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Ansatz
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Ansatz (Ger., "onset"; today, "setup, starting point"; plural: ansätze) is a German noun with several meanings in the English language. The fact that the word Ansatz is found in the English language today suggests that it has been carried by those who have used it frequently,, such as mathematicians and physicists. There was considerable exchange between the English- and German-speaking worlds in the first half of the twentieth century.
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Encyclopedia
Ansatz (Ger., "onset"; today, "setup, starting point"; plural: ansätze) is a German noun with several meanings in the English language. The fact that the word Ansatz is found in the English language today suggests that it has been carried by those who have used it frequently,, such as mathematicians and physicists. There was considerable exchange between the English- and German-speaking worlds in the first half of the twentieth century. Just before, and immediately after the Second World War, mathematicians and physicists of German-language background emigrated to the United States and took up positions in industry and academia. The term "Ansatz" is used in describing solution methods for differential equations.
Definition
More formally, in physics and mathematics, an ansatz is an educated guess that is verified later by its results. An ansatz is the establishment of the starting equation(s) describing a mathematical or physical problem. It can take into consideration boundary conditions. After an ansatz has been established, the equations are solved for the general function of interest. Typically, a word problem starts by writing down the ansatz with subsequent refinement leading to the solution of the problem.
Examples
Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters.
Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the ansatz to most basic problems of thermodynamics.
Another example of ansatz is to suppose the solutions of a homogeneous linear differential equation and difference equation to have, respectively, exponential and power form. More generally one can guess a particular solution of a system of equations and test such an ansatz by direct substitution of the solution in the system of equations.
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