1st Edition

# Delay Ordinary and Partial Differential Equations

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**Delay Ordinary and Partial Differential Equations** is devoted to linear and nonlinear ordinary and partial differential equations with constant and variable delay. It considers qualitative features of delay differential equations and formulates typical problem statements. Exact, approximate analytical and numerical methods for solving such equations are described, including the method of steps, methods of integral transformations, method of regular expansion in a small parameter, method of matched asymptotic expansions, iteration-type methods, Adomian decomposition method, collocation method, Galerkin-type projection methods, Euler and Runge-Kutta methods, shooting method, method of lines, finite-difference methods for PDEs, methods of generalized and functional separation of variables, method of functional constraints, method of generating equations, and more.

The presentation of the theoretical material is accompanied by examples of the practical application of methods to obtain the desired solutions. Exact solutions are constructed for many nonlinear delay reaction-diffusion and wave-type PDEs that depend on one or more arbitrary functions. A review is given of the most common mathematical models with delay used in population theory, biology, medicine, economics, and other applications.

The book contains much new material previously unpublished in monographs. It is intended for a broad audience of scientists, university professors, and graduate and postgraduate students specializing in applied and computational mathematics, mathematical physics, mechanics, control theory, biology, medicine, chemical technology, ecology, economics, and other disciplines.

Individual sections of the book and examples are suitable for lecture courses on applied mathematics, mathematical physics, and differential equations for delivering special courses and for practical training.

1. Delay Ordinary Differential Equations

1.1. First Order Equations. Cauchy Problem. Method of Steps. Exact Solutions

1.1.1. Preliminary Remarks

1.1.2. First-Order ODEs with Constant Delay. Cauchy Problem. Qualitative Features

1.1.3. Exact Solutions to a First-Order Linear ODE with Constant

Delay. The Lambert *W *Function and Its Properties

1.1.4. First-Order Nonlinear ODEs with Constant Delay That Admit Linearization or Exact Solutions

1.1.5. Method of Steps. Solution of the Cauchy Problem for a

First-Order ODE with Constant Delay

1.1.6. Equations with Variable Delay. ODEs with Proportional Delay .

1.1.7. Existence and Uniqueness of Solutions. Suppression of

Singularities in Solving Blow-Up Problems

1.2. Second- and Higher-Order Delay ODEs. Systems of Delay ODEs

1.2.1. Basic Concepts. The Cauchy Problem

1.2.2. Second-Order Linear Equations. The Cauchy Problem. Exact Solutions

1.2.3. Higher-Order Linear Delay ODEs

1.2.4. Linear Systems of First- and Second-Order ODEs with Delay.

The Cauchy Problem. Exact Solutions

1.3. Stability (Instability) of Solutions to Delay ODEs

1.3.1. Basic Concepts. General Remarks on Stability of Solutions to

LinearDelayODEsStabilityofSolutionstoLinearODEswithaSingleConstantDelay

1.3.2. Stability of Solutions to Linear ODEs with Several Constant

Delays

1.3.3. Stability Analysis of Solutions to Nonlinear Delay ODEs by the

First ApproximationExact and Approximate Analytical Solution Methods for Delay ODEs

1.3.4. Using Integral Transforms for Solving Linear Problems

1.3.5. Representation of Solutions as Power Series in the Independent Variable

1.3.6. Method of Regular Expansion in a Small Parameter

1.3.7. Method of Matched Asymptotic Expansions. Singular

Perturbation Problems with a Boundary Layer

1.3.8. Method of Successive Approximations and Other Iterative

MethodsGalerkin Type Projection Methods. Collocation Method

2. Linear Partial Differential Equations with Delay

2.1. Properties and Specific Features of Linear Equations and Problems with Constant Delay

2.1.1. Properties of Solutions to Linear Delay Equations

2.1.2. General Properties and Qualitative Features of Delay Problems

2.2. Linear Initial-Boundary Value Problems with Constant Delay

2.2.1. First Initial-Boundary Value Problem for One-Dimensional

Parabolic Equations with Constant Delay

2.2.2. Other Problems for a One-Dimensional Parabolic Equation

with Constant Delay

2.2.3. Problems for Linear Parabolic Equations with Several Variables

and Constant Delay

2.2.4. Problems for Linear Hyperbolic Equations with Constant Delay

2.2.5. Stability and Instability Conditions for Solutions to Linear

Initial-Boundary Value Problems

2.3. Hyperbolic and Differential-Difference Heat Equations

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

2.4. Linear Initial-Boundary Value Problems with Proportional Dela

2.4.1. Preliminary Remarks

2.4.2. First Initial-Boundary Value Problem for a Parabolic Equation

with Proportional Delay

2.4.3. Other Initial-Boundary Value Problems for a Parabolic

Equation with Proportional Delay

2.4.4. Initial-Boundary Value Problem for a Linear Hyperbolic

Equation with Proportional DelayAnalytical Methods and Exact Solutions to Delay PDEs. Part I

2.5. Remarks and Definitions. Traveling Wave Solutions

2.5.1. Preliminary Remarks. Terminology. Classes of Equations

Concerned

2.5.2. States of Equilibrium. Traveling Wave Solutions. Exact

Solutions in Closed From

2.5.3. Traveling Wave Front Solutions to Nonlinear Reaction-

Diffusion Type Equations

2.6. Multiplicative and Additive Separable Solutions

2.6.1. Preliminary Remarks. Terminology. Examples

2.6.2. Delay Reaction-Diffusion Equations Admitting Separable

Solutions

2.6.3. Delay Klein–Gordon Type Equations Admitting Separable

Solutions

2.6.4. Some Generalizations

2.7. Generalized and Functional Separable Solutions

2.7.1. Generalized Separable Solutions

2.7.2. Functional Separable Solutions

2.7.3. Using Linear Transformations to Construct Generalized and Functional Separable Solutions

2.8. Method of Functional Constraints

2.8.1. General Description of the Method of Functional Constraints

2.8.2. Exact Solutions to Quasilinear Delay Reaction-Diffusion

Equations

2.8.3. Exact Solutions to More Complicated Nonlinear Delay

Reaction-Diffusion Equations

2.8.4. Exact Solutions to Nonlinear Delay Klein–Gordon Type Wave Equations 194

3. Analytical Methods and Exact Solutions to Delay PDEs. Part II

3.1. Methods for Constructing Exact Solutions to Nonlinear Delay PDEs

Using Solutions to Simpler Non-Delay PDEs

3.1.1. The First Method for Constructing Exact Solutions to Delay

PDEs. General Description and Simple Examples

3.1.2. Using the First Method for Constructing Exact Solutions to Nonlinear Delay PDEs

3.1.3. The Second Method for Constructing Exact Solutions to Delay

PDEs. General Description and Simple Examples

3.1.4. Employing the Second Method to Construct Exact Solutions to Nonlinear Delay PDEs

3.2. Systems of Nonlinear Delay PDEs. Generating Equations Method

3.2.1. GeneralDescriptionoftheMethodandApplicationExamples

3.2.2. Quasilinear Systems of Delay Reaction-Diffusion Equations

and Their Exact Solutions

3.2.3. Nonlinear Systems of Delay Reaction-Diffusion Equations and

Their Exact Solutions

3.2.4. Some Generalizations

3.3. Reductions and Exact Solutions of Lotka–Volterra Type Systems and

More Complex Systems of PDEs with Several Delays

3.3.1. Reaction-Diffusion Systems with Several Delays. The Lotka– Volterra System

3.3.2. Reductions and Exact Solutions of Systems of PDEs with

Different Diffusion Coefficients (*a*_{1} /= *a*_{2})Reductions and Exact Solutions of Systems of PDEs with Equal Diffusion Coefficients (*a*_{1} = *a*_{2})

3.3.3. SystemsofDelayPDEsHomogeneousintheUnknownFunctions

3.4. Nonlinear PDEs with Proportional Arguments. Principle of Analogy of Solutions 250

3.4.1. Principle of Analogy of Solutions

3.4.2. Exact Solutions to Quasilinear Diffusion Equations with Proportional Delay

3.4.3. Exact Solutions to More Complicated Nonlinear Diffusion

Equations with Proportional Delay

3.4.4. Exact Solutions to Nonlinear Wave-Type Equations with Proportional Delay

3.5. UnstableSolutionsandHadamardIll-PosednessofSomeDelayProblems

3.5.1. Solution Instability for One Class of Nonlinear PDEs with

Constant Delay

3.5.2. Hadamard Ill-Posedness of Some Delay Problems

4. Numerical Methods for Solving Delay Differential Equations

4.1. Numerical Integration of Delay ODEs

4.1.1. Main Concepts and Definitions

4.1.2. QualitativeFeaturesoftheNumericalIntegrationofDelayODEs

4.1.3. Modified Method of Steps

4.1.4. Numerical Methods for ODEs with Constant Delay

4.1.5. Numerical Methods for ODEs with Proportional Delay. Cauchy Problem

4.1.6. Shooting Method (Boundary Value Problems)

4.1.7. Integration of Stiff Systems of Delay ODEs Using the

Mathematica Software

4.1.8. Test Problems for Delay ODEs. Comparison of Numerical and

Exact SolutionsNumerical Integration of Delay PDEs

4.1.9. Preliminary Remarks. Method of Time-Domain Decomposition

4.1.10. Method of Lines—Reduction of a Delay PDE to a System of

Delay ODEs

4.1.11. Finite Difference Methods

4.2. Construction, Selection, and Usage of Test Problems for Delay PDEs

4.2.1. Preliminary Remarks

4.2.2. Main Principles for Selecting Test Problems

4.2.3. Constructing Test Problems

4.2.4. Comparison of Numerical and Exact Solutions to Nonlinear

Delay Reaction-Diffusion Equations

4.2.5. Comparison of Numerical and Exact Solutions to Nonlinear

Delay Klein–Gordon Type Wave Equations

5. Models and Delay Differential Equations Used in Applications

5.1. Models Described by Nonlinear Delay ODEs

5.1.1. Hutchinson’s Equation—a Delay Logistic Equation

5.1.2. Nicholson’s Equation

5.1.3. Mackey–Glass Hematopoiesis Model

5.1.4. Other Nonlinear Models with Delay

5.2. Models of Economics and Finance Described by ODEs

5.2.1. The Simplest Model of Macrodynamics of Business Cycles

5.2.2. Model of Interaction of Three Economical Parameters

5.2.3. Delay Model Describing Tax Collection in a Closed Economy

5.3. Models and Delay PDEs in Population Theory

5.3.1. Preliminary Remarks

5.3.2. Diffusive Logistic Equation with Delay

5.3.3. Delay Diffusion Equation Taking into Account Nutrient

Limitation

5.3.4. Lotka–Volterra Type Diffusive Logistic Model with Several

Delays

5.3.5. Nicholson’s Reaction-Diffusion Model with Delay

5.3.6. Model That Takes into Account the Effect of Plant Defenses on

a Herbivore Population

5.4. Models and Delay PDEs Describing the Spread of Epidemics and Development of Diseases

5.4.1. Classical SIR Model of Epidemic Spread

5.4.2. Two-Component Epidemic SI Model

5.4.3. Epidemic Model of the New Coronavirus Infection

5.4.4. Hepatitis B Model

5.4.5. Model of Interaction Between Immunity and Tumor Cells

5.5. Other Models Described by Nonlinear Delay PDEs

5.5.1. Belousov–Zhabotinsky Oscillating Reaction Model

5.5.2. Mackey–Glass Model of Hematopoiesis

5.5.3. Model of Heat Treatment of Metal Strips

5.5.4. Food Chain Model

### Biography

**Andrei D. Polyanin**, D.Sc., Ph.D., Professor, is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgarian as well as over 210 research papers and three patents.

**Vsevolod G. Sorokin**, Ph.D., graduated from the Department of Applied Mathematics of the Bauman Moscow State Technical University in 2014. He has been working as a research scientist in the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences since 2015. He defended his PhD thesis on mathematical modelling and numerical integration of reaction–diffusion systems with delay in the Bauman Moscow State Technical University in 2018. Doctor Sorokin has published about 30 research papers.

**Alexei I. Zhurov**, Ph.D., is an outstanding scientist in nonlinear mechanics, mathematical physics, computer algebra, biomechanics, and morphometrics. He graduated with honors from the Department of Airphysics and Space Research of the Moscow Institute of Physics and Technology in 1990. Since then has become a member of staff of the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, where he received his PhD in mechanics and fluid dynamics in 1995 and has become a senior research scientist since 1999. Since 2001, he has joined Cardiff University as a research scientist in the area of biomechanics and morphometrics. Doctor Zhurov has published over 120 research papers and three books, including Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics by A.D. Polyanin, V.F. Zaitsev, and A.I. Zhurov (Fizmatlit, 2005; in Russian).