PREFACE ......................................................... 5
LIST OF ABBREVIATIONS AND SYMBOLS ............................... 6
1. INTRODUCTION ................................................. 7
1.1. Classes of functional differential equations ............ 7
1.2. Applications of functional differential equations in
science and engineering ................................. 9
1.3. Direct approximate methods ............................. 13
1.4. Iterative methods for solution of functional
differential equations ................................. 16
1.5. Direct approximate methods versus iterative methods .... 20
2. TWO-STEP RUNGE KUTTA METHODS AS DIRECT NUMERICAL METHODS
FOR DELAY DIFFERENTIAL EQUATIONS ............................ 23
2.1. Introduction ........................................... 23
2.2. Stability of TSRK methods .............................. 25
2.3. Order conditions ....................................... 32
2.4. Order conditions for CTSRK methods ..................... 34
2.5. Construction of TSRK methods with a given stability
polynomial ............................................. 35
2.5.1. Examples of construction of three stage TSRK
methods of order four and four stage TSRK
methods of order five ........................... 40
2.5.2. An example of implementation of
the constructed four stage explicit TSRK
method of order and stage order p = q = 5 ....... 43
2.5.3. Numerical experiments ........................... 48
2.6. Construction and implementation of TSRK methods using
Nordsieck representation ............................... 50
2.6.1. Error propagation ............................... 51
2.6.2. Starting procedure .............................. 53
2.6.3. Computation of approximations to the Nordsieck
vector z(tn+1,hn) and hnp+ly{p+l)(tn+l) ............. 53
2.6.4. Computation of ỹn and hn+1ƒ(Ỹ[n]) ................. 56
2.6.5. An example of construction of three stage TSRK
methods of order and stage order three .......... 57
2.6.6. Derivation of continuous explicit TSRK methods
of order three .................................. 59
2.6.7. Numerical experiments ........................... 62
2.7. Construction of implicit stiffly accurate TSRK
methods ................................................ 67
2.7.1. Continuous extensions to stiffly accurate TSRK
methods ......................................... 70
2.7.2. Numerical experiments ........................... 72
2.8. Highly stable parallel two-step Runge-Kutta methods
and their P-stable continuous extensions ............... 73
2.8.1. Construction of P-stable TSRK methods for DDEs
with s = 1 and p = g = 2 ........................ 74
2.9. Construction of P-stable TSRK methods for DDEs with
s = 2 and p = q = 3 .................................... 75
2.9.1. Construction of P-stable TSRK methods for DDEs
with s = 3 and p = q = 4 ........................ 78
2.9.2. Numerical experiments ........................... 80
3. ITERATIVE METHODS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS
AND DELAY-DEPENDENT ERROR ESTIMATES ......................... 82
3.1. Convergence of WR methods for functional differential
systems of equations and the existence and uniqueness
of the solution on the whole interval I ................ 82
3.1.1. Convergence and error estimates of WR methods
for functional differential systems of
equations ....................................... 82
3.1.2. The convergence of the perturbed continuous-
time WR methods ................................. 91
3.1.3. The existence and uniqueness of the solution
on the whole interval ........................... 93
3.2. Delay-dependent error estimates for WR methods ......... 95
3.2.1. Error estimates for WR methods .................. 95
3.2.2. Error estimates for nonnegative m ............... 99
3.2.3. Error estimates for nonpositive m .............. 101
3.2.4. Other error estimates for WR methods ........... 102
3.2.5. Error estimates for special cases .............. 103
3.2.6. A general case of WR methods ................... 105
3.2.7. A discussion of the results and other
remarks ........................................ 112
3.3. Examples .............................................. 114
3.4. Delay-dependent error estimates for WR methods for
neutral functional differential systems ............... 118
3.4.1. Introductory remarks ........................... 118
3.4.2. Existence of WR iterations ..................... 120
3.4.3. Convergence of WR iterations ................... 122
3.4.4. Delay dependent error estimates ................ 126
3.5. Error estimates for special cases ..................... 130
3.5.1. Numerical examples ............................. 133
3.6. Convergence V iterative methods for general
differential algebraic systems ........................ 137
3.6.1. Existent and uniqueness of solution.
Convergence of WR methods ...................... 137
3.6.2. Special cases of problem (3.189)-(3.190) and
other remarks .................................. 143
3.6.3. Existence and uniqueness of solution to
quasi-linear system and convergence of WR
methods ........................................ 146
3.6.4. The convergence of iterations of the Gauss-
Seidel and other types ......................... 150
3.7. A one sided Lipschitz condition. The existence and
uniqueness of a solution and the convergence of WR
methods ............................................... 153
3.7.1. Comments and examples .......................... 155
REFERENCES .................................................... 160
SUMMARY IN ENGLISH ............................................ 168
SUMMARY IN POLISH ............................................. 170
|
