Fundamental system (mathematics)

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As a fundamental system in which is Analysis each base of that vector space Denoted a homogeneous solutions from the set of linear ordinary differential equation system is composed.

Is a fundamental system, so is by definition

the set of solutions of this homogeneous system of differential equations.

Knowledge of a fundamental system is a prerequisite for the method of varying the constants in order to construct a special solution of inhomogeneous linear differential equation systems of the first order and inhomogeneous linear differential equations of higher order.

Fundamental system, (main) fundamental matrix and Wronsky determinant

Homogeneous linear system of differential equations of the first order

A linear homogeneous system of differential equations of the first order is given

with and the matrix whose coefficients are. The solutions of this differential equation system are sought in the differentiation class of the continuously differentiable functions .

If this differential equation has two different solutions, the sums and multiples with real factors are also solutions. The solution set is therefore a real subspace in the space of all continuously differentiable functions.

If the coefficients of the matrix are continuous functions, Picard-Lindelöf's existence and uniqueness theorem can be applied. According to this, on the one hand, every solution of the differential equation is already uniquely determined by its value at the starting point of the interval and, on the other hand, every initial value problem with any initial value for this differential equation system can be uniquely solved. It follows that the solution space is -dimensional.

Definitions

Each basis of this -dimensional solution space is called the fundamental system of the linear differential equation system. Usually the system of solution functions for which the initial value is the -th canonical unit vector is chosen as the basis .

If it is a fundamental system, the matrix is called the fundamental matrix and its determinant is called the Wronsky determinant . If the identity matrix is for one , it is also called the main fundamental matrix in the point .

The fundamental matrix is ​​also the solution of a homogeneous ordinary (matrix-valued) differential equation, namely of

The solution space of the original homogeneous system im is then . If the main fundamental matrix is even in , then the initial value problem is solved .

The fundamental matrix is invertible for each . Liouville's formula applies to the Wronsky determinant .

Higher order homogeneous linear differential equation

Just as in the first-order case, the solution space of a higher-order linear system is also a vector space, and each basis thereof is further called a fundamental system.

To define the fundamental matrix of a scalar linear differential equation -th order

First consider the corresponding system of differential equations of the first order, consisting of equations

With

Hint: The connection is that the scalar equation -th order solves if and only if the above system's solution is first order.

As a fundamental matrix of

one denotes every fundamental matrix of the system of the first order

Of course, the principal fundamental matrix is called in if is the identity matrix. is still called the Wronsky determinant .

The above reduction of the equation to a system of the first order yields: If a fundamental system is, then

a fundamental matrix.

Construction of a fundamental system

In the general case, it is difficult to construct fundamental systems. This is only possible through a special structure of the differential equation. This includes the scalar differential equation of the first order, systems of differential equations of the first order with constant coefficients, higher-order differential equations with constant coefficients, or Euler's differential equation . If a solution of the homogeneous differential equation of high order is known, one can use the reduction method of d'Alembert to reduce the equation to a differential equation with an order lower by one.

First order linear differential equation

Let it be an antiderivative of . Then

a fundamental system of .

First order linear differential equation system with constant coefficients

In the case of a linear differential equation with constant coefficients

one first determines the Jordan normal form of the matrix and a corresponding Jordan basis . If there is a complex eigenvalue with the associated basis vectors , then one should choose the basis vectors in the Jordan basis in such a way that they occur as basis vectors .

Now one goes through each chain of main vectors individually: Is a (complete) main vector chain to the eigenvalue , i. H.

,

so they carry the (main vector) solutions to the fundamental system

general

at. After going through all the main vector chains, one has then set up a (possibly complex) fundamental system.

Higher order linear differential equation with constant coefficients

A fundamental system for a scalar linear differential equation -th order with constant coefficients

can by solving the characteristic equation with the characteristic polynomial

respectively. Let be the zeros (different in pairs) of with multiples . Then the zero to the (complex) fundamental system carries the linearly independent solutions

at.

[To explain the way of speaking: If, with the help of the above transformation, the scalar equation -th order is traced back to a system of differential equations of the first order, the coefficient matrix has exactly the characteristic polynomial that was given here.]

Real fundamental system

In the above way, one always obtains linearly independent solutions, which, however, can sometimes be complex-valued - the complex solutions always occur in conjugate complex pairs, since the differential equation was real. Now with also and both are (real) solutions, since the differential equation is linear . One can therefore replace every pair of complex conjugate solutions in the (complex) fundamental system with real solutions . In this way you get a real fundamental system. Note Euler's formula here .

Periodic system of differential equations of the first order

For the system

With a periodic continuous coefficient matrix one cannot explicitly construct a fundamental system - however, Floquet's theorem makes a statement about the structure of the fundamental matrices of this system.

Examples

First order linear differential equation system with constant coefficients

Consider the system of differential equations

The matrix has 1 as a simple eigenvalue and 2 as a double eigenvalue. Their eigenspaces are For the main vector chain with eigenvalue 2, one still needs

For example, choose

Then the main vector of the first stage must be chosen. It results as a fundamental system with

Higher order linear differential equation with constant coefficients

Look now

This differential equation has as a characteristic polynomial which has the four zeros . Therefore you get a complex fundamental system first

Thus one obtains as a real fundamental system

literature

  • Carmen Chicone: Ordinary Differential Equations with Applications . 2nd Edition. In: Texts in Applied Mathematics , 34th Springer-Verlag, 2006, ISBN 0-387-30769-9 .
  • Harro Heuser: Ordinary differential equations . Teubner, 1995, p. 250.
  • Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).