Separation approach

The separation approach or product approach is used to solve partial differential equations with several variables.

General

The solution is assumed to be a product of the form:

${\displaystyle u(t,x)=X(x)\cdot T(t)}$

can represent. A printout is obtained by deriving and inserting the separated functions and in the output function ${\displaystyle X}$ ${\displaystyle T}$

${\displaystyle \Phi (x,X,X',X'')=\lambda =\Psi (t,T,T',T'')}$

This equation can be converted into two ordinary differential equations that can be solved with the help of the boundary conditions. The solution found does not have to be the only solution to the output function.

example

The one-dimensional wave equation is to be solved

${\displaystyle {\frac {\partial ^{2}y(x,t)}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}y(x,t)}{\partial x^{2}}}}$.

The separation approach with : ${\displaystyle y(x,t)=f(x)g(t)}$

${\displaystyle {\frac {\partial ^{2}f(x)g(t)}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}f(x)g(t)}{\partial x^{2}}}}$

performs on

${\displaystyle f(x){\frac {d^{2}g(t)}{dt^{2}}}=c^{2}{\frac {d^{2}f(x)}{dx^{2}}}g(t)}$

Now follows the “separation of the variables” with division by with the assumption inside the area. ${\displaystyle f(x)g(t)}$ ${\displaystyle y(x,t)\neq 0}$

${\displaystyle {\frac {1}{g(t)}}{\frac {d^{2}g(t)}{dt^{2}}}=c^{2}{\frac {1}{f(x)}}{\frac {d^{2}f(x)}{dx^{2}}}}$

Simplification of the notation and results ${\displaystyle f''(x)={\frac {d^{2}f(x)}{dx^{2}}}}$ ${\displaystyle g''(t)={\frac {d^{2}g(t)}{dt^{2}}}}$

${\displaystyle {\frac {g''(t)}{g(t)}}=c^{2}{\frac {f''(x)}{f(x)}}}$

The equation can only be true if both sides of the equation are constant since they depend on different variables. So

${\displaystyle {\frac {g''(t)}{g(t)}}=c^{2}{\frac {f''(x)}{f(x)}}=\lambda }$

This leads to the following ordinary second order differential equations

${\displaystyle f''(x)={\frac {\lambda }{c^{2}}}f(x)}$

${\displaystyle g''(t)=\lambda g(t)}$

Which are now solvable depending on the parameter and the boundary conditions, inserting the individual solutions in results in the solution of the partial differential equation. ${\displaystyle \lambda }$ ${\displaystyle y(x,t)}$

literature

• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 ( Graduate studies in mathematics 19).

Web links

• Mathematics online course at the University of Stuttgart: Separation approach