods are found for example in the classical book
Gustafsson, Bertil, Heinz-Otto Kreiss, and Joseph Oliger. Time dependent problems and difference methods. Vol. 24. John Wiley & Sons, 1995
while for more modern finite element methods one recent reference is
Giraldo, Francis X. An Introduction to Element-Based Galerkin Methods on Tensor-Product Bases: Analysis, Algorithms, and Applications. Vol. 24. Springer International Publishing, 2020.
When you use separation of variables to solve a PDE you get an ODE for each part.
Do not know why you use the term abstract.
Thank you Juan Weisz for the answer.
A functional analytic representation of a differential equation is called abstract formulation. The ODE which we get after the abstract formulation of a PDE is called abstract ODE. Here I wanted to know which method we can use to convert a PDE into an abstract ODE.
Particularly, how we do the abstract formulation of Navier-Stokes equation using Helmholtz Projection.
Thank you
If what is meant by "abstract" relates to ODE on functions spaces, e.g., Hilbert or Banach, etc., then the theory had been around for quite awhile is one of the main drivers for the theory of semigroups of operators. The setting is an mapping from an interval of the real one where for each t, A(t) is an unbounded closed densely defined operator on the appropriate function space. That is A could be an operator associated with a PDE on the appropriate Sobolev space. The the equation x'=A(t)x is an ODE on this function space.
https://en.wikipedia.org/wiki/Abstract_differential_equation
The classical heat and wave equations can be viewed as ODE on the appropriate Banach or Hilbert spaces and the research in this area has been quite active since the 1940's.
https://www.ams.org/books/surv/049/surv049-endmatter.pdf
There are several issues that need to be address. One is often these problems are ill-posed. The second issue is establishing the regularity that insures that weak solutions lead to strong solutions. A good place to maybe start is the 1948 work of Einar Hille where he introduced semigroups of operators, to address the Cauchy problem in Banach spaces. This theory mimics much of the theory of ODE with the appropriate conditions on the mapping t-> A(t).
https://archive.org/details/functionalanalys017173mbp/mode/2up
Much of the theory of ODE in Banach spaces mimic that for ODE on finite dimensional space, however, care has to be taken because unlike finite dimensional space, the operators A(t) are not bounded. The second issue is the methods involved involve weak solutions and special conditions are required to establish that the weak solutions are strong solutions with the appropriate regularity.
This approach was key to the Sampson-Eells theorem where Joe Sampson and Jim Wells proved the existence of harmonic mappings between manifolds under certain conditions by solving a non-linear Cauchy problem for a non-linear "heat equation", replacing boundary conditions with homotopy conditions and using regularity theorems from elliptic PDE to establish regularity of the weak solutions.
https://en.wikipedia.org/wiki/Harmonic_map
https://math.jhu.edu/~js/Math646/eells-sampson.pdf
In a follow on paper, Philip Hartman showed that expanding some techniques from stability theory of ODE to the nonlinear heat equation used by Sampson and Eells, expanded the results for Sampson and Eells and removing some of their "technical" conditions used to obtain the result.
https://math.jhu.edu/~js/Math646/hartman.harmonic.pdf
Some other general references.
Article An existence theorem for ordinary differential equations in ...
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/chap5.pdf
* Separation of variables. Idea: Transform a PDE of 2 variables into a pair of ODEs
http://www.public.asu.edu/~hhuang38/pde_slides_sepvar-heat.pdf
* Transform Methods for Linear PDEs: separation of variables, eigenfunction expansions, spectral repre-sentations.
https://people.bath.ac.uk/eas25/encyc.pdf
* Fourier Transform. 3 variables.
https://math.stackexchange.com/questions/1615069/transform-pde-to-ode-3-variable-case-with-given-boundaries
You can, for example, convert a PDE into an ODE by means of the Fourier transformation. Take the following linear PDE as an example:
(1) ut = auxx,
where a is a constant. Let U(w) be the Fourier transformation of u w.r.t. the spatial coordinate x, i.e. U(w) = F{u(x,t)}(w) then, since F{uxx(x,t)}(w) = (iw)2F{u(x,t)}(w) = -w2F{u(x,t)}(w) = -w2F{u(x,t)}(w), (1) transforms into the linear 1st order ODE:
(2) Ut(w,t) = -aw^2U(w,t).
- Badges
- Science topic
- Similar topics
- Chemistry