Classification of integral equations

Integral equations are classified based on various criteria, such as the limits of integration, the linearity of the equation, the position of the unknown function, and other structural properties. Understanding these classifications is essential for selecting the appropriate method of solving an integral equation.

Classification Criteria

1. Based on the Limits of Integration

   • Fredholm Integral Equations:
     The integration limits are fixed constants $a$ and $b$. The general form is:

     $$\phi(x) = f(x) + \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = x + \int_0^1 (x + t) \phi(t) \, dt$$ Here, the limits of integration are $0$ and $1$.

   • Volterra Integral Equations:
     The integration limits are variable, typically with the upper limit depending on $x$. The general form is:

     $$\phi(x) = f(x) + \lambda \int_a^x K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = x^2 + \int_0^x (x – t) \phi(t) \, dt$$ Here, the upper limit of integration is $x$, making it a Volterra equation.

2. Based on Linearity
   • Linear Integral Equations:
     The unknown function $\phi(x)$ appears linearly. The general form is: $$\phi(x) = f(x) + \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = 2 + \int_0^1 (x + t) \phi(t) \, dt$$
   • Nonlinear Integral Equations:
     The unknown function $\phi(x)$ appears nonlinearly, such as in a power or trigonometric function: $$\phi(x) = f(x) + \lambda \int_a^b K(x, t) [\phi(t)]^2 \, dt$$
     • Example: $$\phi(x) = x + \int_0^x (x – t) [\phi(t)]^2 \, dt$$

3. Based on the Position of the Unknown Function

   • First Kind:
     The unknown function $\phi(x)$ appears only under the integral sign:

     $$\int_a^b K(x, t) \phi(t) \, dt = f(x)$$
     • Example: $$\int_0^1 (x + t) \phi(t) \, dt = x^2$$

   • Second Kind:
     The unknown function $\phi(x)$ appears both inside and outside the integral:

     $$\phi(x) = f(x) + \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = 1 + \int_0^1 (x + t) \phi(t) \, dt$$

   • Third Kind:
     The unknown function $\phi(x)$ is both inside the integral and has a coefficient outside:

     $$g(x) \phi(x) = f(x) + \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$x \phi(x) = x^2 + \int_0^1 (x – t) \phi(t) \, dt$$

4. Based on the Nature of the Kernel $K(x, t)$

   • Symmetric Kernel:
     The kernel satisfies $K(x, t) = K(t, x)$.

     • Example: $$K(x, t) = x^2 + t^2$$

   • Separable Kernel:
     The kernel can be expressed as a product of functions of $x$ and $t$:

     $$K(x, t) = X(x) T(t)$$
     • Example: $$K(x, t) = e^x \sin(t)$$

   • Degenerate Kernel:
     The kernel is approximated by a finite sum of separable functions:

     $$K(x, t) = \sum_{i=1}^n X_i(x) T_i(t)$$
     • Example: $$K(x, t) = x t + x^2 t^2$$

5. Based on Homogeneity

   • Homogeneous Integral Equations:
     The equation has no external forcing function $f(x) = 0$:

     $$\phi(x) = \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = \int_0^1 (x + t) \phi(t) \, dt$$

   • Non-Homogeneous Integral Equations:
     The equation includes an external forcing function $f(x)$:

     $$\phi(x) = f(x) + \lambda \int_a^b K(x, t) \phi(t) \, dt$$
     • Example: $$\phi(x) = x^2 + \int_0^1 (x + t) \phi(t) \, dt$$

Examples for Each Classification

Fredholm Integral Equation of the Second Kind

Volterra Integral Equation of the First Kind

Nonlinear Integral Equation

Symmetric Kernel Example

References

1. Books

   • Tricomi, F.G. Integral Equations. Dover Publications, 1985.
   • Kanwal, Ram P. Linear Integral Equations: Theory and Technique. Birkhäuser, 1996.
   • Porter, David, and Stirling, David S.G. Integral Equations: A Practical Treatment, from Spectral Theory to Applications. Cambridge University Press, 1990.

2. Articles

   • Schmidt, E. “Theory of Integral Equations.” Mathematische Annalen, 1907.

3. Online Resources

   • Wolfram MathWorld: Integral Equations
   • MIT OpenCourseWare – Search for materials on advanced calculus or integral equations.