Statistics is an important branch in academics and it is a lot more than graphs and curves. This field is supplemented by a set of OEDs or Ordinary Differential Equations for the layman’s head.
Over the years, many mathematicians and statisticians have worked out methods to provide an adequate numerical solution of differential equations.
Here we talk about a few of such methods.
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Leonhard Euler
Euler Method
The Euler method was one of the earliest theories in practical maths. It was put forward by the stalwart, Leonhard Euler in his classic treatise Institutionum calculi integralis published in the 18th century. According to this method, the local error is proportional to the square of the step size.
On the other hand, there is the global error which is proportional to the step size. This can be derived through a geometrical description method.
A pendulum
Geometric Integration
There is a whole set of Geometric Integration methods to solve differential equations.
It makes use of geometric properties measuring the exact flow of an OED.
It is usually explained using the flow of a pendulum (a favorite example amongst theoretical physicists). The moving frame method and Hamilton’s equations are further methods that can be used for derivation.
Backward Euler Method
Euler has been such a genius and his theories spawned so many multiple related theories in the past. Here we have another method named after him. The Backward Euler Method is also popularly known as implicit Euler method. It is a quite basic numerical solution to differential equations.
According to mathematical terms, the method yields order one in time. It is called Backward Euler method as it is closely related to the Euler method but is still implicit in the application.
This can be further explained if you look at the derivation of this numerical solution from the classic standard Euler method.
Exponential Integration
Exponential Integration
These particular sets of integrators are helpful in solving differential equations (initial value problems if we talk in particular terms). Certaine and Pope were the experts who made its use widespread from the 1960s onwards. In that sense, it is a quite recent method compared to the other examples in this category.
Both explicit and implicitly constructed exponential integrators can help in solving ordinary differential equations. In addition, they can also be used for solution of partial differential equations (including both hyperbolic and parabolic examples).
Quantized State System Method
Quantized State System Method
These are often abbreviated as QSS methods and are as the name suggest, based on the concept of state quantization. The DEVS formalism has played a major role in their formation and capacity to solve differential equations.
Their greatest advantage is that they can simulate sparse systems with frequent discontinuities. Frequent research in QSS methods has made experts categorize them in three main categories: QSS1, QSS2 & 3 and Backward Quantized State System Method.
These QSS methods have been useful for software equations too later on. The QSS methods can work as numerical solvers in such cases.