Integral Equation — Definition, Formula & Examples

An integral equation is an equation in which the unknown function appears inside an integral. Solving it means finding the function that satisfies the equation for all values of the independent variable.

An integral equation is an equation of the form $f(x) = \lambda \int_a^b K(x, t)\, u(t)\, dt + g(x)$ , where $u(t)$ is the unknown function, $K(x, t)$ is a known kernel, $g(x)$ is a known function, and $\lambda$ is a parameter. The equation is classified as a Fredholm equation if the limits $a$ and $b$ are constants, and as a Volterra equation if the upper limit is the variable $x$ .

Key Formula

f(x) = g(x) + \lambda \int_a^b K(x, t)\, u(t)\, dt

Where:

$u(t)$ = The unknown function to be determined
$K(x, t)$ = The kernel, a known function of two variables
$g(x)$ = A known forcing function
$\lambda$ = A scalar parameter
$a, b$ = Limits of integration (constants for Fredholm, upper limit x for Volterra)

How It Works

To solve an integral equation, you look for a function

u(t)

that makes both sides equal for every value of

x

. Common solution methods include converting the integral equation into a differential equation, using successive approximation (Neumann series), or applying Laplace transforms. The kernel

K(x, t)

encodes how different values of

u

are weighted and mixed together. Many integral equations arise naturally when you reformulate boundary value problems or initial value problems from differential equations.

Worked Example

Problem: Solve the Volterra integral equation of the second kind:

u(x) = 1 + \int_0^x u(t)\, dt

Step 1: Differentiate both sides with respect to

x

. The right side uses the Leibniz rule on the integral.

u'(x) = u(x)

Step 2: This is a first-order ODE. Its general solution is an exponential function.

u(x) = Ce^x

Step 3: Find

C

using the original equation at

x = 0

u(0) = 1 + \int_0^0 u(t)\,dt = 1

. So

C = 1

u(x) = e^x

Answer:

u(x) = e^x

Why It Matters

Integral equations appear in physics (electrostatics, radiative transfer), engineering (signal processing, control theory), and mathematical modeling of hereditary systems where the current state depends on the entire past history. In a differential equations course, converting between differential and integral formulations is a powerful technique for proving existence and uniqueness of solutions.

Common Mistakes

Mistake: Confusing integral equations with integrals that appear while solving differential equations.

Correction: In an integral equation, the unknown function itself sits inside the integral — the integral is part of the equation's structure, not just a step in a solution process.