Laplace transform

About: Laplace transform is a research topic. Over the lifetime, 17104 publications have been published within this topic receiving 284357 citations.

Fractional Calculus: Integral and Differential Equations of Fractional Order

Abstract: We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

An Operational Analysis of Traffic Dynamics

Abstract: The dynamics of a line of traffic composed of n vehicles is studied mathematically. It is postulated that the movements of the several vehicles are controlled by an idealized ``law of separation.'' The law considered in the analysis specifies that each vehicle must maintain a certain prescribed ``following distance'' from the preceding vehicle. This distance is the sum of a distance proportional to the velocity of the following vehicle and a certain given minimum distance of separation when the vehicles are at rest. By the application of this postulated law to the motion of the column of vehicles, the differential equations governing the dynamic state of the system are obtained.The solution of the dynamical equations for several assumed types of motion of the leading vehicle is effected by the operational or Laplace transform method and the velocities and accelerations of the various vehicles are thus obtained. Consideration is given to the use of an electrical analog computer for studying the dynamical e...

Advanced Calculus for Applications

Abstract: 1. Ordinary Differential Equations 1.1 Introduction 1.2 Linear Dependence 1.3 Complete Solutions of Linear Equations 1.4 The Linear Differential Equation of First Order 1.5 Linear Differential Equations with Constant Coefficients 1.6 The Equidimensional Linear Differential Equation 1.7 Properties of Linear Operators 1.8 Simultaneous Linear Differential Equations 1.9 particular Solutions by Variation of Parameters 1.10 Reduction of Order 1.11 Determination of Constants 1.12 Special Solvable Types of Nonlinear Equations 2. The Laplace Transform 2.1 An introductory Example 2.2 Definition and Existence of Laplace Transforms 2.3 Properties of Laplace Transforms 2.4 The Inverse Transform 2.5 The Convolution 2.6 Singularity Functions 2.7 Use of Table of Transforms 2.8 Applications to Linear Differential Equations with Constant Coefficients 2.9 The Gamma Function 3. Numerical Methods for Solving Ordinary Differential Equations 3.1 Introduction 3.2 Use of Taylor Series 3.3 The Adams Method 3.4 The Modified Adams Method 3.5 The Runge-Kutta Method 3.6 Picard's Method 3.7 Extrapolation with Differences 4. Series Solutions of Differential Equations: Special Functions 4.1 Properties of Power Series 4.2 Illustrative Examples 4.3 Singular Points of Linear Second-Order Differential Equations 4.4 The Method of Frobenius 4.5 Treatment of Exceptional Cases 4.6 Example of an Exceptional Case 4.7 A Particular Class of Equations 4.8 Bessel Functions 4.9 Properties of Bessel Functions 4.10 Differential Equations Satisfied by Bessel Functions 4.11 Ber and Bei Functions 4.12 Legendre Functions 4.13 The Hypergeometric Function 4.14 Series Solutions Valid for Large Values of x 5. Boundary-Value Problems and Characteristic-Function Representations 5.1 Introduction 5.2 The Rotating String 5.3 The Rotating Shaft 5.4 Buckling of Long Columns Under Axial Loads 5.5 The Method of Stodola and Vianello 5.6 Orthogonality of Characteristic Functions 5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions 5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations 5.9 Convergence of the Method of Stodola and Vianello 5.10 Fourier Sine Series and Cosine Series 5.11 Complete Fourier Series 5.12 Term-by-Term Differentiation of Fourier Series 5.13 Fourier-Bessel Series 5.14 Legendre Series 5.15 The Fourier Integral 6. Vector Analysis 6.1 Elementary Properties of Vectors 6.2 The Scalar Product of Two Vectors 6.3 The Vector Product of Two Vectors 6.4 Multiple Products 6.5 Differentiation of Vectors 6.6 Geometry of a Space Curve 6.7 The Gradient Vector 6.8 The Vector Operator V 6.9 Differentiation Formulas 6.10 Line Integrals 6.11 The Potential Function 6.12 Surface Integrals 6.13 Interpretation of Divergence. The Divergence Theorem 6.14 Green's Theorem 6.15 Interpretation of Curl. Laplace's Equation 6.16 Stokes's Theorem 6.17 Orthogonal Curvilinear Coordinates 6.18 Special Coordinate Systems 6.19 Application to Two-Dimensional Incompressible Fluid Flow 6.20 Compressible Ideal Fluid Flow 7. Topics in Higher-Dimensional Calculus 7.1 Partial Differentiation. Chain Rules 7.2 Implicit Functions. Jacobian Determinants 7.3 Functional Dependence 7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals 7.5 Taylor Series 7.6 Maxima and Minima 7.7 Constraints and Lagrange Multipliers 7.8 Calculus of Variations 7.9 Differentiation of Integrals Involving a Parameter 7.10 Newton's Iterative Method 8. Partial Differential Equations 8.1 Definitions and Examples 8.2 The Quasi-Linear Equation of First Order 8.3 Special Devices. Initial Conditions 8.4 Linear and Quasi-Linear Equations of Second Order 8.5 Special Linear Equations of Second Order, with Constant Coefficients 8.6 Other Linear Equations 8.7 Characteristics of Linear First-Order Equations 8.8 Characteristics of Linear Second-Order Equations 8.9 Singular Curves on Integral Surfaces 8.10 Remarks on Linear Second-Order Initial-Value Problems 8.11 The Characteristics of a Particular Quasi-Linear Problem 9. Solutions of Partial Differential Equations of Mathematical Physics 9.1 Introduction 9.2 Heat Flow 9.3 Steady-State Temperature Distribution in a Rectangular Plate 9.4 Steady-State Temperature Distribution in a Circular Annulus 9.5 Poisson's Integral 9.6 Axisymmetrical Temperature Distribution in a Solid Sphere 9.7 Temperature Distribution in a Rectangular Parallelepiped 9.8 Ideal Fluid Flow about a Sphere 9.9 The Wave Equation. Vibration of a Circular Membrane 9.10 The Heat-Flow Equation. Heat Flow in a Rod 9.11 Duhamel's Superposition Integral 9.12 Traveling Waves 9.13 The Pulsating Cylinder 9.14 Examples of the Use of Fourier Integrals 9.15 Laplace Transform Methods 9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line 9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters 9.18 Formulation of Problems 9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle 10. Functions of a Complex Variable 10.1 Introduction. The Complex Variable 10.2 Elementary Functions of a Complex Variable 10.3 Other Elementary Functions 10.4 Analytic Functions of a Complex Variable 10.5 Line Integrals of Complex Functions 10.6 Cauchy's Integral Formula 10.7 Taylor Series 10.8 Laurent Series 10.9 Singularities of Analytic Functions 10.10 Singularities at Infinity 10.11 Significance of Singularities 10.12 Residues 10.13 Evaluation of Real Definite Integrals 10.14 Theorems on Limiting Contours 10.15 Indented Contours 10.16 Integrals Involving Branch Points 11. Applications of Analytic Function Theory 11.1 Introduction 11.2 Inversion of Laplace Transforms 11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral 11.4 Conformal Mapping 11.5 Applications to Two-Dimensional Fluid Flow 11.6 Basic Flows 11.7 Other Applications of Conformal Mapping 11.8 The Schwarz-Christoffel Transformation 11.9 Green's Functions and the Dirichlet Problem 11.10 The Use of Conformal Mapping 11.11 Other Two-Dimensional Green's Functions Answers to Problems Index Contents

Fundamentals of Vibrations

Abstract: 1 Concepts from Vibrations 2 Response of Single-Degree-of-Freedom Systems to Initial Excitations 3 Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations 4 Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations 5 Two-Degree-of-Freedom Systems 6 Elements of Analytical Dynamics 7 Multi-Degree-of-Freedom Systems 8 Distributed-Parameter Systems: Exact Solutions 9 Distributed-Parameter Systems: Approximate Mathods 10 The Finite Element Method 11 Nonlinear Oscilations 12 Random Vibrations Appendix A. Fourier Series Appendix B. Laplace Transformation Appendix C. Linear Algebra

Finite spectrum assignment problem for systems with delays

Abstract: In this paper linear systems with delays in state and/or control variables are considered. The objective is to design a feedback law which yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned set of n points in the complex plane. It is shown that in case of systems with delays in control only the problem is solvable if and only if some function space controllability criterion is met. The solution is then easily obtainable by standard spectrum assignment methods, while the resulting feedback law involves integrals over the past control. In case of delays in state variables it is shown that a technique based on the finite Laplace transform, related to a recent work on function space controllability, leads to a constructive design procedure. The resulting feedback consists of proportional and (finite interval) integral terms over present and past values of state variables. Some indications on how to combine these results in case of systems including both state and control delays are given. Sensitivity of the design to parameter variations is briefly analyzed.