I read some papers that study the global existence of a solution to a PDE. Sometimes, I found the notion of the "continuation principle".
What do we mean by the continuation principle? any recommended references and thank you in advance.
Aimen Daoudi
Initial Problem: Start with an initial problem or equation for which you want to find solutions. This could be a PDE or a system of PDEs. Solve in a Restricted Region: Instead of trying to solve the problem over the entire domain immediately, begin by solving it in a smaller, more manageable region where you know techniques to find solutions. Parameterization: Introduce a parameter (often denoted by λ) into the problem. This parameter can represent a physical quantity, a scaling factor, or any other meaningful quantity related to the problem. Continuation Equation: Modify the original problem by adding a term involving the parameter λ. This new equation, called the continuation equation, allows you to study the behavior of solutions as the parameter varies. Solve for a Range of Parameters: Solve the continuation equation for a range of values of λ. This typically involves solving a sequence of related problems, each corresponding to a specific value of λ. Continuation Principle: If you can show that solutions to the continuation equation exist and are continuous with respect to the parameter λ, you can use the continuation principle to extend these solutions to a larger range of parameters. Global Existence: By analyzing the behavior of solutions as λ varies and ensuring continuity and other desired properties, you can establish the global existence of solutions to the original problem over a larger domain. Parameter Variation: Finally, you can study how the solutions change as the parameter λ varies. This can provide valuable insights into the behavior and stability of the solutions under different conditions.
Here's a step-by-step explanation of how the continuation method works:
The continuation method is widely used in various fields of mathematics, physics, engineering, and other disciplines where complex systems are analyzed. It allows researchers to tackle challenging problems by breaking them down into more manageable parts and studying the solutions systematically as parameters vary.
0 votes
0 thanks
- Badges
- Science topic
- Similar topics
- Papers
More Aimen Daoudi's questions See All
Similar questions and discussions