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E-grāmata: Differential Equations: A Linear Algebra Approach

Anindya Dey

Formāts: 626 pages
Izdošanas datums: 26-Sep-2021
Izdevniecība: CRC Press
Valoda: eng
ISBN-13: 9781000436785

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Formāts: 626 pages
Izdošanas datums: 26-Sep-2021
Izdevniecība: CRC Press
Valoda: eng
ISBN-13: 9781000436785

Citas grāmatas par šo tēmu:

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Differential Equations: A Linear Algebra Approach follows an innovative approach of inculcating linear algebra and elementary functional analysis in the backdrop of even the simple methods of solving ordinary differential equations. The contents of the book have been made user-friendly through concise useful theoretical discussions and numerous illustrative examples practical and pathological.

This book follows an innovative approach of inculcating linear algebra and elementary functional analysis in the backdrop of even the simple methods of solving ordinary differential equations. It is made user-friendly through concise useful theoretical discussions and numerous illustrative examples practical and pathological.

Preface

1 A Prelude to Differential Equations

(40)

1.1 Introduction

(6)

1.2 Formulation of Differential Equation---Its Significance

(10)

1.3 Classification of Solutions: General Particular and Singular Solutions

(3)

1.4 More about Solutions of an ODE

(4)

1.5 Existence-Uniqueness Theorem for Cauchy Problem

(6)

1.6 Importance of Lipschitz's condition involved in existence uniqueness theorem in the light of a comparative study between Radiactive decay and Leaky bucket problems

(4)

1.7 First Order Ode and some of its Qualitative Aspects

(7)

2 Equations of First Order and First Degree

(89)

2.1 Introduction

(2)

2.2 Exact Differential Equation

(18)

2.3 Homogeneous Differential Equations

(9)

2.4 Integrating Factor

(10)

2.5 Linear Equations and Bernoulli Equations

(13)

2.6 Integrating Factors Revisited

(2)

2.7 Riccati Equation

(6)

2.8 Application of Differential Equations of First Order

101

(12)

2.9 Orthogonal Trajectories and Oblique Trajectories

113

(17)

3 A Class of First Order Non-Linear Odes

130

(46)

3.1 Introduction

130

(1)

3.2 Non-linear First Order Ode Solvable for p

130

(4)

3.3 Non-linear Ode Solvable for y

134

(1)

3.4 Non-linear Ode Solvable for x

135

(3)

3.5 Existence and Uniqueness Problem

138

(6)

3.6 Envelopes and Other Loci

144

(10)

3.7 Clairaut's Equation and Lagrange's Equation

154

(22)

4 Linear Algebraic Framework in Differential Equations

176

(33)

4.1 Introduction

176

(1)

4.2 Linear Spaces

176

(8)

4.3 Linear Maps or Transformations

184

(1)

4.4 Normed Linear Space

185

(6)

4.5 Bounded Linear Transformation

191

(6)

4.6 Invertible Operators

197

(12)

5 Differential Equations of Higher Order

209

(75)

5.1 Introduction

209

(1)

5.2 Theoretical Aspects

209

(10)

5.3 Wronskian

219

(6)

5.4 Working Rules for Homogeneous Linear Ode

225

(6)

5.5 Few Theoretical Results from Linear Algebra

231

(2)

5.6 Symbolic Operator and Particular Integral

233

(5)

5.7 Method of Variation of Parameters

238

(13)

5.8 Special Methods for finding Particular Integrals

251

(11)

5.9 Method of Undetermined Co-efficients

262

(12)

5.10 Fourier Series Method for Particular Integrals

274

(10)

6 Second Order Linear Ode: Solution Techniques & Qualitative Analysis

284

(96)

6.1 Introduction

284

(1)

6.2 Reduction of Order Method (D'Alembert's Method)

285

(6)

6.3 Method of Inspection for finding one Integral

291

(3)

6.4 Transformation of Second Order Ode by changing the Independent Variable

294

(12)

6.5 Transformation of a Second order Ode by changing the Dependent Variable

306

(3)

6.6 Qualitative Aspects of Second Order Differential Equations

309

(9)

6.7 Exact Second Order Differential Equations

318

(5)

6.8 Adjoint Equation and Self-adjoint Odes

323

(9)

6.9 Sturm-Liouville Problems

332

(25)

6.10 Green's Function Approach to IVP

357

(5)

6.11 Green's Function Approach to BVP

362

(18)

7 Laplace Transformations in Ordinary Differential Equations

380

(58)

7.1 Introduction

380

(1)

7.2 Definition and Anatomy of Laplace Transform

380

(28)

7.3 Laplace transformation technique of solving Ordinary Differential Equations

408

(30)

8 Series Solutions of Linear Differential Equations

438

(90)

8.1 Introduction

438

(1)

8.2 Review of Power Series

439

(5)

8.3 Solutions about Ordinary Points in the Domain

444

(17)

8.4 Solution about Regular Singular Points

461

(7)

8.5 Frobenius Method

468

(41)

8.6 Hypergeometric Equation

509

(5)

8.7 Irregular singular points

514

(14)

9 Solving Linear Systems by Matrix Methods

528

(74)

9.1 Introduction

528

(6)

9.2 Eigenvalue Problems of a square matrix: Diagonalisability

534

(13)

9.3 Solution of Vector Differential Equation using Eigenvalues of Associated Matrix

547

(4)

9.4 Operator Norm and Its use in defining Exponentials of Matrices

551

(6)

9.5 Generalised Eigenvectors: Solution to IVP

557

(8)

9.6 Jordanization of Matrices and Solution of Ode

565

(18)

9.7 Fundamental Matrix and Liouville Theorem

583

(10)

9.8 Alternative Ansatz for Computation of eAx

593

(9)

Appendix

602

(9)

Reference

611

(2)

Index

613

Anindya Dey is Assistant Professor in the Department of Mathematics, St Xaviers Colleges, Kolkata.

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E-grāmata: Differential Equations: A Linear Algebra Approach

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