representaatiomenetelmä

Representaatiomenetelmä, known in English as the representation method or method of representation, is a mathematical technique primarily used in solving differential equations, particularly boundary value problems. It involves finding a solution that can be expressed as an integral or a series involving a special function, often referred to as a "representation function." This function is designed to satisfy certain properties related to the differential equation itself or its associated boundary conditions.

The core idea is to transform a problem involving an unknown function into one involving an integral or a series of known or more easily handled functions. By choosing an appropriate representation function, the differential equation can be simplified, or its solution can be expressed in a more manageable form. This method is particularly useful for linear differential equations and can offer analytical insights into the behavior of solutions.

A common application of the representation method is in the theory of integral equations, where solutions to differential equations are sought in the form of integrals. The representation function often acts as a kernel in these integral equations. The method's effectiveness relies heavily on the judicious selection of the representation function, which is typically determined by the specific form of the differential equation and the boundary conditions it must satisfy. While powerful, it requires a good understanding of functional analysis and integral transforms.