Can we solve non linear boundary value problem or initial value problem by applying Green function technique?
Thanks Miled but all examples in this thesis are of linear type.
Please see
The Green’s Function Method for Solutions of Fourth Order Nonlinear Boundary Value Problem
This thesis has demonstrated that Green’s functions have a wide range of applications with regard to boundary value problems. In particular, existence and uniqueness of so- lutions of a large class of fourth order boundary value problems has been established.
https://trace.tennessee.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=2803&context=utk_gradthes
Thanks Miled but all examples in this thesis are of linear type.
Please see
Dear colleagues,
Masters Theses by Olga A. Teterina , “The Green 's Function Method for Solutions of Fourth Order Nonlinear Boundary Value Problem” (12-2013, University of Tennessee - Knoxville), contains three nonlinear fourth order BVPs given as examples, in accordance with the thesis title . See, pages 40-43 of the thesis.
https://trace.tennessee.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=2803&context=utk_gradthes
Respected Ali sir
All the problems mention in this thesis ( non linear), solved by numerical methods not by Green function. Author find Lipschitz constant only and then draw solution graph and mention he solved using numerical method.Please check
Dear Sudhanshu,
Nevertheless, they are nonlinear BVPs, the point is that "all examples in this thesis are not of linear type".
Now we may go on discussing the rest of the question...
Dear Sudhanshu Aggarwal
Yes, the Green function approach can be used to solve as well as non linear boundary value problem or initial value problem. There are many applications in the litterature.
See, for example, the book of Berger, Melvin S. Nonlinearity and functional analysis. Lectures on nonlinear problems in mathematical analysis. Pure and Applied Mathematics. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977.
Best regards
Dear colleagues,
Since the superposition principle is used in the derivation of Green’s Function, it is applicable only to linear equations.
Nevertheless, the method can be extended to nonlinear equations via the backward and forward propagators and by using the short time expansion of the solution in terms of the nonlinear Green’s function. Please refer to the following:
1-M. Frasca , Green functions and nonlinear systems: Short time expansion, International Journal of Modern Physics. A, 2008, vol. 23, issue 2, pp. 299–308.
2-M. Frasca , Green functions and nonlinear systems , Modern Physics Letters. A, 2007, vol. 22, issue 18, pp. 1293–1299.
3-O. Rey, The Role of the Green’s Function in a Non-linear Elliptic Equation Involving the Critical Sobolev Exponent, Journal of Functional Analysis 89, 1-52 ( 1990)
- Badges
- Science topic