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................................................................................................................... 1. .................................................................... 1.1. ..................................................................................... 1.2. ........................................... 1.3. ......................................... 1.4. ...................................................................... 1.5. .................................................................................... 1.6. ................... 1.7. .................. 1.8. ....................................................... 2. ........................................................................... 2.1. ............................................................................. 2.2. .............................................................. 2.3. ............................................................................ 2.4. ....................................................................... 3. ............................ 3.1. ................................................................................. 3.2. ......................................................................................... 3.3. ....................................................................... 3.4. ......................... 3.5. ........................ 3.6. ................................................................. 4. ................................... 4.1. ............................................................ 4.2. ............................................................ 4.3. .................................... 5. ........................................................................ 5.1. .......................................... 5.2. ........................................................................................................... 5.3. .


.................................................................... 5.4. ................................................................. 5.5. ..................................................................................... 5.6. ......................................... 5.7. ..................................................................... 6. ..................................................... 6.1. .................................................................................... 6.2. ................................................................................................. 6.3. ............. 7. (m, k)-................................................................ 7.1. .................................................................................... 7.2. ......................................................... 7.3. ...................... 7.4. ................................................................................... 7.5. ......................................................... 7.6. . .................................................................................. 7.7. . ................................................................................ 7.8. (3, 2)- ................................................................................................... 8. (m, 3)-............................................. 8.1. ............................................................................................ 8.2. A- (m, 3)- ........................ 8.3. L- (m, 3)- ......................... 8.4. (m, 3)-.............. 8.5. (m, 3)- ........................................................... 9. ................................................................ 9.1. - ............................................................ 9.2. ......................................................................................... 9.3. .................................................................................. 9.4. ....................................................... 9.5. .................................................................................................... 9.6. - ............................................. 9.7. ......................................................................... 9.8. - ....................................... 10. .................................................................................................. 10.1. ................................. 10.2. ............. 10.3. ......... 10.4. .......................................... 10.5. ............................................. 10.6. L- ................................... 10.7. ................. 11. ....... 11.1. ....................................................................... 11.2. ............................................. 11.3. ........................................ 11.4. - ................................................................................. 12. .............................................. 12.1. .................................................... 12.2. ................................................... 12.3. - .............................. 12.4. .................................................... 12.5. ..................................................................................................... 12.6. ......................... 13. .............. 13.1. ........................................................ 13.2. ..................................................................................................... 13.3. ............................................................................................... 13.4. .............................................................................................. 13.5 ................................... 13.6. ....................

................................................ 14. - ................................................................ 14.1. ................................................................ 14.2. ........................................................................... 14.3. ............................................................................................ 14.4. .............................. 14.5. ..................................................................................... ................................................................................ CONTENTS Introduction................................................................................................................ C h a p t e r 1. ON THE DIFFERENTIAL EQUATION THEORY....................... 1.1. Statement of the problem............................................................................ 1.2. Existence and uniqueness of solution.......................................................... 1.3. Dependence of solution on initial data........................................................ 1.4. The Lyapunov stability................................................................................ 1.5. The Lyapunov function............................................................................... 1.6. Stability of linear system equilibrium positions.......................................... 1.7. Stability of nonlinear system equilibrium positions.................................... 1.8. Stability of on-line systems......................................................................... C h a p t e r 2. CONTROL OF ONE-STEP METHOD ACCURACY AND STABILITY........................................................................... 2.1. Basic definitions.......................................................................................... 2.2. Calculation accuracy control....................................................................... 2.3. Stability control........................................................................................... 2.4. Implementation of explicit methods............................................................ C h a p t e r 3. INTEGRATION ALGORITHMS WITH ACCURACY CONTROL BASED ON EXPLICIT METHODS........................... 3.1. The Euler explicit method........................................................................... 3.2. The trapezoid method.................................................................................. 3.3. RungeCutta type methods......................................................................... 3.4. RungeCutta type methods of the second order accuracy........................... 3.5. RungeCutta type methods of the third order accuracy.............................. 3.6. The RungeCuttaMerson method.............................................................. C h a p t e r 4. INTEGRATION ALGORITHMS WITH NUMERICAL SCHEME STABILITY CONTROL................................................ 4.1. Schemes of the second order accuracy........................................................ 4.2. Schemes of the third order accuracy........................................................... 4.3. Schemes of the fourth and fifth order accuracy........................................... C h a p t e r 5. INTEGRATION OF VARIABLE ORDER AND STEP ALGORITHMS.............................................................................. 5.1. Algorithms based on a three-stage system................................................ 5.2. An algorithm based on the stages of the RungeCuttaMerson method....................................................................................................... 5.3. An algorithm based on the stages of the RungeCuttaFelberg method of the fifth order............................................................................ 5.4. An algorithm based on the stages of the RungeCuttaFelberg method of the seventh order...................................................................... 5.5. An algorithm based on the stages of the DormandPrince method of the eighth order.......................................................................................... 5.6. An algorithms based on a two-stage system.............................................. 5.7. Stability polynomials................................................................................. C h a p t e r 6. ROSENBROK-TYPE METHODS................................................ 6.1. Numerical schemes................................................................................... 6.2. Integration algorithms based on the two-stage numerical scheme............ 6.3. Freezing of the Jacobean matrix in the Rosenbrok-type methods............. C h a p t e r 7. THE CLASS OF (m, k)-METHODS............................................. 7.1. Numerical schemes................................................................................... 7.2. Taylors series for method stages.............................................................. 7.3. Numerical schemes with one calculation of the right part........................ 7.4. General concepts....................................................................................... 7.5. Linear problem solution methods.............................................................. 7.6. Nonlinear problem solution methods. Schemes with one calculation of the right part.......................................................................... ................ 7.7. Nonlinear problem solution methods. Schemes with two calculations of the right part.......................................................................................... 7.8. The Jacobean matrix freezing in the (3, 2)-method of stiff problem solution...................................................................................................... C h a p t e r 8. STUDY OF THE (m, 3)-METHODS............................................ 8.1. Notation..................................................................................................... 8.2. The A-stable (m, 3)-method of the fifth order accuracy............................ 8.3. The A-stable (m, 3)-method of the fifth order accuracy............................ 8.4. The theorem of maximum order accuracy of (m, 3)-methods................... 8.5. The theorem of maximum order accuracy of (m, 3)-methods with the Jacobean matrix freezing........................................................................... C h a p t e r 9. HYBRID SYSTEMS..................................................................... 9.1. The discrete continuous model.................................................................. 9.2. The Zenon effect....................................................................................... 9.3. Modes and events...................................................................................... 9.4. Local and global behavior......................................................................... 9.5. Discontinuities........................................................................................... 9.6. The analysis of event-continuous systems................................................. 9.7. The classification of events....................................................................... 9.8. Instrumental analysis of hybrid systems (HS)........................................... C h a p t e r 10. CORRECT DETECTION OF DISCRETE EVENTS.................. 10.1. The hybrid model uncertainty domain.................................................... 10.2. The problem of correct detection of discrete events................................ 10.3. Linearization and the identification method for the localization of events...................................................................................................... 10.4. Provision of asymptotic approximation to the boundary surface in explicit difference schemes..................................................................... 10.5. The Adams method for event detecting................................................... 10.6. The L-stable method for event detecting................................................. 10.7. Event detection by instrumental means................................................... C h a p t e r 11. THE ADAPTIVE METHOD OF STUDYING MODES OF INCREASED STIFFNESS HYBRID SYSTEMS................. 11.1. Stiffness detection................................................................................... 11.2. An implicit method with stiffness control............................................... 11.3. A variable order and step explicit method............................................... 11.4. Instrumental analysis of increased stiffness HS modes........................... C h a p t e r 12. SOFTWARE................................................................................ 12.1. Visual computer models.......................................................................... 12.2. Symbolic computer models..................................................................... 12.3. Structure-symbolic computer models...................................................... 12.4. Computer model implementation............................................................ 12.5. Graphic interpretation of computer experiment results........................... 12.6. Peculiarities of hybrid system programming........................................... C h a p t e r 13. SOFTWARE UNIFICATION..................................................... 13.1. Problem relevance and statement............................................................ 13.2. Formulating differential equations of chemical kinetics......................... 13.3. Syntax...................................................................................................... 13.4. Semantics................................................................................................ 13.5. The specification of complex chemical reactions.................................... 13.6. Peculiarities of unification....................................................................... C h a p t e r 14. INSTRUMENT-ORIENTED ANALYSIS OF HYBRID SYSTEMS................................................................................... 14.1. The automatic tracking system................................................................ 14.2. The ring modulator.................................................................................. 14.3. Biosystems.............................................................................................. 14.4. Computer analysis of high dimension systems........................................ 14.5. Computer simulation of plant tissue growth and differentiation............. References.............................................................................................................. .. .. .. .. .. 07.09. 60 90 1/16. .-. . 28,25. . . 28,25. 100 .

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