Abstract
A general methodology for the stability analysis of discrete approximations of nonstationary PDEs is applied to solve the Kuramoto-Tsuzuki equation, including also the Schr¨odinger problem. Stability regions are constructed for the explicit, backward and symmetrical Euler schemes. The obtained results are applied to solve the Kuramoto-Tsuzuki problem with a non-local integral boundary condition. Results of computational experiments are provided.
Highlights
Non-classical and nonlocal boundary conditions are used in various real-world applications, e.g. parabolic problems in heat conduction and thermodynamics [2,6,10], and pseudo-parabolic problems in underground water flow [1,9], see references given in these papers
The stability analysis of parabolic and pseudoparabolic problems with nonlocal boundary conditions is done by various methods
In papers [5, 7], we have proposed a methodology, when the stability of the specified model is investigated in two steps, and the influence of the differential equation is separated from the analysis of the influence of nonlocal boundary conditions
Summary
Non-classical and nonlocal boundary conditions are used in various real-world applications, e.g. parabolic problems in heat conduction and thermodynamics [2,6,10], and pseudo-parabolic problems in underground water flow [1,9], see references given in these papers. We give a brief review of recent stability analysis methods used for investigation of numerical approximations of parabolic and pseudoparabolic equations with nonlocal boundary conditions. We mention a general technique based on a presentation of the solution of a given PDE with nonlocal boundary conditions as a superposition of solutions of classical boundary value problems For stationary problems, this method is applied in [4], for non-stationary problems in [6, 10] (see references contained therein). One general technique to prove necessary and sufficient stability conditions for non-stationary numerical approximations of differential problems is to apply the eigenvalue criterion for non-normal matrices [7, 13] This stability analysis technique is applied for simple parabolic and pseudo-parabolic problems with nonlocal boundary conditions in [12,14].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
