URAL STATE UNIVERSITY
On the rights of manuscript
KWON Oh Bok
IMPLICIT NUMERICAL METHODS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR SOFTWARE REALIZATION.
01.01.02 - Differential equations
THESIS
submitted for degree of doctor of philosophy
Ekaterinburg - 2000
Contents
Introduction.......................................................... 4
Chapter 1. Implicit one-step methods (with fixed step size) for functional differential equations 12
1. Statement of the problem.......................................... 12
2. Implicit Euler method with piece-wise constant interpolation 13
3. Interpolation and extrapolation of the model pre-history 16
3.1. Interpolating operators.................................... 16
3.2. Extrapolating operators.................................... 19
3.3. Interpolating-extrapolating operator....................... 21
4. Implicit Runge-Kutta-like methods................................. 21
4.1. General scheme ............................................ 22
4.2. Convergence order.......................................... 25
4.3. Approximation order ....................................... 27
4.4. Potential for parallel computation......................... 31
4.5. On the structure of functionals............................ 32
Chapter 2. Numerical algorithms (with variable step size) for functional differential equations 34
5. Description of IRK-methods with variable step size control . 34
6. Convergence order................................................. 37
7. Methods of interpolation and extrapolation of the extended
pre-history of discrete model................................... 39
8. Choice of the step size........................................... 43
9. Numerical modeling of control tirne-delay system.................. 45
Chapter 3. Testing and comparing explicit and implicit Runge-Kutta-like methods for functional differential equations 49
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10. Realization and comparing explicit and implicit numerical methods with fixed step size.......................................... 50
10.1. Testing of explicit and implicit Euler methods .... 50
10.2. High order methods with fixed step size ................... 63
11. Explicit and implicit methods with automatic step size control 66
11.1. Explicit Runge-Kutta-like method of 2(3) order with
automatic step size control.............................. 66
11.2. Implicit trapezoidal method with interpolation and automatic step size control..................................... 67
12. Model examples (medicine and control theory)...................... 79
12.1. HIV-infection model........................................ 79
12.2. Modeling of the linear quadratic control problem with
delay.................................................... 81
Bibliography 86
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Introduction
Investigation of the various phenomena allows one to conclude that future behaviour of many processes depends not only the present state, but also it depends on pre-history. Mathematical description of such processes can be realized on the basis of equations with different types of delays -the so called functional differential equations (FDE), equations with delays, hereditary systems.
The delay (in general form - deviation of the argument in an equation) can cause interesting mathematical phenomena: instability, periodic solutions, limit cycles, appearance and disappearance of stiffness. As the consequence, at present there is the intensification of the investigation of qualitative properties of systems with delays in different directions [22, 39, 40, 54, 77, 78, 79, 100, 117, 124].
The obtained results are applied for modeling and analysis of automatic control processes, stability of motion, mechanics, technological processes, biology, medicine, chemistry, economic and so on (one can find examples [1, 5, 39, 43, 50, 54, 77, 84, 99].
Foundations and fundamental results in the theory of functional differential equations were developed by V.Volterra, R.Bellman, N.N.Krasovskii,
A.D.Myshkis, S.N.Shimanov, L.E.El’sgolts, N.V.Azbelev, H.T.Banks, T.A.Burton, G.A.Kamenskii, C.Corduneanu, M.C.Delfour, R.D.Driver, J.Hale, L.Hatvani, V.B.Kolmanovskii, A.V.Kryazhimskii, A.B.Kurzhanskii, V.Laksmikantham, A.A.Martynyuk, A.Manitius, K.L.Cook, S.B.Norkin, V.R.Nosov, Yu.S.Osipov, A.L.Skubachevskii, S.B.Razumihin and many other mathematicians.
In the present work we follow the functional approach proposed and elaborated by academician N.N.Krasovskii and Sverdlovsk scientific school of mathematics and mechanics.
One of the obstacles for investigating systems with delays consists in the absence of the general analytical representations for FDE solutions. Besides, systems with delays are essentially infinite dimensional systems, and this fact causes principal difficulties for analysis of such systems.
Hence elaboration of the effective numerical methods for FDE and their software realization is the very important and urgent task. Insufficient development of the software programs for simulating and analysis of FDE
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is one of the main obstacles for more wide application of delay models in applied problems.
Let us mention some directions of the theory of numerical methods for FDE.
1. Numerical schemes which are based on the step method [40]. If we consider the system with constant delays
x = f{t,x(t),x(t- t)),
then, substituting known initial function instead of the term x(t - r), we obtain on the interval [£o,£o + r] ODE that can be solved by a standard method. Then we use the obtained on the interval [^o>+ t\ solution as the initial function to find solutions on the interval [£q + r, 4- 2r], and so on.
Advantage of the method consists in a greater simplicity. Unfortunately this method cannot be directly applied to systems with other types of delay (varying delay, distributed delay), besides the method is not suitable for realization of numerical procedures with automatic step size control (the basis of the most algorithms realized in contemporary toolboxes).
2. There are many papers devoted to numerical methods that use the specific form of FDE. Especially there is a big bibliography for systems for systems with lumped time-depending delays
x = f(t,x(t),x(t-r(t))) and for integro-differential Volterra equations
t
x = J f(ty Syx(s))ds.
h
For such systems there were elaborated the analogs of almost all known for ODE numerical methods (see, for example, reviews [33, 56, 19, 12]).
3. The idea of the continuous methods [122, 35, 51] consists in constructing approximate models not only at points £n, but also in all points of the delay interval. Such methods can be used for general systems of the form
± = /(*, £«(*)), *«(•) = {x(t + s), -r < s < 0}.
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Nevertheless even for ODE continuous methods are not usually used in software packages. Especially it is necessary to note that there are no continuous methods among implicit methods.
4. Many types of FDE can be reduced to integral equations [56] and solved by standard for integral equations methods. However, in contrary to ODE, the integral equations are solved by non-positional methods, i.e. for positional methods at point t one can use information about some part of trajectory, meanwhile for solving integral equation we calculate equation on the whole interval. This fact is the obstacle of application such methods in the positional control problems for FDE.
5. Systems with delays can be approximated by ODE systems of a high dimension. The method is based on the ideas of [80, 113]. The accuracy of the method is not very high, and the is applied for low dimension control problems [14, 15].
6. Functional approach [79, 54], representation of FDE as differential equations in Banach spaces and the semi-group approach are very fruitful for theoretical investigating FDE. However the approaches do not allow one to elaborate effective numerical methods because of the difficulties of calculating derivatives of functionals.
7. In the works [70, 71, 72] a new approach to constructing numerical methods for FDE
x = f(t, x(t),xt{-)), xt(-) = {x(t +s),-r < s < 0}. (0.1)
was elaborated. The approach is based on:
1) separation of finite dimensional and infinite dimensional components in FDE structure;
2) constructing with respect to the finite dimensional component the complete analogs known for ODE numerical algorithms;
3) interpolating (with the given properties) of the discrete model prehistory;
4) application of the special technique of calculating derivatives of functionals (along FDE solutions) (these methods are called i-smooth calculus
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