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An Introduction to Partial Differential Equations

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An Introduction to Partial Differential Equations R.S. Johnson An introduction to partial differential equations 2 Download free eBooks at bookboon.com An introduction to partial differential equations © 2012 R.S. Johnson & bookboon.com ISBN 978-87-7681-969-9 3 Download free eBooks at bookboon.com An introduction to partial differential equations Contents Contents Part I 10 First-order partial differential equations 10 List of examples 11 Preface 12 1 Introduction 13 1.1 Types of equation 13 Exercises 1 14 2 The quasi-linear equation 15 2.1 Of surfaces and tangents 15 2.2 The Cauchy (or initial value) problem 18 2.3 The semi-linear and linear equations 22 2.4 The quasi-linear equation in n independent variables 23 Exercises 2 24 GET THERE FASTER Some people know precisely where they want to go. Others seek the adventure of discovering uncharted territory. 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OUR WORLD 4 Click on the ad to read more Download free eBooks at bookboon.com An introduction to partial differential equations Contents 3 The general equation 25 3.1 Geometry again 25 3.2 The method of solution 27 3.3 The general PDE with Cauchy data 34 3.4 The complete integral and the singular solution 36 Exercises 3 47 Answers 48 Part II 49 Partial differential equations: classification and canonical forms 49 List of Equations 50 Preface 52 1 Introduction 53 1.1 Types of equation 53 2 First-order equations 55 2.1 The linear equation 56 2.2 The Cauchy problem 58 5 Click on the ad to read more Download free eBooks at bookboon.com An introduction to partial differential equations Contents 2.3 The quasi-linear equation 59 Exercises 2 62 3 The wave equation 63 3.1 Connection with first-order equations 63 3.2 Initial data 65 Exercises 3 70 4 The general semi-linear partial differential equation in two independent variables 71 4.1 Transformation of variables 71 4.2 Characteristic lines and the classification 73 4.3 Canonical form 76 4.4 Initial and boundary conditions 81 Exercises 4 83 5 Three examples from fluid mechanics 84 5.1 The Tricomi equation 84 5.2 General compressible flow 85 5.3 The shallow-water equations 88 5.4 Appendix: The hodograph transformation 90 Exercise 5 93 In the past four years we have drilled 81,000 km That’s more than twice around the world. 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What will you be? careers.slb.com 6 Click on the ad to read more Download free eBooks at bookboon.com An introduction to partial differential equations Contents 6 Riemann invariants and simple waves 94 6.1 Shallow-water equations: Riemann invariants 95 6.2 Shallow-water equations: simple waves 96 Answers 98 Part III 99 Partial differential equations: method of separation of variables and similarity & travelling-wave solutions 99 List of Equations 100 Preface 102 1 Introduction 103 1.1 The Laplacian and coordinate systems 103 1.2 Overview of the methods 106 2 The method of separation of variables 107 2.1 Introducing the method 107 2.2 Two independent variables: other coordinate systems 110 2.3 Linear equations in more than two independent variables 114 Could you think of 101 new things to do with eggs and oil? Hellmann’s is one of Unilever’s oldest brands having been popular for over 100 years. If you too share a passion for discovery and innovation we will give you the tools and opportunities to provide you with a challenging career. Are you a great scientist who would like to be at the forefront of scientific innovations and developments? Then you will enjoy a career within Unilever Research & Development. For challenging job opportunities, please visit www.unilever.com/rdjobs. 7 Click on the ad to read more Download free eBooks at bookboon.com An introduction to partial differential equations Contents 2.4 Nonlinear equations 118 Exercises 2 121 3 Travelling-wave solutions 122 3.1 The classical, elementary partial differential equations 123 3.2 Equations in higher dimensions 125 3.3 Nonlinear equations 127 Exercises 3 131 4 Similarity solutions 132 4.1 Introducing the method 132 4.2 Continuous (Lie) groups 136 360° 4.3 Similarity solutions of other equations 139 4.4 More general solutions from similarity solutions 144 Exercises 4 360° 148 thinking. Answers 149 Index thinking. 150 360° thinking . 360° thinking. Discover the truth at www.deloitte.ca/careers Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. © Deloitte & Touche LLP and affiliated entities. 8 Discover the truth at www.deloitte.ca/careersClick on the ad to read more Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities. An introduction to partial differential equations Preface to these three texts Preface to these three texts The three texts in this one cover, entitled ‘First-order partial differential equations’ (Part I), ‘Partial differential equations: classification and canonical forms’ (Part II) and ‘Partial differential equations: method of separation of variables and similarity & travelling-wave solutions’ (Part III), are three of the ‘Notebook’ series available as additional and background reading to students at Newcastle University (UK). These three together present an introduction to all the ideas that are usually met in a fairly comprehensive study of partial differential equations, as encountered by applied mathematicians at university level. The material in some of Part I, and also some of Part II, is likely to be that encountered by all students; the rest of the material expands on this, going both further and deeper. The aim, therefore, has been to present the standard ideas on a broader canvas (but as relevant to the methods employed in applied mathematics), and to show how the subject can be developed. All the familiar topics are here, but the text is intended, primarily, to broaden and expand the experience of those who already have some knowledge and interest in the subject. Each text is designed to be equivalent to a traditional text, or part of a text, which covers the relevant material, but in a way that moves beyond an elementary discussion. The development is based on careful derivations and descriptions, supported by many worked examples and a few set exercises (with answers provided). The necessary background is described in the preface to each Part, and there is a comprehensive index, covering the three parts, at the end. 9 Download free eBooks at bookboon.com An introduction to partial differential equations Part I Part I First-order partial differential equations 10 Download free eBooks at bookboon.com An introduction to partial differential equations List of examples List of examples List of Examples HereHere is isa lista list of theof variousthe various examples examples that are thatdiscussed are disin thiscussed text. in this text. 2 General solution of 22yuxy+= uu yu ………………………………………….....p.16 yy22− 2 Show u( x , y )=+ e F [(1 xu )e ] is a solution of 22yuxy+= uu yu ….…….…..p.16 2 General solution of xuxy+=2 xuu u and solutions with: (a) ux= 2 on yx=+21; (b) ux= 2 2 on yx= 3 3 ; (c) ux= 2 on yx=−3 1………………………………......p.20 2 General solution of uxy+=2 xu u ……………………………………..…………..p.22 General solution of yuxy−= xu2 xyu …………………………………….………..p.23 x z xu General solution of uuuxxyz+e + e −+ (2 e )e = 0 ………………………..…....p.23 Solve ODEs that describe uuxy= u i.e. pq= u ……………………………….…...p.32 2 Solution of uuxy= u with ut= on xt= , yt=+1 ………………………….…...p.34 22 Solution of uuxy+=1 with ut= λ on xyt==……………………………….….p.35 Find complete integral of uuxy= u……………………………………………..…..p.41 22 Find complete integral, general solution, singular solution of uuxy+ =+12 u…....p.42 2 Solution of xuxy u+= yu y 1 with ut= on xt= 2 , y = 0………………………….p.44 22 Find complete integral of y() uxy−+ u uu y = 0 and solutions with (a) ut= 3 on xt= 2 , yt= ; (b) ut= 2 on xt= 2 , y = 0…………………………...p.45 11 Download free eBooks at bookboon.com An introduction to partial differential equations Preface Preface This text is intended to provide an introduction to the standard methods that are used for the solution of first-order partial differential equations. Some of these ideas are likely to be introduced, probably in a course on mathematical methods during the second year of a degree programme with, perhaps, more detail in a third year. The material has been written to provide a general – but broad – introduction to the relevant ideas, and not as a text closely linked to a specific module or course of study.
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