Ansatz

An approach referred to in mathematics and physics, a heuristic method for solving an equation or a system of equations. One approach is referred to in the English literature as " educated guess ". First, the assumption is made that the solution has a specific function form, for example a polynomial or exponential function, and this function has a number of unspecified parameters, which correspond to the number of equations. This function is used in the equations to be solved. This results in a system of algebraic equations for the free parameters, which is usually much easier to solve than the original equations. The word approach also referred particularly to the specific assumption about the solution function as such, that is about. Approaches default, used to be called according to the form of its solution function such as Exponentialansatz or power series approach.

Often the investigated problem provides no clear evidence for the choice of an approach. Therein lies an important limitation of this method. Sometimes an approach, however, is the only possible way to solve an equation, but also in many other cases, the cost of the solution through an approach can be significantly reduced. The performance of the mathematician or physicist is to derive an approach in a creative way either from the form of equations or physical problems from the properties of the observed system. Many problems can be solved with this already common approaches, for others the design of a new or the combination of existing approaches is needed.

The recognition method is particularly in the integral calculus and in solving differential equations is important because it other than in the differential calculus is no clearly defined solution method here. Solutions through approaches are fundamentally different in this respect from other standard methods of substitution or integration by parts to simplify the problem by modifying the output equations.

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

Example

The differential equation can be obviously solved by an exponential function, as these up to a pre-factor remains the same in their argument by deriving by a variable. Therefore, one can make the ansatz that is. Insertion into the equation, and its approach is as always in accordance greater than zero, this equation can only be solved through. Thus, a solution is given by.

If in addition to the differential equation is given even an initial condition, for example, two parameters must be present in the neck, so that the resulting algebraic system of equations is overdetermined, ie has more equations than variables. One possible approach would be. The insertion of this approach in which two equations result and then, and it follows the solution function.

Known approaches

• Exponentialansatz (Example: inhomogeneous wave equation in classical mechanics )
• Polynomialansatz
• Power series approach
• Product of powers approach
• Separation approach (Example: Three-dimensional harmonic oscillator ( quantum mechanics) and hydrogen atom in quantum mechanics )