Difference between Green's function and fundamental solution

Question

Hey everyone! 👋 I'm a bit confused about Green's functions and fundamental solutions. They seem similar, but my professor says they're different. 🤔 Can anyone explain the key differences between them in a simple way? Thanks!

janet_delgado · Accepted Answer

📚 Understanding Green's Function and Fundamental Solution
Let's break down the difference between Green's functions and fundamental solutions. While both are essential tools for solving differential equations, they serve slightly different purposes and are defined in distinct ways. Here's a detailed look:

🔍 Definition of Green's Function
A Green's function is a solution to a differential equation with a point source, subject to specific boundary conditions. It provides the influence of a point source at one location on the solution at another location, considering the constraints imposed by the boundaries.

💡 Definition of Fundamental Solution
A fundamental solution, on the other hand, is a solution to a differential equation with a point source (Dirac delta function) but without considering specific boundary conditions. It represents the most basic or 'fundamental' response of the equation to a localized impulse.

🧪 Comparison Table

Feature
      Green's Function
      Fundamental Solution

Boundary Conditions
      Satisfies specific boundary conditions.
      Does not necessarily satisfy any specific boundary conditions.

Application
      Used to solve inhomogeneous differential equations with specified boundary conditions.
      Used to find a general solution to the inhomogeneous equation, often as a building block for finding solutions with boundary conditions.

Uniqueness
      Unique for a given set of boundary conditions.
      Not necessarily unique; different fundamental solutions can exist.

Equation Solved
      $L[G(x, s)] = \delta(x - s)$, with boundary conditions, where $L$ is a linear differential operator.
      $L[E(x, s)] = \delta(x - s)$, without specific boundary conditions, where $L$ is a linear differential operator.

Domain
      Defined on a domain with boundaries where boundary conditions are applied.
      Defined on the entire domain or space where the differential equation is valid, without regard to boundaries.

📝 Key Takeaways

🌍 Scope: Green's functions are tailored to specific problems with boundary conditions, while fundamental solutions are more general.
  🔢 Boundary Impact: The key distinction is that Green's functions incorporate boundary conditions, making them directly applicable to solving boundary value problems. Fundamental solutions do not, and thus require additional steps to enforce boundary conditions.
  💡 Application Strategy: Think of a fundamental solution as the raw response of the equation, and the Green's function as that response shaped by the boundaries of the problem.

Difference between Green's function and fundamental solution

1 Answers

📚 Understanding Green's Function and Fundamental Solution

🔍 Definition of Green's Function

💡 Definition of Fundamental Solution

🧪 Comparison Table

📝 Key Takeaways

Join the discussion

Feature	Green's Function	Fundamental Solution
Boundary Conditions	Satisfies specific boundary conditions.	Does not necessarily satisfy any specific boundary conditions.
Application	Used to solve inhomogeneous differential equations with specified boundary conditions.	Used to find a general solution to the inhomogeneous equation, often as a building block for finding solutions with boundary conditions.
Uniqueness	Unique for a given set of boundary conditions.	Not necessarily unique; different fundamental solutions can exist.
Equation Solved	$L[G(x, s)] = \delta(x - s)$, with boundary conditions, where $L$ is a linear differential operator.	$L[E(x, s)] = \delta(x - s)$, without specific boundary conditions, where $L$ is a linear differential operator.
Domain	Defined on a domain with boundaries where boundary conditions are applied.	Defined on the entire domain or space where the differential equation is valid, without regard to boundaries.