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# Laplace transform applied to differential equations

About: Laplace transform applied to differential equations is a research topic. Over the lifetime, 3786 publications have been published within this topic receiving 74886 citations.

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2,605 citations

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TL;DR: In this article, the authors introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus, and derive the analytical solutions of the most simple linear integral and differential equations in 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.

1,281 citations

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01 Jan 1970

TL;DR: In this paper, the Laplace Transform is used to solve the problem of linear differential equations with constant coefficients, which is a special case of the problem we are dealing with here, and the results are shown to be valid for large values of x.

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

1,169 citations

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TL;DR: A numerical inversion method for Laplace transforms, based on a Fourier series expansion developed by Durbin [5], is presented in this article, where the disadvantage of the inversion methods of that type, the encountered dependence of discretization and truncation error on the free parameters, is removed by the simultaneous application of a procedure for the reduction of the Discretization error, a method for accelerating the convergence of the Fourier Series and a procedure that computes approximately the "best" choice of the free parameter.

1,044 citations

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TL;DR: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method, and the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t.

Abstract: An accurate method is presented for the numerical inversion of Laplace transform, which is a natural continuation to Dubner and Abate's method. (Dubner and Abate, 1968). The advantages of this modified procedure are twofold: first, the error bound on the inverse f{t) becomes independent of t, instead of being exponential in t; second, and consequently, the trigonometric series obtained for fit) in terms of F(s) is valid on the whole period 2T of the series. As it is proved, this error bound can be set arbitrarily small, and it is always possible to get good results, even in rather difficult cases. Particular implementations and numerical examples are presented.

953 citations