Solution of Equation using Integral Transform

From ProofWiki
Jump to navigation Jump to search

Proof Technique

Suppose it is necessary to solve a function $f \left({x}\right)$ of some variable $x$.

Suppose also that $f \left({x}\right)$ has to satisfy certain conditions expressed as equations which involve $f \left({x}\right)$.

It may very well be the case that these equations are difficult to solve in the domain of $f$.

By application of an integral operator $T$, it is often possible to transform the equations specifying $f \left({x}\right)$ into an equivalent set of equations in $F \left({p}\right)$ which are easier to solve.

If a solution for $F \left({p}\right)$ can be found, by use of the appropriate inversion theorem, $F \left({p}\right)$ can then be converted back into the required solution for $f \left({x}\right)$.

Problem: Equations involving $f \left({x}\right)$ $\xrightarrow {\text{Integral operator } T}$ Equations involving $F \left({p}\right)$
$\Big \downarrow$ Solve these equations
Solution for $f \left({x}\right)$ $\xleftarrow {\text{Inverse integral operator } T^{-1} }$ Solution for $F \left({p}\right)$

The usual problem space for which integral transforms are most fruitfully used is the field of differential equations.

An integral transform will in this circumstance be used to convert a differential equation with its boundary conditions or initial conditions into a series of algebraic equations, which are in general more easily solved.

The appropriate use of the inverse integral operator converts the solution of this algebraic equation system back into that of the problem domain.

It is necessary to select of an appropriate integral transform which allows this to be accomplished.