🪟Partial Differential Equations Unit 9 – Variational Calculus & Integral Equations

Key Concepts and Definitions

• Variational calculus branch of mathematical analysis that deals with finding extrema (maxima or minima) of functionals, which are mappings from a set of functions to the real numbers
• Functional assigns a real number to each function in a certain set of functions
• Integral equations equations in which the unknown function appears under an integral sign
• Fredholm integral equation type of integral equation with fixed limits of integration
  • First kind Fredholm equation has the unknown function only inside the integral
  • Second kind Fredholm equation has the unknown function both inside and outside the integral
• Volterra integral equation type of integral equation with variable limits of integration
• Green's function used to solve inhomogeneous differential equations with specified initial conditions or boundary conditions
• Euler-Lagrange equation necessary condition for a function to be a stationary point of a functional

Historical Context and Applications

• Variational calculus has its roots in the 17th and 18th centuries with the works of Johann Bernoulli, Leonhard Euler, and Joseph-Louis Lagrange
• Developed as a tool to solve optimization problems in mechanics, such as finding the path of least action (principle of least action)
• Integral equations first studied systematically by Niels Henrik Abel in the 19th century while investigating the tautochrone problem
• Ivar Fredholm made significant contributions to the theory of integral equations in the early 20th century, leading to the classification of Fredholm equations
• Applications in various fields, including:
  • Physics (quantum mechanics, electromagnetism, and optics)
  • Engineering (control theory, signal processing, and inverse problems)
  • Economics (optimization of resource allocation)
  • Biology (population dynamics and epidemiology)

Variational Principles

• Fundamental idea behind variational calculus is to find a function that minimizes or maximizes a given functional
• Principle of least action states that the path taken by a system between two points is the one that minimizes the action functional
  • Action functional defined as the integral of the Lagrangian (difference between kinetic and potential energy) over time
• Fermat's principle in optics states that light travels along the path that minimizes the optical path length
• Variational principles provide a unifying framework for understanding physical laws and deriving equations of motion
• Variational problems often lead to differential equations (Euler-Lagrange equations) that describe the optimal solution
• Constraints can be incorporated into variational problems using Lagrange multipliers, which leads to constrained optimization

Euler-Lagrange Equations

• Euler-Lagrange equation is a necessary condition for a function to be a stationary point (extremum) of a functional

• For a functional $J[y] = \int_{a}^{b} F(x, y, y') dx$, the Euler-Lagrange equation is given by:

  $\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0$

• Solutions to the Euler-Lagrange equation are called extremals and represent the functions that minimize or maximize the functional

• Boundary conditions must be specified to obtain a unique solution to the Euler-Lagrange equation

• Generalizations of the Euler-Lagrange equation exist for functionals involving higher-order derivatives and multiple functions

• Noether's theorem relates symmetries of a variational problem to conservation laws (e.g., conservation of energy, momentum, and angular momentum)

Types of Integral Equations

• Fredholm integral equations have fixed limits of integration and are of the form:

  $\phi(x) = f(x) + \lambda \int_{a}^{b} K(x, t) \phi(t) dt$

  • First kind Fredholm equation: $f(x) = 0$
  • Second kind Fredholm equation: $f(x) \neq 0$

• Volterra integral equations have variable limits of integration and are of the form:

  $\phi(x) = f(x) + \lambda \int_{a}^{x} K(x, t) \phi(t) dt$

• Singular integral equations contain kernels with singularities, such as the Cauchy principal value integral

• Nonlinear integral equations involve the unknown function in a nonlinear manner inside the integral

• Integro-differential equations combine integral and differential operators, such as the Boltzmann equation in kinetic theory

Solution Methods for Integral Equations

• Analytical methods:
  • Separation of variables for kernels that can be written as a product of functions of $x$ and $t$
  • Laplace transform for Volterra equations with convolution kernels
  • Fourier transform for Fredholm equations with difference kernels
  • Series expansion methods (Neumann series, resolvent kernel)
• Numerical methods:
  • Quadrature methods (trapezoidal rule, Simpson's rule) to discretize the integral equation into a linear system
  • Iterative methods (successive approximations, Fredholm alternative)
  • Galerkin methods project the integral equation onto a finite-dimensional subspace using basis functions
  • Collocation methods enforce the integral equation at a set of discrete points
• Green's function method expresses the solution as an integral involving the Green's function and the inhomogeneous term
  • Green's function satisfies the homogeneous equation with a delta function as the inhomogeneous term
  • Constructed using eigenfunction expansions or integral transforms

Connections to Partial Differential Equations

• Many partial differential equations (PDEs) can be reformulated as integral equations using Green's functions or fundamental solutions
• Boundary value problems for elliptic PDEs (Laplace, Poisson) lead to Fredholm integral equations
  • Solution expressed as a boundary integral involving the Green's function and boundary data
• Initial value problems for parabolic PDEs (heat, diffusion) lead to Volterra integral equations
  • Solution expressed as a space-time integral involving the fundamental solution and initial data
• Hyperbolic PDEs (wave) can be reduced to integral equations using the method of characteristics or Fourier analysis
• Integral equation formulations offer advantages in certain situations:
  • Reduction of dimensionality (boundary element method)
  • Treatment of unbounded domains and singular sources
  • Incorporation of jump conditions and interface problems

Practice Problems and Examples

1. Find the extremals of the functional $J[y] = \int_{0}^{1} (y'^2 - y^2) dx$ subject to the boundary conditions $y(0) = 0$ and $y(1) = 1$.

2. Derive the Euler-Lagrange equation for the functional $J[y] = \int_{a}^{b} F(x, y, y', y'') dx$.

3. Solve the Fredholm integral equation of the second kind:

   $\phi(x) = f(x) + \lambda \int_{0}^{1} e^{x+t} \phi(t) dt$

   using the resolvent kernel method.

4. Use the Laplace transform to solve the Volterra integral equation:

   $\phi(x) = x + \int_{0}^{x} (x - t) \phi(t) dt$

5. Find the Green's function for the boundary value problem:

   $y'' + k^2 y = f(x), \quad y(0) = y(L) = 0$

   and express the solution as an integral involving the Green's function.

6. Reformulate the Dirichlet problem for the Laplace equation in a bounded domain as a Fredholm integral equation using the Green's function.

7. Use the method of successive approximations to solve the nonlinear Volterra integral equation:

   $\phi(x) = x^2 + \int_{0}^{x} xt \phi(t)^2 dt$

   starting with the initial approximation $\phi_0(x) = x^2$.