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Existence of Solutions to Systems of Ordinary Differential Equations |
The solution of differential equations has been the heart and soul of the physical sciences since the time of Isaac Newton. The question of whether or not a solution actually exists for an equation involving a derivative is vital. Fortunately for linear equations constructive proofs of the existence of solutions are readily available. One goal of the analysis is to be able to say that for a system of n linear equations there exists n independent solutions.
For a square matrix M define its expontential function exp(M) which maps square matrices into square matrices of the same dimension as
where I is the identity matrix of the same dimensions as M. Questions of convergence will be dealt with later.
For a matrix multiplied by a scalar t the definition reduces to
A set of linear differential equations of the form
can be expressed as
where X(t) is an n dimensional column vector and A is the n×n matrix of the coefficients a_{i,j}. The solution requires the initial conditions X(t)=X_{0} be satisfied.
Now consider the exponential matrix function
The derivative of the right-hand side (RHS) of the above with respect to t gives
The matrix A can be factored as a premultiplier from each term so
What is left in the brackets is none other than exp(At). Therefore
Now consider X(t) = exp(At)X_{0}. Differentiation by t shows that
Thus the function exp(At)X_{0} satisfies the system of differential equations dX/dt = AX and the initial conditions.
Consider a system of equations of the form
where C is a vector of constants. If A has an inverse then the system can be converted into the form
where D=A^{-1}C.
Let Y(t)=X(t)+D. Then the system becomes
This system has the solution Y(t)=exp(At)Y(0) and thus
The systems previously considered all had constant coefficients. If any coefficient is a non-trivial function of time then the system
has the solution
If A(t) has an inverse for all t then the solution to the inhomogeneous system
where D(t)=A(t)^{-1}C(t).
A square matrix M can be represented as
Where P is an orthogonal matrix; i.e., P^{-1}=P^{T}; i.e., the inverse of P is equal to the transpose of P. The matrix Λ is a diagonal martrix. Thus
This means that
The values of elements of Λ are called the eigenvalues of the matrix M and P is the matrix of its eigenvectors.
One small complication is that the eigenvalues of a matrix may be nonreal. That is no major difficulty in that exp(x+iy) is well defined. It is exp(x)(cos(y)+isin(y)). A more serious complication occurs when an eigenvalue occurs with a multiplicity greater than unity. If an eigenvalue is repeated then the second occurence does not constitute another independent solution.
If an eigenvalue λ occurs with a multiplicity p then
are all solutions.
(To be continued.)
A differential equation of order k can be converted into a system of k first order equations. This just involves defining new variables such that
Suppose a system of ordinary differential equations is representeed as
One approach to a solution is by iteration. Start with an arbitrary X_{0}(t) then construct X_{1}(t) by integration of
and likewise for X_{2}(t) and beyond as
The question of convergence can be examined by looking at the differences y_{n}=X_{n}(t)−X_{n-1}(t). Then
The RHS of the above can be approximated by (∂F/∂X)·y_{n}(t). Thus
The y's would go asymptotically to zero if the eigenvalues of the matrices (∂F/∂X) are all of a negative real part.
(To be continued.)
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