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Contents
Preface xi
Notations and Remarks xv
Authors xvii
1. Delay Ordinary Differential Equations 1
1.1. First Order Equations. Cauchy Problem. Method of Steps. Exact Solutions 1
1.1.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2. First-Order ODEs with Constant Delay. Cauchy Problem.
Qualitative Features . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.3. Exact Solutions to a First-Order Linear ODE with Constant
Delay. The Lambert WFunction and Its Properties . . . . . . . 4
1.1.4. First-Order Nonlinear ODEs with Constant Delay That Admit
Linearization or Exact Solutions . . . . . . . . . . . . . . . . . 12
1.1.5. Method of Steps. Solution of the Cauchy Problem for a
First-Order ODE with Constant Delay . . . . . . . . . . . . . . 13
1.1.6. Equations with Variable Delay. ODEs with Proportional Delay . 17
1.1.7. Existence and Uniqueness of Solutions. Suppression of
Singularities in Solving Blow-Up Problems . . . . . . . . . . . 23
1.2. Second- and Higher-Order Delay ODEs. Systems of Delay ODEs . . . 28
1.2.1. Basic Concepts. The Cauchy Problem . . . . . . . . . . . . . . 28
1.2.2. Second-Order Linear Equations. The Cauchy Problem. Exact
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.2.3. Higher-Order Linear Delay ODEs . . . . . . . . . . . . . . . . 34
1.2.4. Linear Systems of First- and Second-Order ODEs with Delay.
The Cauchy Problem. Exact Solutions . . . . . . . . . . . . . . 38
1.3. Stability (Instability) of Solutions to Delay ODEs . . . . . . . . . . . . 42
1.3.1. Basic Concepts. General Remarks on Stability of Solutions to
Linear Delay ODEs . . . . . . . . . . . . . . . . . . . . . . . . 42
1.3.2. Stability of Solutions to Linear ODEs with a Single Constant Delay 43
1.3.3. Stability of Solutions to Linear ODEs with Several Constant
Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.3.4. Stability Analysis of Solutions to Nonlinear Delay ODEs by the
First Approximation . . . . . . . . . . . . . . . . . . . . . . . 52
1.4. Exact and Approximate Analytical Solution Methods for Delay ODEs . 55
1.4.1. Using Integral Transforms for Solving Linear Problems . . . . . 55
1.4.2. Representation of Solutions as Power Series in the Independent
Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
v
vi CO NT E NT S
1.4.3. Method of Regular Expansion in a Small Parameter . . . . . . . 69
1.4.4. Method of Matched Asymptotic Expansions. Singular
Perturbation Problems with a Boundary Layer . . . . . . . . . . 71
1.4.5. Method of Successive Approximations and Other Iterative
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1.4.6. Galerkin Type Projection Methods. Collocation Method . . . . 80
2. Linear Partial Differential Equations with Delay 85
2.1. Properties and Specific Features of Linear Equations and Problems with
Constant Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.1.1. Properties of Solutions to Linear Delay Equations . . . . . . . . 85
2.1.2. General Properties and Qualitative Features of Delay Problems . 91
2.2. Linear Initial-Boundary Value Problems with Constant Delay . . . . . . 91
2.2.1. First Initial-Boundary Value Problem for One-Dimensional
Parabolic Equations with Constant Delay . . . . . . . . . . . . 91
2.2.2. Other Problems for a One-Dimensional Parabolic Equation
with Constant Delay . . . . . . . . . . . . . . . . . . . . . . . 96
2.2.3. Problems for Linear Parabolic Equations with Several Variables
and Constant Delay . . . . . . . . . . . . . . . . . . . . . . . . 102
2.2.4. Problems for Linear Hyperbolic Equations with Constant Delay 107
2.2.5. Stability and Instability Conditions for Solutions to Linear
Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . 109
2.3. Hyperbolic and Differential-Difference Heat Equations . . . . . . . . . 113
2.3.1. Derivation of the Hyperbolic and Differential-Difference Heat
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.3.2. Stokes Problem and Initial-Boundary Value Problems for the
Differential-Difference Heat Equation . . . . . . . . . . . . . . 115
2.4. Linear Initial-Boundary Value Problems with Proportional Delay . . . . 119
2.4.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 119
2.4.2. First Initial-Boundary Value Problem for a Parabolic Equation
with Proportional Delay . . . . . . . . . . . . . . . . . . . . . 120
2.4.3. Other Initial-Boundary Value Problems for a Parabolic
Equation with Proportional Delay . . . . . . . . . . . . . . . . 122
2.4.4. Initial-Boundary Value Problem for a Linear Hyperbolic
Equation with Proportional Delay . . . . . . . . . . . . . . . . 124
3. Analytical Methods and Exact Solutions to Delay PDEs. Part I 127
3.1. Remarks and Definitions. Traveling Wave Solutions . . . . . . . . . . . 127
3.1.1. Preliminary Remarks. Terminology. Classes of Equations
Concerned . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1.2. States of Equilibrium. Traveling Wave Solutions. Exact
Solutions in Closed From . . . . . . . . . . . . . . . . . . . . 130
3.1.3. Traveling Wave Front Solutions to Nonlinear Reaction-
Diffusion Type Equations . . . . . . . . . . . . . . . . . . . . 134
3.2. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . 139
3.2.1. Preliminary Remarks. Terminology. Examples . . . . . . . . . 139
CON TE N TS vii
3.2.2. Delay Reaction-Diffusion Equations Admitting Separable
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.2.3. Delay Klein–Gordon Type Equations Admitting Separable
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.2.4. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . 149
3.3. Generalized and Functional Separable Solutions . . . . . . . . . . . . . 154
3.3.1. Generalized Separable Solutions . . . . . . . . . . . . . . . . . 154
3.3.2. Functional Separable Solutions . . . . . . . . . . . . . . . . . 159
3.3.3. Using Linear Transformations to Construct Generalized and
Functional Separable Solutions . . . . . . . . . . . . . . . . . 162
3.4. Method of Functional Constraints . . . . . . . . . . . . . . . . . . . . 166
3.4.1. General Description of the Method of Functional Constraints . . 166
3.4.2. Exact Solutions to Quasilinear Delay Reaction-Diffusion
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.4.3. Exact Solutions to More Complicated Nonlinear Delay
Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . 177
3.4.4. Exact Solutions to Nonlinear Delay Klein–Gordon Type Wave
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4. Analytical Methods and Exact Solutions to Delay PDEs. Part II 201
4.1. Methods for Constructing Exact Solutions to Nonlinear Delay PDEs
Using Solutions to Simpler Non-Delay PDEs . . . . . . . . . . . . . . 201
4.1.1. The First Method for Constructing Exact Solutions to Delay
PDEs. General Description and Simple Examples . . . . . . . . 201
4.1.2. Using the First Method for Constructing Exact Solutions to
Nonlinear Delay PDEs . . . . . . . . . . . . . . . . . . . . . . 203
4.1.3. The Second Method for Constructing Exact Solutions to Delay
PDEs. General Description and Simple Examples . . . . . . . . 207
4.1.4. Employing the Second Method to Construct Exact Solutions to
Nonlinear Delay PDEs . . . . . . . . . . . . . . . . . . . . . . 209
4.2. Systems of Nonlinear Delay PDEs. Generating Equations Method . . . 212
4.2.1. General Description of the Method and Application Examples . 212
4.2.2. Quasilinear Systems of Delay Reaction-Diffusion Equations
and Their Exact Solutions . . . . . . . . . . . . . . . . . . . . 215
4.2.3. Nonlinear Systems of Delay Reaction-Diffusion Equations and
Their Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . 218
4.2.4. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . 222
4.3. Reductions and Exact Solutions of Lotka–Volterra Type Systems and
More Complex Systems of PDEs with Several Delays . . . . . . . . . . 225
4.3.1. Reaction-Diffusion Systems with Several Delays. The Lotka–
Volterra System . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.3.2. Reductions and Exact Solutions of Systems of PDEs with
Different Diffusion Coefficients (a16=a2) . . . . . . . . . . . . 226
4.3.3. Reductions and Exact Solutions of Systems of PDEs with Equal
Diffusion Coefficients (a1=a2) . . . . . . . . . . . . . . . . . 239
viii CO NT E NT S
4.3.4. Systems of Delay PDEs Ho mogeneous in the Unknown Functions 247
4.4. Nonlinear PDEs with Proportional Arguments. Principle of Analogy of
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
4.4.1. Principle of Analogy of Solutions . . . . . . . . . . . . . . . . 250
4.4.2. Exact Solutions to Quasilinear Diffusion Equations with
Proportional Delay . . . . . . . . . . . . . . . . . . . . . . . . 253
4.4.3. Exact Solutions to More Complicated Nonlinear Diffusion
Equations with Proportional Delay . . . . . . . . . . . . . . . . 257
4.4.4. Exact Solutions to Nonlinear Wave-Type Equations with
Proportional Delay . . . . . . . . . . . . . . . . . . . . . . . . 263
4.5. Unstable Solutions and Hadamard Ill-Posedness of Some Delay Problems 270
4.5.1. Solution Instability for One Class of Nonlinear PDEs with
Constant Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 270
4.5.2. Hadamard Ill-Posedness of Some Delay Problems . . . . . . . 271
5. Numerical Methods for Solving Delay Differential Equations 273
5.1. Numerical Integration of Delay ODEs . . . . . . . . . . . . . . . . . . 273
5.1.1. Main Concepts and Definitions . . . . . . . . . . . . . . . . . . 273
5.1.2. Qualitative Features of the Numerical Integration of Delay ODEs 275
5.1.3. Modified Method of Steps . . . . . . . . . . . . . . . . . . . . 278
5.1.4. Numerical Methods for ODEs with Constant Delay . . . . . . . 279
5.1.5. Numerical Methods for ODEs with Proportional Delay. Cauchy
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
5.1.6. Shooting Method (Boundary Value Problems) . . . . . . . . . . 287
5.1.7. Integration of Stiff Systems of Delay ODEs Using the
Mathematica Software . . . . . . . . . . . . . . . . . . . . . . 290
5.1.8. Test Problems for Delay ODEs. Comparison of Numerical and
Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.2. Numerical Integration of Delay PDEs . . . . . . . . . . . . . . . . . . 296
5.2.1. Preliminary Remarks. Method of Time-Domain Decomposition 296
5.2.2. Method of Lines—Reduction of a Delay PDE to a System of
Delay ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
5.2.3. Finite Difference Methods . . . . . . . . . . . . . . . . . . . . 302
5.3. Construction, Selection, and Usage of Test Problems for Delay PDEs . . 309
5.3.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 309
5.3.2. Main Principles for Selecting Test Problems . . . . . . . . . . . 309
5.3.3. Constructing Test Problems . . . . . . . . . . . . . . . . . . . 310
5.3.4. Comparison of Numerical and Exact Solutions to Nonlinear
Delay Reaction-Diffusion Equations . . . . . . . . . . . . . . . 316
5.3.5. Comparison of Numerical and Exact Solutions to Nonlinear
Delay Klein–Gordon Type Wave Equations . . . . . . . . . . . 322
CON TE N TS ix
6. Models and Delay Differential Equations Used in Applications 327
6.1. Models Described by Nonlinear Delay ODEs . . . . . . . . . . . . . . 327
6.1.1. Hutchinson’s Equation—a Delay Logistic Equation . . . . . . . 327
6.1.2. Nicholson’s Equation . . . . . . . . . . . . . . . . . . . . . . . 330
6.1.3. Mackey–Glass Hematopoiesis Model . . . . . . . . . . . . . . 333
6.1.4. Other Nonlinear Models with Delay . . . . . . . . . . . . . . . 336
6.2. Models of Economics and Finance Described by ODEs . . . . . . . . . 340
6.2.1. The Simplest Model of Macrodynamics of Business Cycles . . 340
6.2.2. Model of Interaction of Three Economical Parameters . . . . . 341
6.2.3. Delay Model Describing Tax Collection in a Closed Economy . 342
6.3. Models and Delay PDEs in Population Theory . . . . . . . . . . . . . . 343
6.3.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . 343
6.3.2. Diffusive Logistic Equation with Delay . . . . . . . . . . . . . 344
6.3.3. Delay Diffusion Equation Taking into Account Nutrient
Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
6.3.4. Lotka–Volterra Type Diffusive Logistic Model with Several
Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
6.3.5. Nicholson’s Reaction-Diffusion Model with Delay . . . . . . . 347
6.3.6. Model That Takes into Account the Effect of Plant Defenses on
a Herbivore Population . . . . . . . . . . . . . . . . . . . . . . 349
6.4. Models and Delay PDEs Describing the Spread of Epidemics and
Development of Diseases . . . . . . . . . . . . . . . . . . . . . . . . . 350
6.4.1. Classical SIR Model of Epidemic Spread . . . . . . . . . . . . 350
6.4.2. Two-Component Epidemic SI Model . . . . . . . . . . . . . . 353
6.4.3. Epidemic Model of the New Coronavirus Infection . . . . . . . 354
6.4.4. Hepatitis B Model . . . . . . . . . . . . . . . . . . . . . . . . 355
6.4.5. Model of Interaction Between Immunity and Tumor Cells . . . 357
6.5. Other Models Described by Nonlinear Delay PDEs . . . . . . . . . . . 358
6.5.1. Belousov–Zhabotinsky Oscillating Reaction Model . . . . . . . 358
6.5.2. Mackey–Glass Model of Hematopoiesis . . . . . . . . . . . . . 359
6.5.3. Model of Heat Treatment of Metal Strips . . . . . . . . . . . . 360
6.5.4. Food Chain Model . . . . . . . . . . . . . . . . . . . . . . . . 361
6.5.5. Models of Artificial Neural Networks . . . . . . . . . . . . . . 362
References 365
Index 399
Preface
Linear and nonlinear differential equations with delay (ordinary and partial) or, sim-
ply, delay differential equations∗are often used for mathematical modeling of phe-
nomena and processes in various areas of theoretical physics, mechanics, control the-
ory, biology, biophysics, biochemistry, medicine, ecology, economics, and technical
applications.
Let us list a few factors that lead one to introduce delay into mathematical models
described by differential equations. For example, in biology and biomechanics,
delays are due to the limited speed of transmission of nerve and muscle reactions in
living tissues. In medicine, when one deals with the spread of infectious diseases,
the delay time is determined by the incubation period (the time interval between
initial contact with an infectious agent and appearance of the first signs or symptoms
of the disease). In population dynamics, delays arise because individuals participate
in reproduction only after reaching a certain age. In control theory, delays are
usually associated with the finite speed of signal propagation and the limited speed
of technological processes.
The presence of a delay in mathematical models and differential equations is a
complicating factor, which, as a rule, leads to a narrowing of the stability region of
the solutions obtained. Studying and solving ordinary differential equations (ODEs)
with delay is comparable in complexity to studying and solving partial differential
equations (PDEs) without delay.
The book details qualitative features of delay differential equations and presents
typical statements of initial value and initial-boundary value problems for them.
Exact, approximate analytical, and numerical methods for solving such equations are
described. In addition to differential equations with constant delay, equations with
proportional delay (of the pantograph type) are studied, as well as more complex
equations with a general variable delay or several delays. The presentation of the
theoretical material comes with examples of the practical application of the methods
to obtain desired solutions.
The book reviews the most common mathematical models with delay used in
population theory, biology, medicine, and other applications.
Analytical solutions to Cauchy-type linear problems for first- and second-order
ODEs and systems of ODEs with constant or proportional delays are presented.
Some classes of nonlinear first-order delay ODEs that admit linearization or exact
solutions are considered. The issues of stability and instability of solutions to ODEs
with delay are discussed.
The most common analytical and numerical methods for solving initial and
boundary value problems for ODEs with constant or variable delay are described.
∗In the literature, there is also a longer alternative name: differential equations with delayed argument.
xi
xii PR EFAC E
These include the method of steps, methods of integral transforms, method of regular
expansion in a small parameter, method of matched asymptotic expansions, iterative
methods, Adomian decomposition method, homotopy analysis method, collocation
method, Galerkin-type projection methods, Euler and Runge–Kutta methods, shoot-
ing method, methods based on the use of the Mathematica package, and more.
We use the method of separation of variables to obtain Fourier series solutions
in space variables of linear initial-boundary value problems for parabolic and hyper-
bolic PDEs with constant or proportional delay and different boundary conditions.
Numerical methods for solving initial-boundary value problems for linear and non-
linear delay PDEs are also presented. The most attention is paid to the method of
lines, which relies on reducing a delay PDE to a system of delay ODEs. Finite-
difference methods based on an implicit scheme, a weighting scheme, a scheme of
increased order of accuracy, and more are considered. The time domain decompo-
sition method, which generalizes the method of steps used to solve delay ODEs,
is also discussed. We formulate the basic principles for constructing and selecting
test problems for assessing the adequacy and estimating the accuracy of numerical
methods and approximate analytical methods for solving delay PDEs.
The general solutions to nonlinear delay PDEs cannot be obtained even in the
simplest cases. Therefore, when studying such equations, one usually has to search
and analyze their particular solutions, usually called exact solutions.
The book pays much attention to the description and practical application of
methods for constructing exact solutions to nonlinear equations of mathematical
physics with delay. These are the methods of generalized and functional separa-
tion of variables, the method of functional constraints, the method of generating
equations, the principle of the analogy of solutions, and others. Notably, the vast
majority of analytical methods that successfully allow one to find exact solutions
of nonlinear partial differential equations without delay are either inapplicable to
constructing exact solutions of nonlinear PDEs with constant or variable delay or
have a minimal area of applicability. Equations of mathematical physics with two
independent variables and a delay have the following essential qualitative features:
(i) PDEs with constant delay do not admit self-similar solutions, unlike PDEs without
delay, many of which do, and (ii) PDEs with proportional delay in either independent
variable do not have traveling wave solutions, unlike simpler PDEs without delay,
which often have.
The book considers many nonlinear reaction-diffusion and wave-type equations
with delay dependent on one or several arbitrary functions or involving several free
parameters. Such equations are the most difficult to analyze, and their exact solutions
can be used to test numerical and approximate analytical methods for solving related
initial-boundary value problems and estimate the errors of the methods.
The book contains much new material that has not previously been published in
monographs.
The authors tried to avoid using special terminology whenever possible to maxi-
mize the circle of potential readers with different mathematical backgrounds. There-
fore, some results are described in a schematic and simplified manner, which suffices
for practical applications. Many sections can be read independently, making it easier
PRE FACE xiii
to work with the material. A detailed table of contents allows the reader to find the
desired information quickly.
The authors thank A. V. Aksenov for the discussions and valuable remarks.
The authors hope that the book will be helpful for a wide range of scientists,
university professors, and graduate and postgraduate students specializing in applied
mathematics, mathematical physics, computational mathematics, mechanics, control
theory, biology, biophysics, biochemistry, medicine, chemical technology, and ecol-
ogy. In addition, individual sections of the book, methods and examples can be used
in teaching applied mathematics, mathematical physics, and functional differential
equations to deliver special courses and perform practical exercises.
Authors
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