Bellman R.E., Kalaba R.E. Quasilinearization and Nonlinear Boundary-Value Problems
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Elsevier, 1965. - 218 pages.Modern electronic computers can provide the numerical solution of systems of one thousand simultaneous nonlinear ordinary differential equations, given a complete set of initial conditions, with accuracy and speed. It follows that as soon as physical, economic, engineering, and biological problems are reduced to the task of solving initial value problems for ordinary differential equations, they are well on their way to complete solution. Hence, the great challenge to the mathematician is to find ways to construct the requisite theories and formulations, approximations and reductions. Sometimes, as in planetary theory, the basic equations are ordinary differential equations. In this case, however, some of the fundamental problems do not provide us with a full set of initial conditions. In fact, the basic problem of orbit determination is to determine a complete set of initial conditions, given certain angular measurements made at various times. In other areas, particularly in mathematical physics and mathematical biology, the underlying equations are partial differential equations, differential-integral equations, differential-difference equations, and equations of even more complex type. Many preliminary analyses of the precise questions to be asked, and the analytical procedures to be used, must be made in order to exploit fully the powerful computational capabilities now available. Whereas in earlier times the preferred procedure was to simplify so as to obtain linear functional equations, the current aim is to transform all computational questions to initial value problems for ordinary differential equations, be they linear or nonlinear. The theories of dynamic programming and invariant imbedding achieve this in a number of areas through the introduction of new state variables and the use of semigroup properties in space, time, and structure. Quasilinearization achieves this objective by combining linear approximation techniques with the capabilities of the digital computer in various adroit fashions. The approximations are carefully constructed to yield rapid convergence, and monotonicity as well, in many cases.
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