2.4 Green's functions and integral equations

Green's functions are powerful tools for solving complex differential equations. They represent the response of a system to an impulse, allowing us to convert tricky problems into more manageable integral equations.

By constructing Green's functions and applying them to boundary value problems, we can simplify the solution process. This approach provides valuable insights into system behavior and opens up new solving techniques.

Green's Functions and Integral Equations

Green's functions and differential equations

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• Green's functions represent impulse response of linear differential operators, used to solve inhomogeneous differential equations
• Provide general solution to inhomogeneous equation, convert differential equations into integral equations
• Satisfy homogeneous equation except at singularity, incorporate boundary conditions
• Simplify solution process for complex differential equations, enable study of systems with arbitrary source terms

Construction of Green's functions

• Variation of parameters technique constructs particular solutions to inhomogeneous equations
• Construction steps:

1. Solve homogeneous equation
2. Apply jump conditions at singularity
3. Enforce boundary conditions

• Handle various boundary value problems (Dirichlet, Neumann, mixed)
• Exhibit symmetry properties (reciprocity relation, self-adjointness of differential operator)

Integral equations from boundary problems

• Convert boundary value problems to integral equations using Green's functions
• Express solution as integral involving Green's function, incorporate inhomogeneous term and boundary conditions
• Resulting equations (Fredholm, Volterra) often easier to solve than original differential equation
• Provide insights into solution behavior

Solving techniques for integral equations

• Neumann series expands solution as infinite series, requires convergence criteria
• Iterative methods (successive approximations, Picard iteration) approximate solutions
• Numerical techniques discretize integral equation, use quadrature methods for approximating integrals
• Analyze solutions with existence and uniqueness theorems, error estimates and convergence rates