Differential and Difference Equations with Applications in Queueing Theory

Aliakbar Montazer Haghighi, PhD, is Regent Professor, Professor, and former Head of the Department of Mathematics at Prairie View A&M University. He’s the Co-founder and Founding Editor-in-Chief of Applications and Applied Mathematics: An International Journal (AAM). Dimitar P. Mishev, PhD, is Professor in the Department of Mathematics at Prairie View, A&M University. His research is focused on differential and difference equations and queueing theory.

• About the Authors xiiiPreface to the Second Edition xv1 Introduction 11.1 Introduction 11.2 Functions of a Real Variable 11.3 Some Properties of Differentiable Functions 31.4 Functions of More Than One Real Variable 31.4.1 The Chain Rule for Real Multivariable Functions 41.5 Function of a Complex Variable 71.5.1 Complex Numbers and Their Properties 71.5.2 Properties of a Complex Variable z 91.5.3 Complex Variables and Functions of Complex Variables 101.5.4 Some Particular Functions of Complex Variables 121.6 Differentiation of Functions of Complex Variables 121.6.1 Partial Differentiation of Functions of Complex Variables 131.7 Vectors 151.7.1 Dot (or Scalar or Inner) Product of Vectors and Some of Its Properties 171.7.2 The Cross Product (or Vector Product) of Vectors and Some of Its Properties 181.7.3 Directional Derivatives and Gradient Vectors 191.7.4 Eigenvalues and Eigenvectors 24Exercises 252 Transforms 312.1 Introduction 312.2 Fourier Series 322.3 Convergence of Fourier Series 392.4 Fourier Transform 402.4.1 Continuous Fourier Transform 442.4.2 Discrete Fourier Transform 482.4.3 Some Properties of a Fourier Transform 482.4.4 Fast Fourier Transform 492.5 Laplace Transform 502.5.1 Properties of Laplace Transform 512.5.1.1 Linearity 512.5.1.2 Existence of Laplace Transform 522.5.1.3 Uniqueness of the Laplace Transforms 532.5.1.4 The First Shifting or s-Shifting 542.5.1.5 Time Delay 542.5.1.6 Laplace Transform of Derivatives 562.5.1.7 Laplace Transform of Integral 562.5.1.8 The Second Shifting or t-Shifting Theorem 572.5.1.9 Laplace Transform of Convolution of Two Functions 592.5.2 Partial Fraction and Inverse Laplace Transform 632.6 Integral Transform 682.7 Ƶ-Transform 69Notes 70Exercises 753 Ordinary Differential Equations 813.1 Introduction and History of Ordinary Differential Educations 813.2 Basics Concepts and Definitions 813.3 Existence and Uniqueness 873.4 Separable Equations 893.4.1 Method of Solving Separable Ordinary Differential Equations 903.5 Linear Ordinary Differential Equations 983.5.1 Method of Solving a Linear First-Order Differential Equation 993.6 Exact Ordinary Differential Equations 1023.7 Solution of the First ODE by Substitution Method 1123.7.1 Substitution Method 1133.7.2 Reduction to Separation of Variables 1163.8 Applications of the First-Order ODEs 1173.9 Second-Order Homogeneous Ordinary Differential Equation 1223.9.1 Solution of the Homogenous Second-Order Homogeneous Ordinary Differential Equation with Constant Coefficients, Equation (3.9.3) 1233.10 The Second-Order Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients 1383.10.1 Method of Undetermined Coefficients 1403.10.2 Variation of Parameters Method 1473.11 Laplace Transform Method 1503.12 Cauchy–Euler Equation Differential Equation 1573.12.1 The Second-Order Homogenous Cauchy–Euler Equation 1573.12.2 Solving the Second-Order Homogeneous Cauchy–Euler Equation Using x = et or t = ln |x| 1583.13 Elimination Method to Solve Differential Equations 1603.14 Solution of Linear ODE Using Power Series 163Exercises 1684 Partial Differential Equations 1734.1 Introduction 1734.2 Basic Terminologies for Partial Differential Equations 1744.3 Some Particular Functions Used in Partial Differential Equations 1764.4 Types of Boundary Conditions for a Partial Differential Equation 1784.5 Solution for a Partial Differential Equation 1814.5.1 Methods of Finding Solution for a Partial Differential Equation 1824.6 Linear, Semi-linear, and Quasi-linear Partial Differential Equations 1844.6.1 Examples and Solutions of One- and Two-Dimensional Linear and Quasi-linear Partial Differential Equations of the First, Second, and Third Order 1884.6.2 Characteristics Equation Method with Steps 1894.7 Solution of Wave Partial Differential Equation, First and Second Orders, with Different Methods 1974.8 A One-Dimensional, Second-Order Heat (or Parabolic) Equations 211Exercises 2195 Differential Difference Equations 2235.1 Introduction 2235.2 Basic Terms 2255.3 Linear Homogeneous Difference Equations with Constant Coefficients 2285.3.1 Recursive Method 2295.3.2 Characteristic Equation Method 2305.4 Linear Nonhomogeneous Difference Equations with Constant Coefficients 2355.4.1 Characteristic Equation Method 2365.4.1.1 Case 1: a = 1 2365.4.1.2 Case 2: a ≠ 1 2375.4.1.3 Case 3:a=−1 2385.4.1.4 Case 4: a > 1 2395.4.1.5 Case 5: 0 < a < 1 2395.4.1.6 Case 6: −1 < a < 0 2405.4.1.7 Case 7: a