book

Differential Equation Solutions with MATLAB®

Name: Differential Equation Solutions with MATLAB®
ISBN: 9783110675313

by Dingyü Xue, Tsinghua University Tsinghua University Press

April 2020

Intermediate to advanced

451 pages

13h 48m

English

De Gruyter

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Preface
1 An introduction to differential equations
1.1 Introduction to differential equation modeling1.1.1 Modeling of an electric circuit1.1.2 Modeling in mechanical systems1.1.3 Models in social systems1.2 A brief history of differential equations1.3 Outline and main topics in the book1.4 Exercises
2 Analytical solutions of ordinary differential equations
2.1 Analytical solutions of first-order differential equations2.1.1 Differential equation solvable by simple integrals2.1.2 Homogeneous differential equations2.1.3 Inhomogeneous linear differential equations2.1.4 Nonlinear differential equations with separable variables2.2 Special functions and second-order differential equations2.2.1 Gamma function2.2.2 Hypergeometric functions2.2.3 Bessel differential equations2.2.4 Legendre differential equations and functions2.2.5 Airy functions2.3 Solutions of linear differential equations with constant coefficients2.3.1 Mathematical modeling of linear constant-coefficient differential equations2.3.2 Laplace transform-based solutions2.3.3 Solutions of inhomogeneous differential equations2.3.4 Solutions of differential equations with nonzero initial values2.4 Analytical solutions of ordinary differential equations2.4.1 Analytical solutions of simple differential equations2.4.2 Analytical solutions of high-order linear differential equations with constant coefficients2.4.3 Analytical solutions of linear time-varying differential equations2.4.4 Solutions of time-varying differential equation sets2.4.5 Solutions of boundary value problems2.5 Solutions of linear matrix differential equations2.5.1 Analytical solutions of linear state space equations2.5.2 Direct solutions of state space models2.5.3 Solution of Sylvester differential equation2.5.4 Kronecker product-based solutions of Sylvester differential equations2.6 Analytical solutions to special nonlinear differential equations2.6.1 Solvable nonlinear differential equations2.6.2 Nonlinear differential equations where analytical solutions are not available2.7 Exercises
3 Initial value problems
3.1 Initial value descriptions for first-order explicit differential equations3.1.1 Mathematical forms of initial value problems3.1.2 Existence and uniqueness of solutions3.2 Implementation of fixed-step numerical algorithms3.2.1 Euler’s method3.2.2 Second-order Runge–Kutta algorithm3.2.3 Fourth-order Runge–Kutta algorithm3.2.4 Gill’s algorithm3.2.5 The mth order Runge–Kutta algorithm3.2.6 Multistep algorithms and implementation3.3 Variable-step numerical algorithms and implementations3.3.1 Measures to increase efficiency3.3.2 An introduction to variable-step algorithms3.3.3 The 4/5th order Runge–Kutta variable-step algorithm3.3.4 The differential equation solver provided in MATLAB3.3.5 Solutions of differential equations with additional parameters3.3.6 Avoiding the use of additional parameters3.4 Validations of numerical solutions3.4.1 Validation of the computation results3.4.2 Dynamic manipulation of intermediate results3.4.3 More accurate solvers3.4.4 Step-sizes and fixed-step display3.4.5 Demonstrations of high-order nonlinear differential equations3.5 Exercises
4 Standard form conversions of ordinary differential equations
4.1 Conversion method for a single high-order differential equation4.1.1 Conversion of explicit equations4.1.2 Solutions of time-varying differential equations4.1.3 Singularities in differential equations4.1.4 State augmentation for constant parameters4.2 Conversions of complicated high-order differential equations4.2.1 Equations containing the square of the highest-order derivative4.2.2 Equations containing odd powers4.2.3 Equations containing nonlinear operations4.3 Conversions of differential equation sets4.3.1 Simple explicit differential equation sets4.3.2 Limitations with fixed-step methods4.3.3 Simple implicit differential equations4.3.4 Even more complicated nonlinear differential equations4.4 Conversions for matrix differential equations4.4.1 Conversion and solutions of differential equations in matrix form4.4.2 Sylvester differential equations4.4.3 Riccati differential equations4.5 Conversions of a class of Volterra integro-differential equations4.6 Exercises
5 Special differential equations
5.1 Stiff differential equations5.1.1 Time constants in linear differential equations5.1.2 Demonstrations of stiff phenomena5.1.3 Direct solution of stiff differential equations5.1.4 Stiffness detection5.1.5 Fixed-step solution of stiff differential equations5.2 Implicit differential equations5.2.1 Mathematical description of implicit differential equations5.2.2 Consistent initial value transformation5.2.3 Direct solution of implicit differential equations5.2.4 Implicit differential equations with multiple solutions5.3 Differential-algebraic equations5.3.1 General form of differential-algebraic equations5.3.2 Indices of differential-algebraic equations5.3.3 Semi-explicit differential-algebraic equations5.3.4 Limitations of the direct solver5.3.5 Implicit solvers for differential-algebraic equations5.3.6 Index reduction for differential-algebraic equations5.4 Switched differential equations5.4.1 Linear switched differential equations5.4.2 Zero-crossing detection and event handling5.4.3 Nonlinear switched differential equations5.4.4 Discontinuous differential equations5.5 Linear stochastic differential equations5.5.1 Transfer function model for linear stochastic differential equations5.5.2 Erroneous methods in continuous stochastic system simulation5.5.3 Discretizing stochastic linear systems5.6 Exercises
6 Delay differential equations
6.1 Numerical solutions of delay differential equations with constant delays6.1.1 Ordinary and delay differential equations6.1.2 Delay differential equation with zero history functions6.1.3 Nonzero history functions6.2 Differential equations with variable delays6.2.1 Variable delay models6.2.2 Time-dependent delays6.2.3 State-dependent delays6.2.4 Delay differential equations with generalized delays6.3 Neutral-type delay differential equations6.3.1 Neutral-type equations6.3.2 Neutral-type differential equations with variable delays6.4 Volterra differential equations with delays6.5 Exercises
7 Properties and behaviors of ordinary differential equations
7.1 Stability of differential equations7.1.1 Stability of linear differential equations with constant coefficients7.1.2 Routh–Hurwitz stability criterion7.1.3 Lyapunov function and Lyapunov stability7.1.4 Autonomous conversion of time-varying differential equations7.1.5 Stability assessment of nonlinear systems by examples7.1.6 Stability assessment of complicated systems with simulation methods7.2 Special behaviors of differential equations7.2.1 Limit cycles7.2.2 Periodic solutions7.2.3 Chaos and attractors7.2.4 Poincaré mapping7.3 Linearization approximation of differential equations7.3.1 Equilibrium points7.3.2 Linearization of nonlinear differential equations7.3.3 Properties of equilibria7.4 Bifurcation of differential equations7.5 Exercises
8 Fractional-order differential equations
8.1 Definitions and numerical computation in fractional calculus8.1.1 Definitions of fractional calculus8.1.2 Relationships and properties of different fractional calculus definitions8.1.3 Numerical computation for Grünwald–Letnikov derivatives8.1.4 Numerical computation for Caputo derivatives8.2 Analytical solution of linear commensurate-order differential equations8.2.1 Mittag-Leffler functions8.2.2 Commensurate-order linear differential equations8.2.3 An important Laplace transform formula8.2.4 Partial fraction expansion-based analytical approach8.3 Numerical solutions of linear fractional-order differential equations with constant coefficients8.3.1 A closed-form solution8.3.2 Riemann–Liouville differential equations8.3.3 Caputo differential equations8.3.4 Computing equivalent initial values8.3.5 High precision algorithms8.4 Solution of nonlinear fractional-order differential equations8.4.1 Predictor method8.4.2 Corrector method8.4.3 Matrix method for implicit Caputo differential equations8.5 Exercises
9 Block diagram solutions of ordinary differential equations
9.1 Simulink essentials9.1.1 An introduction to Simulink9.1.2 Relevant blocks in Simulink9.2 Block diagram modeling methodologies in differential equations9.2.1 Integrator chains and key signal generation9.2.2 Differential equation description methods9.2.3 Solutions of differential equations9.2.4 Algorithms and parameter settings9.3 Modeling examples of differential equations9.3.1 Simple differential equations9.3.2 Differential-algebraic equations9.3.3 Switched differential equations9.3.4 Discontinuous differential equations9.3.5 Delay differential equations9.3.6 Delay differential equations with nonzero history functions9.3.7 Stochastic differential equations9.4 Simulink modeling of fractional-order differential equations9.4.1 Fractional-order operator block9.4.2 Modeling and solution of Riemann–Liouville differential equations9.4.3 Block for Caputo derivatives9.4.4 Modeling and solving of Caputo differential equations9.4.5 Fractional-order delay differential equations9.5 Exercises

10 Boundary value problems for ordinary differential equations
10.1 Standard boundary value problems10.2 Shooting methods in two-point boundary value problems for second-order differential equations10.2.1 Shooting algorithms for linear time-varying differential equations10.2.2 Finite difference algorithm for linear differential equations10.2.3 Shooting algorithms for nonlinear differential equations10.3 Two-point boundary value problems for high-order differential equations10.3.1 Direct solver in MATLAB10.3.2 Solution of simple boundary value problems10.3.3 Descriptions of complicated boundary conditions10.3.4 Boundary value problems with undetermined coefficients10.3.5 Boundary value problems with semi-infinite intervals10.3.6 Differential equations with floating boundary values10.3.7 Boundary value problems of integro-differential equations10.4 Optimization-based boundary value problem solutions10.4.1 Illustrative example to simple boundary value problems10.4.2 Boundary value problems for implicit differential equations10.4.3 Boundary value problems for delay differential equations10.4.4 Multipoint boundary value problems10.4.5 Floating boundary value problems revisited10.4.6 Boundary value problems for block diagrams10.4.7 Boundary value problems for fractional-order differential equations10.5 Exercises
11 Partial differential equations
11.1 Numerical solutions of diffusion equations11.1.1 Mathematical form and analytical solutions of one-dimensional diffusion equations11.1.2 Discretizing diffusion equations11.1.3 Inhomogeneous diffusion equations11.1.4 Multidimensional diffusion equations11.2 Several special forms of partial differential equations11.2.1 Classification of partial differential equations11.2.2 Eigenvalue-type partial differential equations11.2.3 Classification of boundary conditions11.3 User interface of typical two-dimensional partial differential equations11.3.1 An introduction of the user interface11.3.2 Geometric region design11.3.3 Boundary condition description11.3.4 Examples of partial differential equation solutions11.3.5 Other solution display methods11.3.6 Partial differential equations with functional coefficients11.4 Solutions of partial differential equations11.4.1 Creating a blank partial differential equation object11.4.2 Statement description of geometric regions11.4.3 Boundary and initial condition descriptions11.4.4 Partial differential equations descriptions11.4.5 Numerical solutions of partial differential equations11.5 Exercises
Bibliography
Subject Index
Sachregister

Content preview from Differential Equation Solutions with MATLAB®

8 Fractional-order differential equations

It is known that $dny∕dxn$ is used to represent the nth order derivative of y with respect to x. What if $n=1∕2$ ? This was the question asked by a French mathematician Guillaume François Antoine L’Hôpital (1661–1704) to one of the inventors of calculus, Gottfried Wilhelm Leibniz, 300 years ago.^[70] This marked the beginning of fractional calculus. Strictly speaking, the term “fractional-order” is misused, proper names should be “noninteger-order” or “arbitrary-order”. The irrational number $\sqrt{2}$ can be used as the order, but it is not a fraction. Since the term “fractional-order” is already widely used in the research community, it is also used in this book, while it really means arbitrary-order.

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Differential Equations Workbook For Dummies®

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Differential Equation Solutions with MATLAB®

by Dingyü Xue, Tsinghua University Tsinghua University Press

8 Fractional-order differential equations

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