Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Calculus
Setting Up Inverse Problems
Reply to thread
Message
[QUOTE="member 428835, post: 6008416"] Hi PF! I'm reading an article and there is a differential equation cast as an operator equation: $$f_n-d_x^2 f_n = \lambda f$$ where ##f_n = \partial_n f##, which is derivative of ##f## normal to a given parameterized curve. The author casts the ODE as $$B[f_n] = \lambda A[f_n]:\\ B[f_n] \equiv f_n-d_x^2 f_n,\\A[f_n] \equiv f.$$ So if we simply take ##A^{-1}## and ##B^{-1}## to both sides of the operator equation we have $$A^{-1}[f_n] = \lambda B^{-1}[f_n]$$ However, if we rewrite the operator equation as $$B[f_n] = \lambda f \implies\\ f_n = \lambda B^{-1}f.$$ Recall ##A[f_n] \equiv f \implies A^{-1}[f] \equiv f_n##. Then the rewritten operator equation is $$A^{-1}[f] = \lambda B^{-1}[f].$$ Notice one inverse equation operates on ##f## and the other on ##f_n##. How can I tell which is correct? (I know it is ##f##, just not sure where the logic went wrong). PS sorry, I can't figure out how to label equations here. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Calculus
Setting Up Inverse Problems
Back
Top