Converting BVP to Fredholm integral equations

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A Boundary Value Problem (BVP) involves solving a differential equation with conditions specified at the boundaries of the domain. Many BVPs, especially linear ones, can be reformulated as Fredholm integral equations. This conversion leverages integral equation theory to simplify certain problems and offers numerical and analytical advantages.

Mathematical Background

Boundary Value Problem (BVP)

A typical BVP for a second-order linear differential equation is written as:

d2y(x)dx2+p(x)dy(x)dx+q(x)y(x)=f(x),x∈[a,b],\frac{d^2y(x)}{dx^2} + p(x) \frac{dy(x)}{dx} + q(x)y(x) = f(x), \quad x \in [a, b],

with boundary conditions:

y(a)=α,y(b)=β.y(a) = \alpha, \quad y(b) = \beta.

Here:

  • p(x)p(x), q(x)q(x), and f(x)f(x) are given functions,
  • [a,b][a, b] is the domain of the problem,
  • α\alpha and β\beta are boundary values.

Fredholm Integral Equation

A Fredholm integral equation of the second kind is given by:

y(x)=f(x)+λ∫abK(x,t)y(t) dt,y(x) = f(x) + \lambda \int_a^b K(x, t)y(t) \, dt,

where:

  • K(x,t)K(x, t) is the kernel,
  • f(x)f(x) is a known function,
  • y(x)y(x) is the unknown function to be solved.

Conversion Procedure

Step 1: Green’s Function Representation

To convert a BVP into a Fredholm integral equation, a Green’s function is used. The Green’s function G(x,t)G(x, t) satisfies:

  1. The differential equation: d2G(x,t)dx2+p(x)dG(x,t)dx+q(x)G(x,t)=δ(x−t),\frac{d^2G(x, t)}{dx^2} + p(x)\frac{dG(x, t)}{dx} + q(x)G(x, t) = \delta(x – t), where δ(x−t)\delta(x – t) is the Dirac delta function.
  2. The boundary conditions: G(a,t)=0,G(b,t)=0.G(a, t) = 0, \quad G(b, t) = 0.

Step 2: Integral Representation

The solution y(x)y(x) of the BVP can be expressed as:

y(x)=∫abG(x,t)f(t) dt.y(x) = \int_a^b G(x, t)f(t) \, dt.

Step 3: Reformulation as Fredholm Integral Equation

For a second-order equation, y(x)y(x) is expressed in terms of the Green’s function and an integral involving itself:

y(x)=f(x)+∫abK(x,t)y(t) dt,y(x) = f(x) + \int_a^b K(x, t)y(t) \, dt,

where K(x,t)K(x, t) is derived from G(x,t)G(x, t).

Example

Problem: Converting a BVP to a Fredholm Integral Equation

Consider the BVP:

d2y(x)dx2−y(x)=ex,x∈[0,1],\frac{d^2y(x)}{dx^2} – y(x) = e^x, \quad x \in [0, 1],

with boundary conditions:

y(0)=0,y(1)=0.y(0) = 0, \quad y(1) = 0.

Step 1: Construct Green’s Function

The corresponding homogeneous differential equation is:

d2G(x,t)dx2−G(x,t)=δ(x−t),\frac{d^2G(x, t)}{dx^2} – G(x, t) = \delta(x – t),

with boundary conditions:

G(0,t)=0,G(1,t)=0.G(0, t) = 0, \quad G(1, t) = 0.

The solution for G(x,t)G(x, t) depends on the relative positions of xx and tt:

G(x,t)={A(t)ex+B(t)e−x,x<t,C(t)ex+D(t)e−x,x>t.G(x, t) = \begin{cases} A(t)e^x + B(t)e^{-x}, & x < t, \\ C(t)e^x + D(t)e^{-x}, & x > t. \end{cases}

Boundary conditions yield specific forms for the constants A(t)A(t), B(t)B(t), C(t)C(t), and D(t)D(t), ensuring continuity at x=tx = t and proper handling of the delta function.

Step 2: Write the Integral Equation

The solution y(x)y(x) is:

y(x)=∫01G(x,t)et dt.y(x) = \int_0^1 G(x, t)e^t \, dt.

Step 3: Reformulate as Fredholm Equation

Substituting y(x)y(x) into the original BVP using the Green’s function and simplifying gives:

y(x)=ex+∫01G(x,t)y(t) dt,y(x) = e^x + \int_0^1 G(x, t)y(t) \, dt,

where K(x,t)=G(x,t)K(x, t) = G(x, t).

Numerical Considerations

  1. Discretization:
    The integral equation can be discretized for numerical computation using quadrature rules. Divide the interval [a,b][a, b] into nn subintervals:

    y(xi)=f(xi)+∑j=1nK(xi,tj)y(tj)wj,y(x_i) = f(x_i) + \sum_{j=1}^n K(x_i, t_j)y(t_j)w_j,

    where wjw_j are weights.

  2. Matrix Representation:
    The integral equation transforms into a linear system:

    y=f+λKy,\mathbf{y} = \mathbf{f} + \lambda \mathbf{K}\mathbf{y},

    where K\mathbf{K} is the matrix of kernel values K(xi,tj)K(x_i, t_j).

Applications

  • Physics: Solving problems in quantum mechanics (e.g., Schrödinger equation).
  • Engineering: Heat conduction and structural analysis.
  • Mathematics: Spectral theory and eigenvalue problems.

References

  1. Books

    • Tricomi, F.G. Integral Equations. Dover Publications, 1985.
    • Kanwal, Ram P. Linear Integral Equations: Theory and Technique. Birkhäuser, 1996.
    • Stakgold, I., and Holst, M. J. Green’s Functions and Boundary Value Problems. Wiley, 2011.

  2. Articles

    • Arfken, G.B., and Weber, H.J. “Boundary Value Problems and Green’s Functions.” Mathematical Methods for Physicists, 6th Edition.
    • Numerical solutions to Fredholm integral equations using collocation methods, SIAM Review, 2002.

  3. Online Resources

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