Approach (mathematics)

In mathematics and physics, an approach describes a heuristic method for solving an equation or a system of equations .

In the English-language literature, one approach is referred to as educated guess (for example: "well-founded assumption") in addition to the German term as a loan word .

It is first made the assumption that the solution function has a certain form, e.g. B. is a polynomial or an exponential function , and that this function has a number of indefinite parameters which correspond to the number of equations. This function is then inserted into the equations to be solved. This results in a system of algebraic equations for the free parameters, which are usually much easier to solve than the original equations. The word approach also specifically designates the concrete assumption about the solution function as such, for example . Approaches used as standard are determined according to the form of their solution function e.g. B. referred to as exponential approach or power series approach . ${\displaystyle f(x)=A\cdot \exp(-x)+B}$

Often the problem being investigated does not provide any clear indications for the choice of an approach. This is an important limitation of this procedure. Sometimes, however, an approach is the only possible method of solving an equation, but in many other cases the effort for the solution can be significantly reduced by an approach. The achievement of the mathematician or physicist is to creatively derive an approach either from the form of the equations or, in the case of physical problems, from the properties of the observed system. Many problems can be solved with approaches that are already in use, others require the drafting of a new one or a combination of existing approaches.

The approach method is particularly important in integral calculus and when solving differential equations , since there are no clearly specified solution methods here other than in differential calculus . In this context, solutions based on approaches are to be fundamentally differentiated from other standard methods of substitution or partial integration , which simplify the problem by modifying the initial equations.

The word approach in the form described here has found its way into the English language as a loan word . It is often found in the scientific publications of the international mathematician and physicist community , written in English .

example

The differential equation

${\displaystyle x'(t)=-3\,x(t)}$

can evidently be solved by an exponential function, since it remains the same in its argument except for a prefactor by deriving it from a variable. The following approach is therefore promising:

${\displaystyle x(t)=\exp(A\,t)}$

Inserting into the equation gives

${\displaystyle A\,x(t)=-3\,x(t)}$,

and since, according to his approach, is always greater than zero, this equation can only be solved by. A solution function is thus given by ${\displaystyle x}$ ${\displaystyle A=-3}$

${\displaystyle x(t)=\exp(-3\,t)}$.

If an initial condition is given in addition to the differential equation , e.g. B. , two parameters must be present in the approach so that the resulting algebraic system of equations is not overdetermined, i.e. has more equations than variables. One possible approach would then be ${\displaystyle x(0)=2}$

${\displaystyle x(t)=C\,\exp(A\,t)}$.

Substituting this approach into the two equations gives

${\displaystyle A\,C\,x(t)=-3\,C\,x(t)}$

and

${\displaystyle C=2}$,

and the solution function follows

${\displaystyle x(t)=2\exp(-3\,t)}$.

Well-known approaches

• Exponential approach (application example: inhomogeneous oscillation equation in classical mechanics )
• Polynomial approach
• Power series approach
• Potency product approach
• Separation approach (application example: three-dimensional harmonic oscillator (quantum mechanics) and hydrogen atom in quantum mechanics )

Individual evidence

1. ↑ https://www.merriam-webster.com/dictionary/ansatz
2. ^ John Dowden: The Theory of Laser Materials Processing. Springer Science & Business Media, 2009, ISBN 9781402093401 , p. 2. limited preview in Google book search
3. ^ R. Shankar: Basic Training in Mathematics. Springer Science & Business Media, 1995, ISBN 9780306450358 , p. 100. Restricted preview in the Google book search