A PROPOSAL FOR THE NUMERICAL SOLUTION OF THE BOUNDARY VALUE PROBLEM OF THE LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION SYSTEM
1968 Volume 24 Issue 12 Pages 579-582
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1968 Volume 24 Issue 12 Pages 579-582
The boundary value problem of the linear homogeneous differential equation can be completely solved by obtaining eigen-value of the coefficient matrix. However, for the numerical solution, various ideas are also used in practice. One proposal is offered in this paper. It is an application of the difference method in such a way to replace the differential equation by the difference equation with an approximate transition matrix.
The boundary value problem is generally expressed as follow
It is rquired to determine the unknown under conditions of the given. By using an approximate transition matrix, eq. is rewritten as follows: is used, where N is the dividing number of the time interval T and j (j=0, …, N) the numbering of any lattice point. can be calaulated numerically step by step, thus the boundary value problem is solved. If A(t) is constant in time, eq. simply becomes and the following eq. (5) is derived for the evaluation of error.
If it is desired to limit the relative error given within α (α_??_1) then n and N may be determined by eq. (6):
