Integral equations

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• ConceptIntegral equation

  An integral equation is an equation in which the unknown function appears under an integral sign. The typical linear example is $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$, where $f$ and the kernel $K$ are given and $\varphi$ is unknown. Many problems for differential equations, such as initial value problems and boundary value problems, can be rewritten as integral equations. The integral form often needs less smoothness and suits fixed point and operator methods.

• ConceptIntegral operator

  An integral operator is a linear map between spaces of functions that acts by integration against a kernel: $(Tf)(x) = \int K(x, y) f(y)\, dy$. Integral operators are the infinite-dimensional analogue of matrices. They appear as integral transforms, as solution operators of differential equations, and in Volterra integral equations and Fredholm integral equations. With a square-integrable kernel they are compact operators, which gives them a spectral theory close to that of matrices.

• ConceptKernel of an integral equation

  The kernel of a linear integral equation is the function $K(x, s)$ of two variables that multiplies the unknown function inside the integral. In $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ it is $K$. The properties of the kernel, such as continuity, symmetry or the form of its singularity, decide which theory applies to the equation.

• ConceptLinear integral equation

  A linear integral equation is an integral equation in which the unknown function enters linearly, both outside and inside the integral. Its general form is $a(x) \varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$ with a given kernel $K$. The solutions form an affine space, so linear algebra and operator theory apply. Fredholm and Volterra equations are the main classes.

• ConceptNonlinear integral equation

  A nonlinear integral equation is an integral equation that is not a linear integral equation. Typical forms are the Urysohn equation $\varphi(x) = f(x) + \int_a^b H(x, s, \varphi(s)) \, ds$ and the Hammerstein equation $\varphi(x) = f(x) + \int_a^b K(x, s) g(s, \varphi(s)) \, ds$. They arise from nonlinear differential equations, such as initial value problems for $y' = F(x, y)$. The main tools are fixed point theorems, such as the Banach fixed point theorem.

• ConceptIntegral equation of the first kind

  An integral equation of the first kind is a linear integral equation in which the unknown function appears only under the integral: $\int K(x, s) \varphi(s) \, ds = f(x)$. Solving it means inverting an integral operator. Such equations are typically ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$. Inverse problems, such as recovering a signal from blurred data, lead to equations of the first kind.